diff --git a/-NE2T4oBgHgl3EQfQQbP/content/tmp_files/2301.03769v1.pdf.txt b/-NE2T4oBgHgl3EQfQQbP/content/tmp_files/2301.03769v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..61220bd7a2acc0176ba160fb79e249885647364b --- /dev/null +++ b/-NE2T4oBgHgl3EQfQQbP/content/tmp_files/2301.03769v1.pdf.txt @@ -0,0 +1,856 @@ +Learning from What is Already Out There: +Few-shot Sign Language Recognition with Online Dictionaries +Maty´aˇs Boh´aˇcek1,2 and Marek Hr´uz1 +1 Department of Cybernetics and New Technologies for the Information Society, +University of West Bohemia, Pilsen, Czech Republic +2 Gymnasium of Johannes Kepler, Prague, Czech Republic +Abstract— Today’s sign language recognition models require +large training corpora of laboratory-like videos, whose collec- +tion involves an extensive workforce and financial resources. +As a result, only a handful of such systems are publicly +available, not to mention their limited localization capabili- +ties for less-populated sign languages. Utilizing online text-to- +video dictionaries, which inherently hold annotated data of +various attributes and sign languages, and training models +in a few-shot fashion hence poses a promising path for the +democratization of this technology. In this work, we collect +and open-source the UWB-SL-Wild few-shot dataset, the first +of its kind training resource consisting of dictionary-scraped +videos. This dataset represents the actual distribution and +characteristics of available online sign language data. We select +glosses that directly overlap with the already existing datasets +WLASL100 and ASLLVD and share their class mappings to +allow for transfer learning experiments. Apart from providing +baseline results on a pose-based architecture, we introduce a +novel approach to training sign language recognition models +in a few-shot scenario, resulting in state-of-the-art results on +ASLLVD-Skeleton and ASLLVD-Skeleton-20 datasets with top- +1 accuracy of 30.97 % and 95.45 %, respectively. +I. INTRODUCTION +Sign languages (SLs) are natural language systems based +on manual articulations and non-manual components, serving +as the primary means of communication among d/Deaf +communities. While they allow one to convey identical +semantics as the written and spoken language, they operate in +a distinctively more variable gestural-visual modality. There +are currently over 70 million people worldwide whose native +language is one of the approximately 300 SLs that exist [1]. +Nevertheless, no publicly available SL translation system +has been introduced so far. This hinders d/Deaf people’s +ability to use their natural form of communication when +working with technology or interacting with people that do +not sign. Although the problem of automatic SL Recognition +(SLR) has been addressed for many years, it is far from +being solved. Modern solutions utilizing deep learning show +promise, and neural networks might help tear these barriers +down. +There are two prevalent topics related to SLs pursued +in the literature - SL Synthesis and SLR. The first one’s +objective is to translate written language into SL, typically by +animating avatars. The second is intended to translate videos +of performed signs into the written form of a language. It can +This work has been accepted and scheduled for publication at the Face +& Gestures 2023 conference. 979-8-3503-4544-5/23/$31.00 ©2023 IEEE +be further divided into isolated SLR, which recognizes single +sign lemmas out of a known set of glosses, and continuous +SLR, translating unconstrained signing utterances. In this +paper, we attend to the task of few-shot isolated SLR. +The current methods can be generally divided into two +main approaches differing in the means of input repre- +sentations; the appearance-based and the pose-based. The +first prevalent stream of works uses a sequence of RGB +images, optionally complemented with the depth channel. +These methods reach state-of-the-art results but are more +computationally demanding. The second approach performs +an intermediate step of first estimating a body pose sequence +which is then fed into an ensuing recognition model. These +systems tend to be more lightweight and would thus be +more suitable for applications on conventional consumer +technology, e.g., laptops or mobile phones. +Multiple model training and evaluation datasets have been +published over recent years. Generally large-scale in size of +glosses and instances, they vary primarily in the originating +SL and the manners of data collection. It is essential to con- +sider that, unlike with many tasks in the Natural Language +Processing (NLP) domain, no organic sources of potential SL +training data (such as the internet and printed media in the +case of NLP) yield vast amounts of training instances daily. +It hence takes a dedicated, tailored effort to record a SLR +dataset. Such an operation is costly and requires specialists +from multiple fields at once, making it strenuous and risky +to begin with. Accordingly, languages with a smaller user +base receive less attention. +Some of the few resources that contain SL data with +built-in annotations are online text-to-video dictionaries. +We believe they will be crucial in minimizing barriers +in constructing future SLR systems, especially for niche +regional contexts. We thus focus on training models using +data scraped from such websites. As these services usually +contain a few repetitions per sign lemma, such a configu- +ration comprises a few-shot training paradigm. To account +for the lack of a diverse, high-repetitive dataset, we utilize +SPOTER [7], a pose-based Transformer [34] architecture +for SLR. We hypothesize that it will learn faster since it +considers only pre-selected information necessary for such +a classification, which is much smaller in dimension than +raw RGB video. Appearance-based methods, contrastingly, +glutted by the large volume of additional sensory infor- +mation, need more data to generalize sturdily, as observed +arXiv:2301.03769v1 [cs.CV] 10 Jan 2023 + +by Boh´aˇcek et al. [7]. We further investigate the ability +of models to learn across different datasets and introduce +boosting training mechanisms. The main contributions of this +work include: +• Introducing and open-sourcing UWB-SL-Wild: a new +dataset for few-shot SLR obtained from public SL +dictionary data, provided with class mappings to already +existing SLR datasets; +• Proposing Validation Score-Conscious Training proce- +dure which adaptively augments and re-trains for classes +that are identified as under-performing during training; +• Establishing the state-of-the-art results on the ASLLVD- +Skeleton and ASLLVD-Skeleton-20 datasets. +II. RELATED WORK +This section reviews the existing datasets and methods +for isolated SLR. As low-instance training has not yet been +explored to a greater extent for this task, we consider +the overlaps to few-shot or zero-shot gesture and action +recognition. +A. Datasets +Multiple datasets of isolated signs have been published and +studied in the literature. We summarize the prominent ones in +Table I. Purdue RVL-SLLL ASL Database [23], containing +1, 834 videos across 104 classes within the American Sign +Language (ASL), was one of the first to encompass a larger +vocabulary. LSA64 [28] for the Argentinian Sign language is +similar in size, as it contains 3, 200 instances from 64 classes. +Later on, substantially larger corpora started to emerge. +DEVISIGN [12], for instance, provides 24, 000 recordings +spanning 2, 000 glosses from the Chinese sign language. Its +videos were captured in a laboratory-like environment and +were, to the best of our knowledge, the first to provide the +depth information along RGB for this task. MS-ASL [17] +brings a similar scale for the ASL, as it contains 25, 000 RGB +videos from 1, 000 classes. Lastly, the AUTSL [31] dataset +pushed the size and per-class instance ratio even further. It +holds 38, 366 RGB-D recordings spanning 226 classes from +the Turkish SL. +While the available datasets span different geographical +contexts, most research has centered around ASL. We left +out recent datasets, which we consider to capture the most +significant traction within the community, from the introduc- +tory survey and provide their detailed descriptions below. We +later utilize these for experiments and for constructing our +new dataset. +1) WLASL: +Word-level +American +Sign +Language +dataset [21] is a large-scale database of lemmas from +the +ASL +collected +from +multiple +online +sources +and +organizations. The dataset’s gloss totals 2, 000 terms with +their translations to English. The authors provide training, +validation, and test splits. There is an average of over 10 +repetitions in the training set for each class. There are +three primary splits of the dataset depending on the number +of +classes +they +cover: +WLASL100, +WLASL300, +and +WLASL2000. In our experiments, we use the WLASL100 +split only. +TABLE I +SURVEY OF PROMINENT SLR DATASETS. +Dataset +SL +Gloss +Instances +Format +DEVISIGN [12] +CN +2,000 +24,000 +RGB-D +LSA64 [28] +AR +64 +3,200 +RGB +AUTSL [31] +TR +226 +38,336 +RGB-D +RVL-SLLL [23] +US +104 +1,834 +RGB +ASLLVD [24] +US +2,745 +9,763 +RGB/Skelet. +MS-ASL [17] +US +1,000 +25,000 +RGB +WLASL [21] +US +2,000 +21,083 +RGB +2) ASLLVD: American Sign Language Lexicon Video +Dataset [24] holds 2, 745 classes of unique terms in the +ASL. The authors recorded the data in a consistent lab-like +environment with a handful of protagonists. The authors have +not defined training and testing splits, resulting in an average +of nearly 4 repetitions per gloss in the whole set. +3) ASLLVD-Skeleton: Amorim et al. have later created an +abbreviation of the ASLLVD dataset focused on evaluating +pose-based methods. They open-sourced pose estimations of +all the included videos from OpenPose [10] and proposed +fixed training and test splits. The authors also introduced +ASLLVD-Skeleton-20, a smaller subset with only 20 classes, +enabling computationally resource-lighter and more distinc- +tive ablations studies. +B. Sign language recognition +The primal works in SLR have leveraged shallow sta- +tistical modeling such as Hidden Markov Models [32], +[33], which achieved reasonable performance on very small +datasets. A big leap has been observed with the advent of +deep learning. Convolutional Neural Networks (CNNs) were +amidst the first deep architectures employed for this prob- +lem [9], [20], [26], [29]. These were used to construct unitary +representations of the input frames that could be thereafter +used for recognition. Later, various Recurrent Neural Net- +works (RNNs) have been utilized for input encoding as well - +namely Long Short-Term Memory Networks (LSTMs) [13], +[19] or Transformers [8], [29]. The usage of different 3D +CNNs has also been studied extensively (e.g., with I3D [11], +[17], [21]). With the advances in pose estimation, another +stream of approaches has emerged, making use of signer pose +representations at the input. Unlike the previous methods, +these models do not process raw RGB/RGB-D data, but +rather pose representations of the estimated body, hand, and +face landmarks. V´azquez-Enr´ıquez et al. [35] have been +the first to use a Graph Convolutional Network (GCN) +on top of pose sequences, following Yan et al. [38] who +earlier proposed using GCNs for action recognition. Trans- +formers have been recently employed in this regard, as +Boh´aˇcek et al. [7] introduced Pose-based Transformer for +SLR (SPOTER). While the architecture does not surpass +the existing appearance-based approaches in general bench- +marks, the authors have shown that when trained only on +small splits of a training set, SPOTER outperforms even the +appearance-based approaches significantly. Lastly, multiple + +Fig. 1. +Illustrative examples of videos from the used datasets: ASLLVD, WLASL, and our new UWB-SL-Wild. ASLLVD contains videos from a +homogeneous lab environment with few repetitions for each class. WLASL consists of videos captured in multiple settings with a larger instance repetition. +UWB-SL-Wild, on the other hand, contains videos from an online dictionary with only a handful of examples for each class and both inconsistent signers +and recording settings. +ensemble models combining the raw visual data with the +pose estimates [16] have also transpired. +C. Few-shot gesture and action recognition +Both few-shot gesture and action recognition have not +gained extensive traction in literature and are hence not +greatly investigated. Most methods have employed metric +learning, where the similarity between input videos is learned +to classify unfamiliar classes at inference using nearest +neighbors. Bishay et al. [6] have proposed the TARN ar- +chitecture, being the first to incorporate attention mechanism +for this task. More recently, Generative Adversarial Networks +(GANs) have also been studied in this regard [15]. +D. Few- and Zero-shot SLR +Zero-shot SLR has been studied by Bilge et al. [4], [5]. +In both works, the authors propose a pipeline consisting +of multiple RNNs and CNNs exploiting the BERT [14] +representations of given SL lemmas’ textual translations in +corresponding primary written language. This has enabled +zero-shot SLR to a limited, yet promising extent, supposing +the BERT embeddings are available. To the best of our +knowledge, the only work addressing few-shot SLR specif- +ically is introduced in [36]. Therein, Wang et al. leverage +a Siamese Network [18] for feature extraction followed by +K-means and a custom matching algorithm. +III. UWB-SL-WILD +Online SL dictionaries and learning resources are an +excellent fit for in-the-wild training data, as they inherently +dispose of a gloss annotation. However, since the primary +intention with such platforms is not the training of neural +networks, only a limited amount of repetitions can be found +for each gloss (often 2 − 3). To the best of our knowledge, +no available benchmark in the literature can simulate such a +training paradigm, and we thus decided to create one. We +collected a custom dataset called UWB-SL-Wild and are +introducing it in this paper. +There are numerous text-to-video dictionaries available on +the internet1. We decided to use the Sign ASL dictionary as it +1As +an +example, +let +us +mention +Spread +the +Sign +(www. +spreadthesign.com), Signing Savvy (www.signingsavvy.com), +Handspeak (www.handspeak.com), and Sign ASL (www.signasl. +org) websites. +Fig. 2. +Distribution of video repetitions per class in the UWB-SL-Wild +dataset. +introduces the largest variability of signer identities and video +settings due to gathering videos from multiple providers. +The first three websites either contain laboratory-like videos +with a single signer (similar to the already existing datasets), +have a limited vocabulary, or hold other unsuitable video +properties (such as only possessing black-and-white footage). +To allow for transfer learning experiments with the +already-existing datasets, we decided that our dataset’s vo- +cabulary would be equivalent to that of WLASL100. We +then scraped the dataset structure from the Sign ASL portal. +This yielded 307 videos from 100 classes (corresponding to +lemmas in ASL), leaving us with a mean of under 3 repeti- +tions per class. The total distribution of repetitions per class +is depicted in Figure 2. There are 25 unique signers in the +set. Each goes hand-in-hand with a different setting: video +quality, distance and angle from the camera, and background. +While some stand in front of a wall, many sit casually on a +sofa or at a table. We manually annotated the signer identity +in each video and are providing this information along with +the dataset. The 100 classes in UWB-SL-Wild represent +lemmas of frequent terms in ASL, including ordinary objects +(e.g., book, candy, and hat), verbs (e.g., play, enjoy, go), and +other adjectives or particles (e.g., thin, who, blue). Given that +certain signs in ASL dispose of different variations, it may +almost seem as if the signs gathered under a single class +were sometimes completely different. Despite distinctively +unalike in appearance, they still convey identical or highly +similar meanings. This further enlarges the difficulty of +learning on this dataset since some glosses’ sign variations + +ASLLVD (lab recording) +UWB Wild Dataset (wild low-shot data) +WLASL (uniform sources, large-scale) +class mapping +class mapping25 +23 +22 +20 +20 +19 +Number of classes +15 +10 +7 +5 +5 +2 +1 +1 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +Number of training instanceseventually ended up with only a single instance in the entire +set (supposing each of the 2 − 3 videos in a given class +depicts a different variation). While this is not the case for +most classes with just a single variant, a considerable part +of the dataset’s glosses hold at least two versions. We thus +provide manual annotations identifying different variations in +each class. We created a mapping schema of classes between +UWB-SL-Wild, WLASL100, and ASLLVD datasets2. This +enables future researchers to train on and evaluate using these +three datasets. Examples of videos from all three sources can +be seen in Figure 1. We are open-sourcing the UWB-SL- +Wild dataset, including the cross-datasets mappings and pose +estimates of signers in all videos at https://github. +com/matyasbohacek/uwb-sl-wild. +IV. METHODS +This section presents a method that can learn in a few- +shot scenario. We build upon SPOTER [7], as it has shown +substantial promise for training on smaller sets of data, fitting +our few-shot use case. According to the authors, it should +require lower amounts of training data because it is a pose- +based method. We review the pipeline’s key elements and the +changes we have made below. Any unmentioned attributes or +configurations were kept identical. We hence refer the reader +to the original publication for details. +Preprocessing: We first estimate the signer’s pose in all +input video frames. 2-D coordinates of key landmarks are +obtained for the upper body (9), hands (2 × 21), and face +(70). +Augmentations and normalization: We follow the aug- +mentation and normalization procedures from [7] to the full +extent. +A. Architecture +SPOTER is a moderate abbreviation of the Transformer +architecture [34]. The input to the network is a sequence +of normalized and flattened skeletal representations with a +dimension of 242. Learnable positional encoding is added to +the sequence before it is processed further by the standard +Encoder module. The input to the Decoder module is a +single classification query. It is decoded into corresponding +class probabilities by a multi-layer perceptron on top of the +Decoder. +TABLE II +PERFORMANCE COMPARISON ON ASLLVD-SKELETON DATASET +ASLLVD-S +ASLLVD-S-20 +Model +top-1 +top-5 +top-1 +top-5 +HOF [22] +– +– +70.0 +– +BHOF [22] +– +– +85.0 +– +ST-GCN [2] +16.48 +37.15 +61.04 +86.36 +SPOTER [7] +30.77 +52.05 +93.18 +97.72 +SPOTER + VSCT +30.97 +52.87 +95.45 +100.00 +2There were no related videos for 3 classes of WLASL100 split in +SignASL.org. We thus took the following 3 classes from the full WLASL +to compensate for this. +B. Validation score-conscious training +In an attempt to adapt the SLR pipeline for the few- +shot training environment, we propose the Validation Score- +Conscious Training (VSCT). It aims to minimize the classi- +fication error on the fly by identifying the bottleneck classes, +i.e., the classes that get misclassified the most. VSCT adds +the following steps at the end of each epoch of batch gradient +descent: +1) Validation accuracy is calculated for every class within +the set. If a validation split is unavailable, the accuracy +is computed on the training split. +2) The classes are sorted by their performance. A set of +classes Wvsct is found as a proportion of γvsct × c +worst-performing ones, where c is the total number of +classes. +3) Next, a mini-batch is constructed as a random τvsct +share of the training set with classes from Wvsct. +4) Backpropagation is performed yet again on the above- +described mini-batch. However, the parameters of aug- +mentations are drawn from a different distribution. This +allows us to target the problematic classes with better- +suited representations. +γvsct, τvsct, and all VSCT-specific augmentation parame- +ters are constant hyperparameters of a training run. +V. EXPERIMENTS +In this section, we report our results compared to the +already existing methods. We also evaluate our approach on +a newly proposed benchmark leveraging the class mappings +from UWB-SL-Wild and ASLLVD datasets to WLASL100. +A. Implementation details +The SPOTER architecture with VSCT has been imple- +mented in PyTorch [25]. The model’s weights were initial- +ized from a uniform distribution within [0, 1). We trained it +for 130 epochs with an SGD optimizer. The learning rate +was set to 0.001 with no scheduler and both momentum and +weight decay set to 0, following the original implementation. +VSCT hyperparameters differ based on the examined dataset. +For body pose estimation, we used the HRNet-w48 [37] +complemented by a Faster R-CNN [27] for person detec- +tion within the MMPose library [30]. We also leveraged +the Sweep functionality (hyperparameter search) within the +Weights and Biases library [3] to find augmentation and +VSCT hyperparameters. We namely employed the Bayesian +hyperparameter search method 3 and conducted this proce- +dure for each dataset individually. +B. Quantitative results +The results on the ASLLVD-Skeleton dataset, along with +a comparison to the already available methods, are shown +in Table II. We establish an overall state-of-the-art on this +benchmark by achieving 30.97% top-1 and 52.87% top-5 +accuracy on the primary dataset. Our method surpasses the +3For details on this search method, we refer the reader to the official +Weights and Biases documentation available at https://docs.wandb. +ai/guides/sweeps/. + +TABLE III +RESULTS OF THE TRANSFER LEARNING EXPERIMENTS WHERE TRAINING AND EVALUATION WERE PERFORMED ON DIFFERENT DATASETS +ASLLVD → WLASL +UWB-SL-Wild → WLASL +Norm. +Aug. +Bal. sample +VSCT +test +val +test +val + + + + +10.51 +5.94 +8.56 +7.41 + + + + +19.07 +16.62 +14.79 +16.62 + + + + +19.84 +15.73 +15.18 +16.91 + + + + +20.23 +15.13 +16.73 +14.54 + + + + +22.96 +13.95 +18.68 +16.02 +pose-based ST-GCN by a significant margin, almost doubling +the top-1 performance. When evaluated on the much smaller +20-class subsplit, SPOTER+VSCT achieves 95.45% top- +1 and 100.0% top-5 accuracy, which exceeds the so far +best BHOF by more than absolute 10%. Note that all the +models listed in rows 1-5 of Table II use appearance-based +representations. BHOF, for instance, builds upon a block- +based histogram of the incoming videos’ optical flow. +The latter of our evaluation settings makes use of +the class mappings introduced in Section III. We trained +SPOTER+VSCT on ASLLVD or UWB-SL-Wild dataset but +calculated the accuracy on the WLASL100 testing set. We +made the WLASL100 validation split available to the training +procedure for the purposes of per-class statistics computa- +tion within VSCT. The results are presented in Table III. +SPOTER+VSCT achieves a top-1 accuracy of 22.96% when +trained on ASLLVD and 18.68% when trained using UWB- +SL-Wild. +TABLE IV +ABLATION STUDY ON ASLLVD-SKELETON DATASET +ASLLVD-S +Norm. +Aug. +Bal. sample +VSCT +Full +20 cls. + + + + +5.13 +47.73 + + + + +29.18 +86.36 + + + + +30.77 +88.64 + + + + +30.77 +90.91 + + + + +30.97 +95.45 +To provide context to these values, let us consider the +results of Boh´aˇcek et al. [7] who trained and evaluated +SPOTER (without VSCT) on WLASL100. They achieved +63.18%, roughly three times greater accuracy. Their training +set averaged 10.5 repetitions per class, whereas ASLLVD +and UWB-SL-Wild have a mean of 3.6 and 2.9 per-class +instances, respectively. Moreover, UWB-SL-Wild is signifi- +cantly more variable as opposed to the other two datasets +in both unique protagonists and camera settings. While +these cross-dataset results are not nearly comparable to the +standard methods applied for WLASL100 benchmarking, we +believe they attest to the pose-based methods’ ability to +generalize on characteristically distinct few-shot data. +C. Ablation study +We have conducted an ablation study of the individual con- +tributions of normalization, augmentations, and the VSCT +to the above-presented results. We also compare VSCT to +the balanced sampling of classes, which counterbalances the +disproportion of per-class samples in the training set. We +summarize the ablations on the ASLLVD-Skeleton dataset +and its 20-class subset in Table IV. Norm., Aug., and Bal. +sample refer to using normalization, augmentations, and +balanced sampling, respectively, in the given model variant. +The baseline models achieved an accuracy of 5.13% and +46.73%, respectively. We can observe that normalization +itself provides the most significant improvement to 29.18% +and 86.36%, while augmentations provide a slight boost on +top of that, resulting in an accuracy of 30.77% and 88.64%. +With all the previous modules fixed, we test the advan- +tages of using either balanced sampling or VSCT. As for +the complete dataset, balanced sampling does not provide +any performance benefits, whereas VSCT brings a slight +improvement resulting in 30.97% testing accuracy. When +examined on the smaller subset, the balanced sampling +improves the result by a relative 2.6% to 90.91%. VSCT, +nevertheless, still outperforms it by enhancing the result with +a relative 7.7% to the final 95.45% testing accuracy. +The outturn of ablations on the cross-dataset training +experiments is shown in Table III. For both ASLLVD and +UWB-SL-Wild, we conduct the same ablations. The results +mimic the tendencies commented on in the previous exper- +iment. This study suggests that VSCT provides merits to +training on such low-shot data, evincing itself more beneficial +than a standard balanced sampling of classes. +VI. CONCLUSION +We collected and open-sourced a new dataset for SLR +with footage from online text-to-video dictionaries. We con- +structed it with the already-available datasets in mind and +created class mappings to WLASL100 and ASLLVD. To +reflect the attained problem’s few-shot setting, we proposed a +novel procedure of training a neural pose-based SLR system +called Validation Score-Conscious Training. 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Corner- +stone network with feature extractor: a metric-based few-shot model +for chinese natural sign language. Applied Intelligence, 51:7139–7150, +2021. +[37] J. Wang, K. Sun, T. Cheng, B. Jiang, C. Deng, Y. Zhao, D. Liu, +Y. Mu, M. Tan, X. Wang, W. Liu, and B. Xiao. Deep high-resolution +representation learning for visual recognition. IEEE Transactions on +Pattern Analysis and Machine Intelligence, 43:3349–3364, 2021. +[38] S. Yan, Y. Xiong, and D. Lin. Spatial temporal graph convolutional +networks for skeleton-based action recognition. Thirty-second AAAI +conference on artificial intelligence, 2018. + diff --git a/-NE2T4oBgHgl3EQfQQbP/content/tmp_files/load_file.txt b/-NE2T4oBgHgl3EQfQQbP/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..33c3d10a42ad139c6f182e4dd05482b11c3458f7 --- /dev/null +++ b/-NE2T4oBgHgl3EQfQQbP/content/tmp_files/load_file.txt @@ -0,0 +1,663 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf,len=662 +page_content='Learning from What is Already Out There: Few-shot Sign Language Recognition with Online Dictionaries Maty´aˇs Boh´aˇcek1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='2 and Marek Hr´uz1 1 Department of Cybernetics and New Technologies for the Information Society,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' University of West Bohemia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' Pilsen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' Czech Republic 2 Gymnasium of Johannes Kepler,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' Prague,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' Czech Republic Abstract— Today’s sign language recognition models require large training corpora of laboratory-like videos,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' whose collec- tion involves an extensive workforce and financial resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' As a result, only a handful of such systems are publicly available, not to mention their limited localization capabili- ties for less-populated sign languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' Utilizing online text-to- video dictionaries, which inherently hold annotated data of various attributes and sign languages, and training models in a few-shot fashion hence poses a promising path for the democratization of this technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' In this work, we collect and open-source the UWB-SL-Wild few-shot dataset, the first of its kind training resource consisting of dictionary-scraped videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' This dataset represents the actual distribution and characteristics of available online sign language data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' We select glosses that directly overlap with the already existing datasets WLASL100 and ASLLVD and share their class mappings to allow for transfer learning experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' Apart from providing baseline results on a pose-based architecture, we introduce a novel approach to training sign language recognition models in a few-shot scenario, resulting in state-of-the-art results on ASLLVD-Skeleton and ASLLVD-Skeleton-20 datasets with top- 1 accuracy of 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='97 % and 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='45 %, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' INTRODUCTION Sign languages (SLs) are natural language systems based on manual articulations and non-manual components, serving as the primary means of communication among d/Deaf communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' While they allow one to convey identical semantics as the written and spoken language, they operate in a distinctively more variable gestural-visual modality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' There are currently over 70 million people worldwide whose native language is one of the approximately 300 SLs that exist [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' Nevertheless, no publicly available SL translation system has been introduced so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' This hinders d/Deaf people’s ability to use their natural form of communication when working with technology or interacting with people that do not sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' Although the problem of automatic SL Recognition (SLR) has been addressed for many years, it is far from being solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' Modern solutions utilizing deep learning show promise, and neural networks might help tear these barriers down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' There are two prevalent topics related to SLs pursued in the literature - SL Synthesis and SLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' The first one’s objective is to translate written language into SL, typically by animating avatars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' The second is intended to translate videos of performed signs into the written form of a language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' It can This work has been accepted and scheduled for publication at the Face & Gestures 2023 conference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' 979-8-3503-4544-5/23/$31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='00 ©2023 IEEE be further divided into isolated SLR, which recognizes single sign lemmas out of a known set of glosses, and continuous SLR, translating unconstrained signing utterances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' In this paper, we attend to the task of few-shot isolated SLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' The current methods can be generally divided into two main approaches differing in the means of input repre- sentations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' the appearance-based and the pose-based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' The first prevalent stream of works uses a sequence of RGB images, optionally complemented with the depth channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' These methods reach state-of-the-art results but are more computationally demanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' The second approach performs an intermediate step of first estimating a body pose sequence which is then fed into an ensuing recognition model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' These systems tend to be more lightweight and would thus be more suitable for applications on conventional consumer technology, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=', laptops or mobile phones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' Multiple model training and evaluation datasets have been published over recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' Generally large-scale in size of glosses and instances, they vary primarily in the originating SL and the manners of data collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' It is essential to con- sider that, unlike with many tasks in the Natural Language Processing (NLP) domain, no organic sources of potential SL training data (such as the internet and printed media in the case of NLP) yield vast amounts of training instances daily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' It hence takes a dedicated, tailored effort to record a SLR dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' Such an operation is costly and requires specialists from multiple fields at once, making it strenuous and risky to begin with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' Accordingly, languages with a smaller user base receive less attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' Some of the few resources that contain SL data with built-in annotations are online text-to-video dictionaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' We believe they will be crucial in minimizing barriers in constructing future SLR systems, especially for niche regional contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' We thus focus on training models using data scraped from such websites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' As these services usually contain a few repetitions per sign lemma, such a configu- ration comprises a few-shot training paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' To account for the lack of a diverse, high-repetitive dataset, we utilize SPOTER [7], a pose-based Transformer [34] architecture for SLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' We hypothesize that it will learn faster since it considers only pre-selected information necessary for such a classification, which is much smaller in dimension than raw RGB video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' Appearance-based methods, contrastingly, glutted by the large volume of additional sensory infor- mation, need more data to generalize sturdily, as observed arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='03769v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='CV] 10 Jan 2023 by Boh´aˇcek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' We further investigate the ability of models to learn across different datasets and introduce boosting training mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' The main contributions of this work include: Introducing and open-sourcing UWB-SL-Wild: a new dataset for few-shot SLR obtained from public SL dictionary data, provided with class mappings to already existing SLR datasets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' Proposing Validation Score-Conscious Training proce- dure which adaptively augments and re-trains for classes that are identified as under-performing during training;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' Establishing the state-of-the-art results on the ASLLVD- Skeleton and ASLLVD-Skeleton-20 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' RELATED WORK This section reviews the existing datasets and methods for isolated SLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' As low-instance training has not yet been explored to a greater extent for this task, we consider the overlaps to few-shot or zero-shot gesture and action recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' Datasets Multiple datasets of isolated signs have been published and studied in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' We summarize the prominent ones in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' Purdue RVL-SLLL ASL Database [23], containing 1, 834 videos across 104 classes within the American Sign Language (ASL), was one of the first to encompass a larger vocabulary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' LSA64 [28] for the Argentinian Sign language is similar in size, as it contains 3, 200 instances from 64 classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' Later on, substantially larger corpora started to emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' DEVISIGN [12], for instance, provides 24, 000 recordings spanning 2, 000 glosses from the Chinese sign language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' Its videos were captured in a laboratory-like environment and were, to the best of our knowledge, the first to provide the depth information along RGB for this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' MS-ASL [17] brings a similar scale for the ASL, as it contains 25, 000 RGB videos from 1, 000 classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' Lastly, the AUTSL [31] dataset pushed the size and per-class instance ratio even further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' It holds 38, 366 RGB-D recordings spanning 226 classes from the Turkish SL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' While the available datasets span different geographical contexts, most research has centered around ASL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' We left out recent datasets, which we consider to capture the most significant traction within the community, from the introduc- tory survey and provide their detailed descriptions below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' We later utilize these for experiments and for constructing our new dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' 1) WLASL: Word-level American Sign Language dataset [21] is a large-scale database of lemmas from the ASL collected from multiple online sources and organizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' The dataset’s gloss totals 2, 000 terms with their translations to English.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' The authors provide training, validation, and test splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' There is an average of over 10 repetitions in the training set for each class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' There are three primary splits of the dataset depending on the number of classes they cover: WLASL100, WLASL300, and WLASL2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' In our experiments, we use the WLASL100 split only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' TABLE I SURVEY OF PROMINENT SLR DATASETS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' Dataset SL Gloss Instances Format DEVISIGN [12] CN 2,000 24,000 RGB-D LSA64 [28] AR 64 3,200 RGB AUTSL [31] TR 226 38,336 RGB-D RVL-SLLL [23] US 104 1,834 RGB ASLLVD [24] US 2,745 9,763 RGB/Skelet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' MS-ASL [17] US 1,000 25,000 RGB WLASL [21] US 2,000 21,083 RGB 2) ASLLVD: American Sign Language Lexicon Video Dataset [24] holds 2, 745 classes of unique terms in the ASL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' The authors recorded the data in a consistent lab-like environment with a handful of protagonists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' The authors have not defined training and testing splits, resulting in an average of nearly 4 repetitions per gloss in the whole set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' 3) ASLLVD-Skeleton: Amorim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' have later created an abbreviation of the ASLLVD dataset focused on evaluating pose-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' They open-sourced pose estimations of all the included videos from OpenPose [10] and proposed fixed training and test splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' The authors also introduced ASLLVD-Skeleton-20, a smaller subset with only 20 classes, enabling computationally resource-lighter and more distinc- tive ablations studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' Sign language recognition The primal works in SLR have leveraged shallow sta- tistical modeling such as Hidden Markov Models [32], [33], which achieved reasonable performance on very small datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' A big leap has been observed with the advent of deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' Convolutional Neural Networks (CNNs) were amidst the first deep architectures employed for this prob- lem [9], [20], [26], [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' These were used to construct unitary representations of the input frames that could be thereafter used for recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' Later, various Recurrent Neural Net- works (RNNs) have been utilized for input encoding as well - namely Long Short-Term Memory Networks (LSTMs) [13], [19] or Transformers [8], [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' The usage of different 3D CNNs has also been studied extensively (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=', with I3D [11], [17], [21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' With the advances in pose estimation, another stream of approaches has emerged, making use of signer pose representations at the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' Unlike the previous methods, these models do not process raw RGB/RGB-D data, but rather pose representations of the estimated body, hand, and face landmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' V´azquez-Enr´ıquez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' [35] have been the first to use a Graph Convolutional Network (GCN) on top of pose sequences, following Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' [38] who earlier proposed using GCNs for action recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' Trans- formers have been recently employed in this regard, as Boh´aˇcek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' [7] introduced Pose-based Transformer for SLR (SPOTER).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' While the architecture does not surpass the existing appearance-based approaches in general bench- marks, the authors have shown that when trained only on small splits of a training set, SPOTER outperforms even the appearance-based approaches significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' Lastly, multiple Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' Illustrative examples of videos from the used datasets: ASLLVD, WLASL, and our new UWB-SL-Wild.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' ASLLVD contains videos from a homogeneous lab environment with few repetitions for each class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' WLASL consists of videos captured in multiple settings with a larger instance repetition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' UWB-SL-Wild, on the other hand, contains videos from an online dictionary with only a handful of examples for each class and both inconsistent signers and recording settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' ensemble models combining the raw visual data with the pose estimates [16] have also transpired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' Few-shot gesture and action recognition Both few-shot gesture and action recognition have not gained extensive traction in literature and are hence not greatly investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' Most methods have employed metric learning, where the similarity between input videos is learned to classify unfamiliar classes at inference using nearest neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' Bishay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' [6] have proposed the TARN ar- chitecture, being the first to incorporate attention mechanism for this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' More recently, Generative Adversarial Networks (GANs) have also been studied in this regard [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' Few- and Zero-shot SLR Zero-shot SLR has been studied by Bilge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' [4], [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' In both works, the authors propose a pipeline consisting of multiple RNNs and CNNs exploiting the BERT [14] representations of given SL lemmas’ textual translations in corresponding primary written language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' This has enabled zero-shot SLR to a limited, yet promising extent, supposing the BERT embeddings are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' To the best of our knowledge, the only work addressing few-shot SLR specif- ically is introduced in [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' Therein, Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' leverage a Siamese Network [18] for feature extraction followed by K-means and a custom matching algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' UWB-SL-WILD Online SL dictionaries and learning resources are an excellent fit for in-the-wild training data, as they inherently dispose of a gloss annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' However, since the primary intention with such platforms is not the training of neural networks, only a limited amount of repetitions can be found for each gloss (often 2 − 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' To the best of our knowledge, no available benchmark in the literature can simulate such a training paradigm, and we thus decided to create one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' We collected a custom dataset called UWB-SL-Wild and are introducing it in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' There are numerous text-to-video dictionaries available on the internet1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' We decided to use the Sign ASL dictionary as it 1As an example, let us mention Spread the Sign (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' spreadthesign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='com), Signing Savvy (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='signingsavvy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='com), Handspeak (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='handspeak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='com), and Sign ASL (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='signasl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' org) websites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' Distribution of video repetitions per class in the UWB-SL-Wild dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' introduces the largest variability of signer identities and video settings due to gathering videos from multiple providers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' The first three websites either contain laboratory-like videos with a single signer (similar to the already existing datasets), have a limited vocabulary, or hold other unsuitable video properties (such as only possessing black-and-white footage).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' To allow for transfer learning experiments with the already-existing datasets, we decided that our dataset’s vo- cabulary would be equivalent to that of WLASL100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' We then scraped the dataset structure from the Sign ASL portal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' This yielded 307 videos from 100 classes (corresponding to lemmas in ASL), leaving us with a mean of under 3 repeti- tions per class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' The total distribution of repetitions per class is depicted in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' There are 25 unique signers in the set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' Each goes hand-in-hand with a different setting: video quality, distance and angle from the camera, and background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' While some stand in front of a wall, many sit casually on a sofa or at a table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' We manually annotated the signer identity in each video and are providing this information along with the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' The 100 classes in UWB-SL-Wild represent lemmas of frequent terms in ASL, including ordinary objects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=', book, candy, and hat), verbs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=', play, enjoy, go), and other adjectives or particles (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=', thin, who, blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' Given that certain signs in ASL dispose of different variations, it may almost seem as if the signs gathered under a single class were sometimes completely different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' Despite distinctively unalike in appearance, they still convey identical or highly similar meanings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' This further enlarges the difficulty of learning on this dataset since some glosses’ sign variations ASLLVD (lab recording) UWB Wild Dataset (wild low-shot data) WLASL (uniform sources,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' large-scale) class mapping class mapping25 23 22 20 20 19 Number of classes 15 10 7 5 5 2 1 1 0 1 2 3 4 5 6 7 8 9 Number of training instanceseventually ended up with only a single instance in the entire set (supposing each of the 2 − 3 videos in a given class depicts a different variation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' While this is not the case for most classes with just a single variant, a considerable part of the dataset’s glosses hold at least two versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' We thus provide manual annotations identifying different variations in each class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' We created a mapping schema of classes between UWB-SL-Wild, WLASL100, and ASLLVD datasets2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' This enables future researchers to train on and evaluate using these three datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' Examples of videos from all three sources can be seen in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' We are open-sourcing the UWB-SL- Wild dataset, including the cross-datasets mappings and pose estimates of signers in all videos at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' com/matyasbohacek/uwb-sl-wild.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' METHODS This section presents a method that can learn in a few- shot scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' We build upon SPOTER [7], as it has shown substantial promise for training on smaller sets of data, fitting our few-shot use case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' According to the authors, it should require lower amounts of training data because it is a pose- based method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' We review the pipeline’s key elements and the changes we have made below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' Any unmentioned attributes or configurations were kept identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' We hence refer the reader to the original publication for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' Preprocessing: We first estimate the signer’s pose in all input video frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' 2-D coordinates of key landmarks are obtained for the upper body (9), hands (2 × 21), and face (70).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' Augmentations and normalization: We follow the aug- mentation and normalization procedures from [7] to the full extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' Architecture SPOTER is a moderate abbreviation of the Transformer architecture [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' The input to the network is a sequence of normalized and flattened skeletal representations with a dimension of 242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' Learnable positional encoding is added to the sequence before it is processed further by the standard Encoder module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' The input to the Decoder module is a single classification query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' It is decoded into corresponding class probabilities by a multi-layer perceptron on top of the Decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' TABLE II PERFORMANCE COMPARISON ON ASLLVD-SKELETON DATASET ASLLVD-S ASLLVD-S-20 Model top-1 top-5 top-1 top-5 HOF [22] – – 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='0 – BHOF [22] – – 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='0 – ST-GCN [2] 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='48 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='15 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='04 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='36 SPOTER [7] 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='77 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='05 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='18 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='72 SPOTER + VSCT 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='97 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='87 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='45 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='00 2There were no related videos for 3 classes of WLASL100 split in SignASL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' We thus took the following 3 classes from the full WLASL to compensate for this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' Validation score-conscious training In an attempt to adapt the SLR pipeline for the few- shot training environment, we propose the Validation Score- Conscious Training (VSCT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' It aims to minimize the classi- fication error on the fly by identifying the bottleneck classes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=', the classes that get misclassified the most.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' VSCT adds the following steps at the end of each epoch of batch gradient descent: 1) Validation accuracy is calculated for every class within the set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' If a validation split is unavailable, the accuracy is computed on the training split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' 2) The classes are sorted by their performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' A set of classes Wvsct is found as a proportion of γvsct × c worst-performing ones, where c is the total number of classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' 3) Next, a mini-batch is constructed as a random τvsct share of the training set with classes from Wvsct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' 4) Backpropagation is performed yet again on the above- described mini-batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' However, the parameters of aug- mentations are drawn from a different distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' This allows us to target the problematic classes with better- suited representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' γvsct, τvsct, and all VSCT-specific augmentation parame- ters are constant hyperparameters of a training run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' EXPERIMENTS In this section, we report our results compared to the already existing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' We also evaluate our approach on a newly proposed benchmark leveraging the class mappings from UWB-SL-Wild and ASLLVD datasets to WLASL100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' Implementation details The SPOTER architecture with VSCT has been imple- mented in PyTorch [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' The model’s weights were initial- ized from a uniform distribution within [0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' We trained it for 130 epochs with an SGD optimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' The learning rate was set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='001 with no scheduler and both momentum and weight decay set to 0, following the original implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' VSCT hyperparameters differ based on the examined dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' For body pose estimation, we used the HRNet-w48 [37] complemented by a Faster R-CNN [27] for person detec- tion within the MMPose library [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' We also leveraged the Sweep functionality (hyperparameter search) within the Weights and Biases library [3] to find augmentation and VSCT hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' We namely employed the Bayesian hyperparameter search method 3 and conducted this proce- dure for each dataset individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' Quantitative results The results on the ASLLVD-Skeleton dataset, along with a comparison to the already available methods, are shown in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' We establish an overall state-of-the-art on this benchmark by achieving 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='97% top-1 and 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='87% top-5 accuracy on the primary dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' Our method surpasses the 3For details on this search method, we refer the reader to the official Weights and Biases documentation available at https://docs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='wandb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' ai/guides/sweeps/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' TABLE III RESULTS OF THE TRANSFER LEARNING EXPERIMENTS WHERE TRAINING AND EVALUATION WERE PERFORMED ON DIFFERENT DATASETS ASLLVD → WLASL UWB-SL-Wild → WLASL Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' Aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' Bal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' sample VSCT test val test val \x17 \x17 \x17 \x17 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='51 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='94 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='56 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='41 \x13 \x17 \x17 \x17 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='07 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='62 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='79 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='62 \x13 \x13 \x17 \x17 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='84 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='73 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='18 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='91 \x13 \x13 \x13 \x17 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='23 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='13 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='73 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='54 \x13 \x13 \x17 \x13 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='96 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='95 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='68 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='02 pose-based ST-GCN by a significant margin, almost doubling the top-1 performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' When evaluated on the much smaller 20-class subsplit, SPOTER+VSCT achieves 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='45% top- 1 and 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='0% top-5 accuracy, which exceeds the so far best BHOF by more than absolute 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' Note that all the models listed in rows 1-5 of Table II use appearance-based representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' BHOF, for instance, builds upon a block- based histogram of the incoming videos’ optical flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' The latter of our evaluation settings makes use of the class mappings introduced in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' We trained SPOTER+VSCT on ASLLVD or UWB-SL-Wild dataset but calculated the accuracy on the WLASL100 testing set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' We made the WLASL100 validation split available to the training procedure for the purposes of per-class statistics computa- tion within VSCT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' The results are presented in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' SPOTER+VSCT achieves a top-1 accuracy of 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='96% when trained on ASLLVD and 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='68% when trained using UWB- SL-Wild.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' TABLE IV ABLATION STUDY ON ASLLVD-SKELETON DATASET ASLLVD-S Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' Aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' Bal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' sample VSCT Full 20 cls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' \x17 \x17 \x17 \x17 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='13 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='73 \x13 \x17 \x17 \x17 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='18 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='36 \x13 \x13 \x17 \x17 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='77 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='64 \x13 \x13 \x13 \x17 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='77 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='91 \x13 \x13 \x17 \x13 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='97 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='45 To provide context to these values, let us consider the results of Boh´aˇcek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' [7] who trained and evaluated SPOTER (without VSCT) on WLASL100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' They achieved 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='18%, roughly three times greater accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' Their training set averaged 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='5 repetitions per class, whereas ASLLVD and UWB-SL-Wild have a mean of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='6 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='9 per-class instances, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' Moreover, UWB-SL-Wild is signifi- cantly more variable as opposed to the other two datasets in both unique protagonists and camera settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' While these cross-dataset results are not nearly comparable to the standard methods applied for WLASL100 benchmarking, we believe they attest to the pose-based methods’ ability to generalize on characteristically distinct few-shot data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' Ablation study We have conducted an ablation study of the individual con- tributions of normalization, augmentations, and the VSCT to the above-presented results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' We also compare VSCT to the balanced sampling of classes, which counterbalances the disproportion of per-class samples in the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' We summarize the ablations on the ASLLVD-Skeleton dataset and its 20-class subset in Table IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=', Aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=', and Bal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' sample refer to using normalization, augmentations, and balanced sampling, respectively, in the given model variant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' The baseline models achieved an accuracy of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='13% and 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='73%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' We can observe that normalization itself provides the most significant improvement to 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='18% and 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='36%, while augmentations provide a slight boost on top of that, resulting in an accuracy of 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='77% and 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='64%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' With all the previous modules fixed, we test the advan- tages of using either balanced sampling or VSCT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' As for the complete dataset, balanced sampling does not provide any performance benefits, whereas VSCT brings a slight improvement resulting in 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='97% testing accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' When examined on the smaller subset, the balanced sampling improves the result by a relative 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='6% to 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='91%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' VSCT, nevertheless, still outperforms it by enhancing the result with a relative 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='7% to the final 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='45% testing accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' The outturn of ablations on the cross-dataset training experiments is shown in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' For both ASLLVD and UWB-SL-Wild, we conduct the same ablations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' The results mimic the tendencies commented on in the previous exper- iment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' This study suggests that VSCT provides merits to training on such low-shot data, evincing itself more beneficial than a standard balanced sampling of classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' CONCLUSION We collected and open-sourced a new dataset for SLR with footage from online text-to-video dictionaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' We con- structed it with the already-available datasets in mind and created class mappings to WLASL100 and ASLLVD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' To reflect the attained problem’s few-shot setting, we proposed a novel procedure of training a neural pose-based SLR system called Validation Score-Conscious Training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' This procedure analyzes intermediate training results on a validation split and adaptively selects samples from the worst-performing classes to create additional mini-batches for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' We demonstrated VSCT’s merits in several experiments of few- shot learning tasks utilizing the SPOTER model, resulting in a state-of-the-art result on the ASLLVD-Skeleton dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' ACKNOWLEDGEMENT This work was supported by the Ministry of Educa- tion, Youth and Sports of the Czech Republic, Project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' LM2018101 LINDAT/CLARIAH-CZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' Computational resources were supplied by the project ”e- Infrastruktura CZ” (e-INFRA CZ LM2018140).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' REFERENCES [1] World federation of the deaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' https://wfdeaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content='org, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} +page_content=' Accessed: 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE2T4oBgHgl3EQfQQbP/content/2301.03769v1.pdf'} diff --git a/-tAyT4oBgHgl3EQfRPa5/content/tmp_files/2301.00063v1.pdf.txt b/-tAyT4oBgHgl3EQfRPa5/content/tmp_files/2301.00063v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..fb8191f1dcd035f49e9b82f89164bf1a9e118a5c --- /dev/null +++ b/-tAyT4oBgHgl3EQfRPa5/content/tmp_files/2301.00063v1.pdf.txt @@ -0,0 +1,1105 @@ +arXiv:2301.00063v1 [math.PR] 30 Dec 2022 +The Sticky L´evy Process as a solution to a Time Change Equation +Miriam Ram´ırez & Ger´onimo Uribe Bravo +Instituto de Matem´aticas +Universidad Nacional Aut´onoma de M´exico +ABSTRACT. Stochastic Differential Equations (SDEs) were originally devised by Itˆo to provide a path- +wise construction of diffusion processes. A less explored approach to represent them is through Time +Change Equations (TCEs) as put forth by Doeblin. TCEs are a generalization of Ordinary Differential +Equations driven by random functions. We present a simple example where TCEs have some advantage +over SDEs. +We represent sticky L´evy processes as the unique solution to a TCE driven by a L´evy process with +no negative jumps. The solution is adapted to the time-changed filtration of the L´evy process driving +the equation. This is in contrast to the SDE describing sticky Brownian motion, which is known to have +no adapted solutions as first proved by Chitashvili. A known consequence of such non-adaptability for +SDEs is that certain natural approximations to the solution of the corresponding SDE do not converge in +probability, even though they do converge weakly. Instead, we provide strong approximation schemes for +the solution of our TCE (by adapting Euler’s method for ODEs), whenever the driving L´evy process is +strongly approximated. +1. INTRODUCTION AND STATEMENT OF THE RESULTS +Feller’s discovery of sticky boundary behavior for Brownian motion on [0,∞) (in [Fel52, Fel54]) +is, undoubtedly, a remarkable achievement. The discovery is inscribed in the problem of describing +every diffusion processes on [0,∞) that behaves as a Brownian motion up to the time the former first +hits 0. See [EP14] for a historical account and [IM63] for probabilistic intuitions and constructions. +We now consider a definition for sticky L´evy processes associated L´evy processes which only jump +upwards (also known as Spectrally Positive L´evy process and abbreviated SPLP). General information +on SPLPs can be consulted in [Ber96, Ch. VII]. +Definition 1. Let X be a SPLP and X0 stand for X killed upon reaching zero. An extension of X0 will +be c`adl`ag a strong Markov process Z with values in [0,∞) such that X and Z have the same law if +killed upon reaching 0. We say that Z is a L´evy process with sticky boundary at 0 based on X (or a +sticky L´evy process for short) if Z is an extension of X0 for which 0 is regular and instantaneous and +which spends positive time at zero. In other words, if Z0 = 0 then +0 = inf{t > 0 : Zt = 0} = inf{t > 0 : Zt ̸= 0} +and +� ∞ +0 I(Zs = 0)ds > 0 +almost surely. +It is well known that sticky Brownian motion satisfies a stochastic differential equation (SDE) of the +form +(1) +Zt = z+ +� t +0 I(Zs > 0)dBs +γ +� t +0 I(Zs = 0)ds, +t ≥ 0, +2010 Mathematics Subject Classification. +60G51, 60G17, 34F05. +Research supported by UNAM-DGAPA-PAPIIT grant IN114720. +1 + +The Sticky L´evy Process as a solution to a Time Change Equation +2 +where B is a standard Brownian motion, the stickiness parameter γ is strictly positive and I denotes +the indicator function. This equation has no strong solutions, which means that any process satisfying +(1) involves some extra randomness to that of Brownian motion B. This result was conjectured by +Skorohod and initially proved by R. Chitashvili in [Chi89] (later published as [Chi97]) and [War97]. +More recent proofs can be found in [EP14, Bas14] and [HCA17]. In contrast to the representation of +the sticky Brownian motion as a solution to an SDE, we propose a representation of any SPLP with +a sticky boundary as a solution to a TCE. The particularity of our representation is that it does not +require any extra randomness to that generated by the L´evy process driving the equation. In the L´evy +process case, a fundamental hypothesis to construct sticky L´evy processes will be that the sample paths +have unbounded variation on any interval. Equivalently, we can assume that either there is a Gaussian +component or the sum of jumps is absolutely divergent (i.e. ∑s≤t |Xs −Xs−| = ∞ almost surely for some +t > 0). +Theorem 1. Let X be a SPLP adapted to a right-continuous and complete filtration (Ft,t ≥ 0). Assume +that the sample paths of X have unbounded variation. Given a parameter γ > 0 and a point z ≥ 0, there +exists a unique pair of stochastic processes Z = (Zt,t ≥ 0) and C = (Ct,t ≥ 0) satisfying +(2) +Zt = z+XCt +γ +� t +0 I(Zs = 0)ds, +where +Ct = +� t +0 I(Zs > 0)ds, +for every t ≥ 0. For the unique pair (Z,C) verifying Equation (2), it holds that C is a (Ft)-time change +and that Z is adapted to the time-changed filtration ( � +Ft,t ≥ 0) given by � +Ft = FCt. Furthermore, Z is +a sticky L´evy process based on X. +This result attempts to honor the memory of Wolfgang Doeblin, the pioneer of TCEs, because for +historical reasons that can be consulted in [BY02], the representation of diffusion processes suggested +by Doeblin using TCEs is less known than the one given by Kiyosi Itˆo via SDEs. In particular, the +region of applicability of TCEs has not been as carefully delineated as the one for SDEs. Note, however, +that TCEs a priori do not even need the notion of a stochastic integral to be stated and, as showed in +[CPGUB17, CPGUB13], TCEs have much better stability properties than SDEs. +To explain the unbounded variation assumption, it implies that the Dini derivatives of X are infinite +(as proved originally in [Rog68]; see [AHUB20] for an extension and further applications). In other +words, at any given stopping time T (such as the hitting time of zero), we have +−liminf +h→0+ +XT+h −XT +h += limsup +h→0+ +XT+h −XT +h += ∞. +This will aid in proving that 0 is regular and instantaneous for Z. The following (counter)example also +indirectly shows its relevance: the equation +h(t) = β +� t +0 I(h(s) > 0)ds+γ +� t +0 I(h(s) = 0)ds +does not admit solutions if β < 0 < γ. The difficulty with a time-change equation such as (2) is the +discontinuity of the indicator functions of (0,∞) and of {0}. The success in its analysis follows from +an explicit description of a solution in terms of reflection in the sense of Skorohod. This is done for a +deterministic version of (2) in Proposition 3 of Section 2.2. +Sticky L´evy processes are a one parameter family of processes built from the trajectories of X and +are part of the notion of recurrent extensions of X0 analyzed in [RUB22] in terms of three non-negative +constants and a measure on (0,∞). Such processes are called SPLP (with values) in [0,∞). As in Feller’s +result, these parameters describe the domain of the infinitesimal generator L of the corresponding +recurrent extension. A possible boundary condition describing such a domain is given by +f ′(0+) = γ−1L f(0+) + +The Sticky L´evy Process as a solution to a Time Change Equation +3 +for some constant γ > 0. In the Brownian case, this condition corresponds to the so-called sticky +Brownian motion with stickiness parameter γ. Generalizing the Brownian case, we will compute the +boundary condition for the generator of the sticky L´evy process of Theorem 1 in Section 3.3. Gen- +erator considerations are also relevant to explain the assumption on X having no negative jumps: The +generator L of such a L´evy process acts on functions defined on R, but immediately makes sense on +functions only defined on [0,∞). This last assertion is not true for the generator of a L´evy process with +jumps of both signs. +Our second main result exposes a positive consequence of the adaptability of the solution to the TCE +(2). In [Bas14], an equivalent system to the SDE (1) is studied. In particular, it is showed that the non- +existence of strong solutions prevent the convergence in probability of certain natural approximations +to the solutions of the corresponding SDE, even though they converge weakly. In contrast, we present +a simple (albeit strong!) approximation scheme for the solution to the TCE (2). To establish such a +convergence result, we start from an approximation to the L´evy process X which drives the TCE (2). +Theorem 2. Let X be a SPLP with unbounded variation. Let (Z,C) denote the unique solution to the +TCE (2). Consider (Xn,n ≥ 1) a sequence of processes with c`adl`ag paths, such that each Xn is the +piecewise constant extension of some discrete-time process defined on N/n and starts at 0. Suppose +that Xn → X in the Skorohod topology, either weakly or almost surely. Let (zn,n ≥ 1) be a sequence +of non-negative real numbers converging to a point z. Consider the processes Cn and Zn defined by +Cn(0) = Cn(0−) = 0, +Cn(t) = Cn(⌊nt⌋/n−)+(t −⌊nt⌋/n)I(Zn(t) > 0) +(3) +and +Zn(t) = (zn +Xn −γ Id)(Cn(⌊nt⌋/n))+γ⌊nt⌋/n. +(4) +Then Cn →C uniformly on compact sets and Zn → Z in the Skorohod topology. The type of convergence +will be weak or almost sure, depending on the type of convergence of (Xn,n ≥ 1). +Observe that the above procedure corresponds to an Euler-type approximation for the solution to +the TCE (2). If we consider the same equation but now driven by a process for which we could not +guarantee the existence of a solution, our approximation scheme might converge but the limit might not +be solution, as shown in the following simple but illustrative example. Let X = −Id, z = 0 and γ = 1. +Then the approximations proposed in (3) and (4) reduce to +Cn +�2k −1 +n +� += Cn +�2k +n +� += k +n +and +Zn +�k +n +� += +� +0 +if k is even +− 1 +n +if k is odd +for each k ∈ N. These sequences converge to C∗(t) = t/2 and Z∗ = 0, but clearly such processes do +not satisfy TCE (2). In general, TCEs are very robust under approximations; the failure to converge is +related to the fact that the equation that we just considered actually admits no solutions, as commented +in a previous paragraph. +Weak approximation results for sticky Brownian motion or of L´evy processes of the sticky type have +been given in [Yam94] and [HL81]. In the latter reference, reflecting Brownian motion is used, while in +the former, an SDE representation is used. In [BRHC20], the reader will find an approximation of sticky +Brownian motions by discrete space Markov chains and by diffusions in deep-well potentials as well a +numerical study and many references regarding applications. In particular, we find there the following +phrase which highlights why Theorem 2 is surprising: ... there are currently no methods to simulate +a sticky diffusion directly: there is no practical way to extend existing methods for discretizing SDEs +based on choosing discrete time steps, such as Euler-Maruyama or its variants ... to sticky processes... +It is argued that the Markov chain approximation can be extended to multiple sticky Brownian motions. +In the setting of multiple sticky Brownian motions, one can consult [BR20] and [RS15]. We are only + +The Sticky L´evy Process as a solution to a Time Change Equation +4 +aware of a strong approximation of sticky Brownian motion, in terms of time-changed embedded simple +and symmetric random walks, in [Ami91]. +The rest of this paper is structured as follows. We split the proof of Theorem 1 into several parts. In +Section 2 we explore a deterministic version of the TCE (2), which is applied in Section 2.1 to show +a monotonicity property, the essential ingredient to show uniqueness and convergence of the proposed +approximation scheme (Section 2.3). In Section 2.2, we obtain conditions for the existence of the +unique solution to the deterministic version of the TCE (2). The purpose of Section 3.1 is to apply +the deterministic analysis to prove existence and uniqueness of the solution to the TCE (2) and the +approximation Theorem 2. Then in Section 3.2, we verify that the unique process satisfying the TCE +(2) is is measurable with respect to the time-changed filtration and that it is a sticky L´evy process. +Finally in Section 3.3, using stochastic calculus instead of Theorem 2 from [RUB22], we analyze the +boundary behavior of the solution to the proposed TCE to describe the infinitesimal generator of a +sticky L´evy process. +2. DETERMINISTIC ANALYSIS +Following the ideas from [CPGUB13] and [CPGUB17], we start by considering a deterministic +version of the TCE (2). +We will prove that every solution to the corresponding equation satisfies a monotonicity property, +which will be the key in the proof of uniqueness. Assume that Z solves almost surely the TCE (2). +Hence, its paths satisfy an equation of the type +(5) +h(t) = f(c(t))+g(t), +c(t) = +� t +0 I(h(s) > 0)ds. +where f : [0,∞) → R is a c`adl`ag function without negative jumps starting at some non-negative value +and g is an non-decreasing c`adl`ag +function. (Indeed, we can take as f a typical sample path of +t �→ z + Xt − γt and g(t) = γt.) Recall that, f being c`adl`ag , we can define the jump of f at t, denoted +∆ f(t), as f(t) − f(t−). By a solution to (5), we might refer either to the function h (from which c is +immediately constructed), or to the pair (h,c). +We first verify the non-negativity of the function h. +Proposition 1. Let f and g be c`adl`ag and assume that ∆ f ≥ 0, g is non-decreasing and f(0)+g(0) ≥ +0. Then, every solution h to the TCE (5) is non-negative. Furthermore, if g is strictly increasing, the +function c given by c(t) = +� t +0 I(h(s) > 0)ds is also strictly increasing. +Proof. Let h be a solution to (5) and suppose that it takes negative values. Note that h(0) = f(0) + +g(0) ≥ 0 and that h is c`adl`ag without negative jumps. Hence, h reaches (−∞,0) continuously. The +right continuity of f (and then of h) ensures the existence of some non-degenerate interval on which h +is negative. Fix ε > 0 small enough to ensure that τ defined by +τ = inf{t ≥ 0 : h < 0 on (t,t +ε)} +is finite. (Note that, with this definition and the fact that f decreases continuously, we have that h(τ) = +0. ) Given that h is negative on a right neighborhood of τ, then +� τ +0 I(h(s) > 0)ds = +� τ+ε +0 +I(h(s) > 0)ds, +which leads us to a contradiction because +0 = h(τ) = f +�� τ +0 I(h(s) > 0)ds +� ++g(τ) ≤ f +�� τ+ε +0 +I(h(s) > 0)ds +� ++g(τ +ε) = h(τ +ε) < 0. +Hence, h is non-negative. + +The Sticky L´evy Process as a solution to a Time Change Equation +5 +Assume now that g is strictly increasing. By definition, c is non-decreasing. We prove that c is +strictly increasing by contradiction: assume that c(t) = c(s) for some s < t. Then, h = 0 on (s,t) and, +by working on a smaler interval, we can assume that h(s) = h(t) = 0. However, we then get +0 = h(s) = f ◦c(s)+g(s) < f ◦c(s)+g(t) = f ◦c(t)+g(t) = h(t) = 0. +The contradiction implies that c is strictly increasing. +□ +If f−(t) = f(t−), note that the above result and (a slight modification of) its proof also holds for +solutions to the inequality +� t +s I(h(r) > 0)dr ≤ c(t)−c(s) ≤ +� t +s I(h(r) ≥ 0)dr +where h(r) = f− ◦c(r)+g−(r) and f and g satisfy the hypotheses of Proposition 1. These inequalities +are natural when studying the stability of solutions to (5) and will come up in the proof of Theorem 2. +2.1. Monotonicity and Uniqueness. The following comparison result for the solutions to Equation +(5) will be the key idea in the uniqueness proof of Theorem (1). Moreover, we pick up it in Section 2.3, +where it also plays an essential role in the approximation of sticky L´evy processes. +Proposition 2. Let ( f 1,g1) and ( f 2,g2) be pairs of functions satisfying that f i and gi are c`adl`ag , +∆ f i ≥ 0, gi is strictly increasing and f i(0)+gi(0) ≥ 0. Suppose that f 1 ≤ f 2 and g1 ≤ g2. If h1 and h2 +satisfy +hi(t) = f i(ci(t))+gi(t), +ci(t) = +� t +0 I(hi(s) > 0)ds, +for i = 1,2, then we have the inequality c1 ≤ c2. In particular, Equation (5) admits has at most one +solution when g is strictly increasing. +Proof. Fix ε > 0 and define cε(t) = c2(ε +t). Set +τ = inf{t > 0 : c1(t) > cε(t)}. +To get a contradiction, suppose that τ < ∞. The continuity of c1 and cε guarantees that c1(τ) = cε(τ) +and c1 is bigger than cε at some point t of every right neighborhood of τ. At such points, the inequality +cε(t)−cε(τ) < c1(t)−c1(τ) is satisfied. Applying a change of variable, this is equivalent to +(6) +� t +τ I(h2(ε +s) > 0)ds < +� t +τ I(h1(s) > 0)ds. +The assumpions about g1 and g2 imply that g1(τ) < g2(ε +τ). Therefore +0 ≤ h1(τ) = f 1(c1(τ))+g1(τ) < f 2(cε(τ))+g2(ε +τ) = h2(ε +τ). +Thanks to the right continuity of h2, we can choose t close enough to τ such that h2(ε +s) > 0 for every +s ∈ [τ,t). Going back to the inequality (6), we see that +t −τ = +� t +τ I(h2(ε +s) > 0)ds < +� t +τ I(h1(s) > 0)ds ≤ t −τ, +which is a contradiction. Therefore τ = ∞ and we conclude the announced result by letting ε → 0. +In particular, if (h1,c1) and (h2,c2) are two solutions to (5) (driven by the same functions f and +g), then the above monotonicity result (applied twice) implies c1 = c2 and therefore h1 = f ◦c1 +g = +f ◦c2 +g = h2. +□ + +The Sticky L´evy Process as a solution to a Time Change Equation +6 +2.2. Existence. The following variant of a well-known result of Skorohod (cf. [RY99, Chapter VI, +Lemma 2.1]) will be helpful to verify the existence of the unique solution to the TCE (5). +Lemma 1. Let f : [0,∞) → R be a c`adl`ag function with non-negative jumps and f(0) ≥ 0. Then there +exists a unique pair of functions (r,l) defined on [0,∞) which satisfies: r = f + l, r is non-negative, +l is a non-decreasing continuous function that increases only on the set {s : r(s) = 0} and such that +l(0) = 0. Moreover, the function l is given by +l(t) = sup +s≤t +(− f(s)∨0). +Note that the lack of negative jumps of f is fundamental to obtain a continuous process l. +With the above Lemma, we can give a deterministic existence result for equation (5). +Proposition 3. Assume that f is c`adl`ag , ∆ f ≥ 0 and f(0) ≥ 0. Let (r,l) be the pair of processes of +Lemma 1 applied to f. If {t ≥ 0 : r(t) = 0} has Lebesgue measure zero, then, for every γ > 0 there +exists a solution h to +(7) +h = f +�� t +0 I(h(s) > 0)ds +� ++γ +� t +0 I(h(s) = 0)ds. +Equivalently, in terms of Equation (5), the function h satisfies +(8) +h = f γ ◦c+γ Id, +c(t) = +� t +0 I(h(s) > 0)ds. +where f γ(t) = f(t)−γt. +Proof. Applying Lemma 1 to f, we deduce the existence of a unique pair of processes (r,l) satisfying +r(t) = f(t) + l(t) with r is a non-negative function and l a continuous function with non-decreasing +paths such that l(0) = 0 and +(9) +� t +0 I(r(s) > 0)l(ds) = 0. +To construct the solution to the deterministic TCE (7), let us consider the continuous and strictly in- +creasing function a defined by a(t) = t + l(t)/γ for every t ≥ 0. Denote its inverse by c and consider +the composition h = r ◦c. The hypothesis on f implies that +� t +0 I(r(s) = 0)ds = 0 for all t. Therefore, +since r is non-negative, then +t = +� t +0 I(r(s) > 0)ds = +� t +0 I(r(s) > 0)(ds+γ−1l(ds)). +Substituting the deterministic time t for c(t) in the previous expression and using that c is the inverse +of a, we have +c(t) = +� c(t) +0 +I(r(s) > 0)a(ds) = +� t +0 I(h(s) > 0)ds. +Finally, the definition of a and its continuity imply l(t) = γ(a(t)−t), so that +l(c(t)) = γ(t −c(t)) = γ +� t +0 I(h(s) = 0)ds. +Hence, the identity h(t) = r(c(t)) can be written as +h(t) = f +�� t +0 I(h(s) > 0)ds +� ++γ +� t +0 I(h(s) = 0)ds, +as we wanted. +□ + +The Sticky L´evy Process as a solution to a Time Change Equation +7 +2.3. Approximation. It is our purpose now to discuss a simple method to approximate the solution +to the TCE (7). Among the large number of existing discretization schemes, we choose a widely used +method, an adaptation of that of Euler’s. Again, the key to the proof relies deeply on our monotonicity +result. +Proposition 4. Let f be c`adl`ag and satisfy ∆ f ≥ 0, and f(0) ≥ 0. Assume that Equation (7), or +equivalently (8), admits a unique solution denoted by (h,c). Let ˜f n be a sequence of c`adl`ag functions +which converge to f and let f n = ˜f n −γ⌊n·⌋/n. Let cn and hn be given by cn(0) = cn(0−) = 0, +cn(t) = cn(⌊nt⌋/n−)+(t −⌊nt⌋/n)I(hn(t) > 0) +(10) +and +hn(t) = f n(cn(⌊nt⌋/n))+γ⌊nt⌋/n. +(11) +Then hn → h in the Skorohod J1 topology and cn → c uniformly on compact sets. +Note that Propositions 2 and 3 give us conditions for the existence of a unique solution, which is +the main assumption in the above proposition. Also, hn is piecewise on [(k −1)/n,k/n) and, therefore, +cn is piecewise linear on [(k − 1)/n,k/n] and, at the endpoints of this interval, cn takes values in N/n. +Hence, cn(⌊tn⌋/n) = ⌊ncn(t)/n⌋. +The proof of Proposition 4 is structured as follows: we prove that the sequence (cn,n ≥ 1) is +relatively compact. Given (cnj, j ≥ 1) a subsequence that converges to certain limit c∗, we see that +((cnj,hnj), j ≥ 1) also converges and its limit is given by (c∗,h∗), where h∗ = f γ ◦c∗ +γ Id and we re- +call that f γ = f −γ Id. A slight modification of the proof of Proposition 2 implies that the limit (c∗,h∗) +does not depend on the choice of the subsequence (nj, j ≥ 1) and consequently the whole sequence +((cn,hn),n ≥ 1) converges. +Proof of Proposition 4. Since γ Id is continuous, then our hypothesiss ˜f n → f implies that f n → f − +γ Id. (Since addition is not a continuous operation on Skorohod space as in [Bil99, Ex. 12.2], we need +to use Theorem 4.1 in [Whi80] or Theorem 12.7.3 in [Whi02].) +Fix t0 > 0. Note that Equation (10) can be written as +cn(t) = +� t +0 I(hn(s) > 0)ds. +This guarantees that the functions cn are Lipschitz continuous with Lipschitz constant equal to 1. Hence +they are non-decreasing, equicontinuous and uniformly bounded on [0,t0]. It follows from Arzel`a- +Ascoli Theorem that (cn,n ≥ 1) is relatively compact. Let (cnj, j ≥ 1) be a subsequence which con- +verges uniformly in the space of continuous function on [0,t0], let us call c∗ to the limit, which is +non-decreasing and continuous. Actually, c∗ is 1-Lipschitz continuous, so that c∗(t)−c∗(s) ≤ t −s for +s ≤ t. This is a fundamental fact which will be relevant to proving that c = c∗. Since cnj(⌊njt⌋/nj) = +⌊njcnj(t)⌋/nj for every t ≥ 0, we can write hnj = f nj ◦cnj +γ⌊nj·⌋/nj. We now prove that: as j → ∞: +(cnj, f nj ◦cnj) → (c∗, f γ ◦c∗). Indeed, the convergence f n → f γ implies that liminfn→∞ f n(tn) ≥ f γ +−(t) +whenever tn → t. (If a proof is needed, note that Proposition 3.6.5 in [EK86] tells us that the accumu- +lation points of f n(tn) belong to {f γ +−(t), f γ(t)}.) Then, +I( f γ +− ◦c∗(s)+γs > 0) ≤ liminf +j +I( f nj ◦cnj(s)+γ⌊ns⌋/n > 0), +so that, by Fatou’s lemma, +� t +s I( f γ +− ◦c∗(r)+γr > 0)dr ≤ c∗(t)−c∗(s). +But now, arguing as in Proposition 1, we see that f γ +− ◦c∗ +γ Id is non-negative and that c∗ is strictly in- +creasing. Since c∗ is continuous and stricly increasing, Theorem 13.2.2 in [Whi80, p. 430] implies that + +The Sticky L´evy Process as a solution to a Time Change Equation +8 +the composition operation is continuous at ( f γ,c∗), so that f nj ◦cnj → f γ ◦c∗. Since γ Id is continuous, +we see that hnj → h∗ := f γ ◦c∗ +γ Id, as asserted. +Another application of Fatou’s lemma gives +� t +s I( f γ ◦c∗(r)+γr > 0)dr ≤ c∗(t)−c∗(s). +Now, arguing as in the monotonicity result of Proposition 2, we get c ≤ c∗. +Let us obtain the converse inequality c∗ ≤ c by a small adaptation of the proof of the aforementioned +proposition, which then finishes the proof of Theorem 2. Let ε > 0, define ˜c(t) = c(ε + t) and let +τ = inf{t ≥ 0 : c∗(t) > ˜c(t)}. If τ < ∞, note that c∗(τ) = ˜c(τ) and, in every right neighborhood of τ, +there exists t such that c∗(t) > ˜c(t). At τ, observe that +0 ≤ h∗(τ) = f γ ◦c∗(τ)+γτ < f γ ◦ ˜c(τ)+γ(τ +ε) = h(τ +ε). +Thanks to the right continuity of the right hand side, there exists a right neighborhood of τ on which +h(· + ε) is strictly positive and on which, by definition of c, ˜c grows linearly. Let t belong to that +right-neighborhood and satisfy c∗(t) > ˜c(t). Since c∗ is 1-Lipschitz continuous, we then obtain the +contradiction: +(t −τ) = +� t +τ I(h(ε +r) > 0)dr = ˜c(t)− ˜c(τ) < c∗(t)−c∗(τ) ≤ t −τ. +Hence, τ = ∞ and therefore c∗ ≤ ˜c. Since this inequality holds for any ε > 0, we deduce that c∗ ≤ c. +The above implies that c∗ = c and consequently h∗ = h. In other words, the limits c∗ and h∗ do not +depend on the subsequence (nj, j ≥ 1) and then we conclude the convergence of the whole sequence +((cn,hn),n ≥ 1) to the unique solution to the TCE (8). +□ +3. APPLICATION TO STICKY L´EVY PROCESSES +The aim of this section is to apply the deterministic analysis of the preceeding section to prove +Theorems 1 and 2. The easy part is to obtain existence, uniqueness and approximation, while the +Markov property and the fact that the solution Z to Equation (2) is a sticky L´evy process require some +extra (probabilistic) work. We tackle the existence and uniqueness assertions in Theorem 1 and prove +Theorem 2 in Subsection 3.1. Then, we prove the strong Markov property of solutions to Equation 2 in +Subsection 3.2. This allows us to prove that solutions are sticky L´evy processes, thus finishing the proof +of Theorem 1, but leaves open the precise computation of the stickiness parameter (or, equivalently, +the boundary condition for its infinitesimal generator). We finally obtain the boundary condition in +Subsection 3.3. We could use the excursion analysis of [RUB22] to obtain the boundary condition but +decided to also include a different proof via stochastic analysis to make the two works independent. +3.1. Existence, Uniqueness and Approximation. We now turn to the proof of the existence and +uniqueness assertions in Theorem 1. +Proof of Theorem 1, Existence and Uniqueness. Note that uniqueness of Equation (2) is immediate +from Proposition 2 by replacing the c`adl`ag function f by the paths of x+X −γ Id and taking g = γ Id. +To get existence, note that applying Lemma 1 to the paths of X, we deduce the existence of a unique +pair of processes (R,L) satisfying Rt = z + Xt + Lt with R a non-negative process and L a continuous +process with non-decreasing paths such that L0 = 0 and +� t +0 I(Rs > 0)dLs = 0. In fact, we have an explicit +representation of L as +(12) +Lt = sup +s≤t +((−z−Xs)∨0) = −inf +s≤t((z+Xs)∧0). +Note that R corresponds to the process X reflected at its infimum which has been widely studied as a +part of the fluctuation theory of L´evy processes (cf. [Ber96, Ch. VI, VII], [Bin75] and [Kyp14]). + +The Sticky L´evy Process as a solution to a Time Change Equation +9 +From the explicit description of the process L given in (12), it follows that P(Rt = 0) = P(Xt = Xt), +where Xt = infs≤t(Xs ∧ 0). Similarly, we denote Xt = sups≤t(Xs ∨ 0). Proposition 3 from [Ber96, Ch. +VI] ensures that the pairs of variables (Xt −Xt,−Xt) and (Xt,Xt −Xt) have the same distribution under +P. Consequently +P(Xt = Xt) = P((Xt −Xt,−Xt) ∈ {0}×[0,∞)) = P((Xt,Xt −Xt) ∈ {0}×[0,∞)) ≤ P(Xt = 0). +The unbounded variation of X guarantees that 0 is regular for (−∞,0) and for (0,∞) (as mentioned, this +result can be found in [Rog68] and has been extended in [AHUB20]). Hence, for any t > 0, Xt > 0. +We decude that P(Xt = 0) = 1−P(Xs > 0 for some s ≤ t) = 0. Thus, +E +�� ∞ +0 I(Rt = 0)dt +� += +� ∞ +0 P(Xt = Xt)dt = 0. +Therefore, we can apply Proposition 3 to deduce the existence of solutions to Equation (2). +□ +Let us now pass to the proof of 2. +Proof of Theorem 2. As we have stated in Theorem 2, we allow the convergence Xn → X to be weak +or almost surely. Using Skorohod’s representation Theorem, we may assume that it is satisfied almost +surely in some suitable probability space. The desired result follows immediately from Proposition 4 +by considering the paths of f = z+X −γ Id and f n = zn +Xn −γ⌊n·⌋/n. +□ +3.2. Measurability details and the strong Markov property. In order to complete the proof of The- +orem 1, it remains to verify the adaptability of the unique solution to the TCE (2) to the time changed +filtration ( � +Ft,t ≥ 0) and that such a solution is, in fact, a sticky L´evy process based on X. This is the +objective of the current section, which ends the proof of Theorem 1. +By construction the mapping t �→ Ct is continuous and strictly increasing. Furthermore, given that C +is the inverse of the map t �→ t +Lt/γ, we can write +{Ct ≤ s} = {γ(t −s) ≤ Ls} ∈ Fs, +for every t ≥ 0. In other words, the random time Ct is a (Fs)-stopping time, since the filtration is +right-continuous. Therefore the process C is a (Fs)-time change and Z is adapted to the time-changed +filtration ( � +Ft,t ≥ 0). In this sense we say that Z exhibits no extra randomness to that of the original +L´evy process. This contrasts with the SDE describing sticky Brownian motion (cf. [War97, Theorem +1]). +Let us verify that the unique solution Z to (2) is an extension of the killed process X0. By construc- +tion, we see that if Z0 = z > 0, then Z equals X until they both reach zero. Hence Z and X have the +same law if killed upon reaching zero. Let now Z be the unique solution of (2) with Z0 = z = 0. The +concrete construction which proves existence to (2) of Section 2.2 shows that +γ +� t +0 I(Zs = 0)ds = L◦C +where Ct = +� t +0 I(Zs > 0)ds, Lt = −infs≤t Xs. We have already argued that the unbounded variation +hypothesis implies that Lt > 0 for any t > 0 and therefore L∞ > 0 almost surely. As above, recalling +that C is the inverse of Id+L/γ, we see that C∞ = ∞. We conclude that L◦C∞ > 0 almost surely, so that +Z spends positive time at zero. We will now use the unbounded variation of X to guarantee the regular +and instantaneous character of 0 for Z. By construction, the unique solution Z to the TCE (2) is the +process X reflected at its infimum by applying a continuous strictly increasing time change C to it, that +is Z = R◦C where R = X −X. Consequently +P(inf{s > 0 : Zs = 0} = 0) = P(inf{s > 0 : X ◦Cs = X ◦Cs} = 0) = P(inf{s > 0 : Xs = Xs} = 0). + +The Sticky L´evy Process as a solution to a Time Change Equation +10 +Since 0 is regular for (−∞,0) thanks to the unbounded variation hypothesis (meaning that X visits +(−∞,0) immediatly upon reaching 0), we conclude the regularity of 0. Similarly, given the regularity +of 0 for (0,∞) for X, we have +P(inf{s > 0 : Zs > 0} = 0) = P(inf{s > 0 : Xs > Xs} = 0) ≥ P(inf{s > 0 : Xs > 0} = 0) = 1. +Thus, 0 is an instantaneous point. +To conclude the proof of Theorem 1, it now remains to prove the strong Markov property. From the +construction of the unique solution to the TCE (2), we deduce the existence of a measurable mapping Fs +that maps the paths of the L´evy process X and the initial condition z to the unique solution to the TCE +(2) evaluated at time s, that is, Zs = Fs(X,z) for s ≥ 0. Let T be a ( � +Ft)-stopping time. Approximating +T by a decreasing sequence of ( � +Ft)-stopping times (T n,n ≥ 1) taking only finitely many values, we +see that CT is an (Ft)-stopping time. From the TCE (2), we deduce that +ZT+s = ZT +(XC(T+s) −XC(T))+γ +� s +0 I(ZT+r = 0)dr. +Consider the processes ˜C, ˜X and ˜Z given by ˜Cs =C(T +s)−C(T), ˜Xs = XC(T)+s −XC(T) and ˜Zs = ZT+s +respectively. We can write the last equation as +(13) +˜Zs = ZT + ˜X ˜C(s) +γ +� s +0 I( ˜Zr = 0)dr, +and ˜C satisfies ˜Cs = +� s +0 I( ˜Zr > 0)dr for s ≥ 0. In other words, ˜Z is solution to the TCE (2) driven by +˜X with initial condition ZT. Consequently ˜Zs = Fs( ˜X,ZT). Note that ˜X has the same distribution as X +and it is independent of � +FT. Hence, the conditional law of ˜Z given � +FT is that of F(·,ZT). (One could +make appeal to Lemma 8.7 in [Kal21, p. 169] if needed.) This allows us to conclude that Z is a strong +Markov process and concludes the proof of Theorem 1. +3.3. Stickiness and martingales. In this section we aim at describing the boundary condition of the +infinitesimal generator of the sticky L´evy process Z of Theorem 1 by proving the following result. +Proposition 5. Let X be a L´evy process of unbounded variation and no negative jumps and let L be +its infinitesimal generator. For a given z ≥ 0, let Z be the unique (strong Markov) process satisfying the +time-change equation (2): +Zt = z+X� t +0 I(Zs>0)ds +γ +� t +0 I(Zs = 0)ds. +Then, for every f : [0,∞) → R which is of class C2,b and which satisfies the boundary condition +γ f ′(0+) = L f(0+), the process M defined by +Mt = f(Zt)− +� t +0 L f(Zs)ds +is a martingale and +∂ +∂t +���� +t=0 +E( f(Zt)) = L f(z). +Theorem 2 from [RUB22] describes the domain of the infinitesimal generator of any recurrent exten- +sion of X0 (which is proved to be a Feller process) by means of three non-negative constants pc, pd, pκ +and a measure µ on (0,∞). To describe such parameters we note a couple of important facts about the +unique solution to (2). By construction we can see that it leaves 0 continuously. Indeed, if we consider +the left endpoint g of some excursion interval of Z, then Cg is the left endpoint of some excursion inter- +val of the process reflected at its infimum R. Thanks to Proposition 2 from [RUB22], such excursions +start at 0, so Z leaves 0 continuously. Thus, from [RUB22], pc > 0 and µ = 0. Note also that Z has +infinite lifetime because R has it and C is bounded by the identity function, so pκ = 0. Finally, since Z + +The Sticky L´evy Process as a solution to a Time Change Equation +11 +spends positive time at 0, then pd > 0. Theorem 2 from [RUB22] ensures that every function f in the +domain of the infinitesimal generator of Z satisfies +f ′(0+) = pd +pc +L f(0+). +Our proof of Proposition 5 does not require the results from [RUB22]. The main intention is to give +an application of stochastic calculus, since we recall that a classical computation of the infinitesimal +generator for L´evy processes is based on Fourier analysis (cf. [Ber96]). Regarding the generator L , +recall that it can be applied to C2,b functions such as f and that L f is continuous (an explicit expression +is forthcoming). The lack of negative jumps implies that L f is defined even if f is only defined and +C2,b on an open set containing [0,∞). +Proof of Proposition 5. Let Z be the unique solution to the TCE (2) driven by the SPLP X. Itˆo’s formula +for semimartingales [Pro04, Chapter II, Theorem 32] guarantees that for every function f ∈ C2 +0[0,∞): +f(Zt) = f(z)+ +� t +0 f ′(Z− +s )dXCs + +� t +0 γ f ′(Z− +s )I(Z− +s = 0)ds+ 1 +2 +� t +0 f ′′(Z− +s )d[Z,Z]c +s ++∑ +s≤t +(∆ f(Zs)− f ′(Z− +s )∆Zs). +(14) +In order to analyze this expression, we recall the so-called L´evy-Itˆo decomposition, which describes +the structure of any L´evy process in terms of three independent auxiliary L´evy processes, each with a +different type of path behaviour. Consider the Poisson point process N of the jumps of X given by +Nt = ∑ +s≤t +δ(s,∆Xs). +Denote by ν the characteristic measure of N, which is called the L´evy measure of X and fulfills the +integrability condition +� +(0,∞)(1 ∧ x2)ν(dx) < ∞. Then, we write the L´evy-Itˆo decomposition as X = +X(1) + X(2) + X(3), where X(1) = bt + σBt is a Brownian motion independent of N, with diffusion +coefficient σ 2 ≥ 0 and drift b = E[X1 − +� +(0,1] +� +[1,∞) xN(ds,dx)], +X(2) = +� +(0,t] +� +[1,∞) xN(ds,dx) +is a compound Poisson process consisting of the sum of the large jumps of X and finally +X(3) = +� +(0,t] +� +(0,1) x(N(ds,dx)−ν(dx)ds) +is a square-integrable martingale. +Assuming the L´evy-Itˆo decomposition of X and using the next result, whose proof is postponed, we +will see that +� t +0 f ′(Z− +s )dXCs is a semimartingale of the form +(15) +Mt + +� t +0 bf ′(Z− +s )(1−I(Zs = 0))ds+ +� t +0 f ′(Z− +s )dX(2) +Cs , +for some square-integrable martingale M. +Lemma 2. Let C be a (Ft)-time change whose paths are continuous and locally bounded. Let X be a +right-continuous local martingale with respect to (Ft,t ≥ 0). Then the time-changed process XC is a +right-continuous local martingale with respect to the time-changed filtration ( � +Ft,t ≥ 0). +Lemma 2 ensures that the time-changed process (σB + X(3)) ◦C remains a local martingale. Ac- +cording to Theorem 20 from [Pro04, Chapter II], square-integrable local martingales are preserved + +The Sticky L´evy Process as a solution to a Time Change Equation +12 +under stochastic integration provided that the integrand process is adapted and has c`adl`ag paths. Con- +sequently the stochastic integral1 M = f ′(Z−) · (σBC + X(3) +C ) is a ( � +Ft)-local martingale. Thanks to +Corollary 27.3 from [Pro04, Chapter II], we know that a necessary and sufficient condition for a local +martingale to be a square-integrable martingale is that its quadratic variation is integrable. Let us verify +that E[[M,M]t] < ∞ for every t ≥ 0. Theorem 10.17 from [Jac79] implies the quadratic variation of the +time-changed process coincides with the time change of the quadratic variation +� +σBC +X(3) +C ,σBC +X(3) +C +� +t = +� +σB+X(3),σB+X(3)� +Ct , +t ≥ 0. +Given that the Brownian motion B is independent of X(3), the quadratic variation is σ 2Ct + +� +X(3),X(3)� +Ct, +which is bounded by σ 2t + +� +X(3),X(3)� +t. Thus +E[[M,M]t] ≤ ∥f ′∥2 +∞E +�� +σB+X(3),σB+X(3)� +Ct +� +≤ ∥f ′∥2 +∞ +� +σ 2t +t +� +(−1,1) x2 ν(dx) +� +< ∞. +This verifies the decomposition (15). Later we will deal with the last term of this decomposition. +Coming back to Itˆo’s formula (14), we need to calculate the term corresponding to the integral with +respect to the continuous part of the quadratic variation of Z. First, we decompose the variation as +[Z,Z]s = [XC,XC]s +2[XC,γ(Id−C)]s +γ2[Id−C,Id−C]s, +for every s ≥ 0. The first term is [X,X]Cs. Given the finite variation of γ(Id−C) and the continuity +of C, Theorem 26.6 from [Kal02] implies that almost surely the other two terms are zero. Thereby +[Z,Z]s = [X,X]Cs for every s ≥ 0 and +1 +2 +� t +0 f ′′(Z− +s )d[Z,Z]c +s = 1 +2 +� t +0 σ 2 f ′′(Z− +s )(1−I(Zs = 0))ds. +Now we analyze the last term on the right-hand side from (14), which corresponds to the jump part. +Let us note that the discontinuities of f ◦Z derive from the discontinuities of Z, which are caused by +the jumps of X ◦C, in other words +{s ≤ t : |∆ f(Zs)| > 0}⊆{s ≤ t : ∆Zs > 0} = {s ≤ t : ∆(X ◦C)s > 0}. +Making the change of variable r = Cs, the sum of the jumps in (14) can be written as +(16) +∑ +r≤Ct +(∆ f(Z ◦Ar)− f ′(Z− ◦Ar)∆(Z ◦Ar)), +where A denotes the inverse of C. We claim that A is a ( � +Ft)-time change. Indeed, splitting in the cases +r < t and r ≥ t, we see that {At ≤ s}∩{Cs ≤ r} = {t ≤Cs ≤ r} ∈ Fr for any r ≥ 0. Exercise 1.12 from +[RY99, Chapter V] ensures that the time-changed filtration ( � +FAt,t ≥ 0) is in fact (Ft,t ≥ 0). Thus, for +any continuous function g, the process (g(Z− +At),t ≥ 0) is (Ft)-predictable. +We return to (15) to put together the sum of the jumps in (16) and the stochastic integral ( f ′ ◦Z−)· +(X(2) ◦C). For this purpose, it is convenient to rewrite the last integral as ( f ′ ◦Z− ◦A ◦C) · (X(2) ◦C) +and apply Lemma 10.18 from [Jac79] to deduce that ( f ′ ◦Z−) · (X(2) ◦C) = (( f ′ ◦Z− ◦A) · X(2)) ◦C. +1We use both notations +� Hs dXs and H ·X to refer to the stochastic integral. + +The Sticky L´evy Process as a solution to a Time Change Equation +13 +Consequently +� t +0 f ′(Z− +s )dX(2) +Cs + ∑ +s≤Ct +(∆ f(Z ◦As)− f ′(Z− ◦As)∆(Z ◦As)) += +� Ct +0 +� +(0,∞) +� +f(Z− +As +x)− f(Z− +As)− f ′(Z− +As)xI(x ∈ (0,1)) +� +(N(ds,dx)−ν(dx)ds) ++ +� Ct +0 +� +(0,∞) +� +f(Z− +As +x)− f(Z− +As)− f ′(Z− +As)xI(x ∈ (0,1)) +� +ν(dx)ds. +(17) +Define the process M by +Mt =− +� t +0 +� +[1,∞) +� +f(Z− +As +x)− f(Z− +As) +� +ν(dx)ds ++ +� t +0 +� +[1,∞) +� +f(Z− +As +x)− f(Z− +As) +� +N(ds,dx) ++ +� t +0 +� +(0,1) +� +f(Z− +As +x)− f(Z− +As)− f ′(Z− +As)x +� +(N(ds,dx)− ds). +Since ν is a L´evy measure, then +E +�� t +0 +� +[1,∞) +�� f(Z− +As +x)− f(Z− +As) +�� ds +� +≤ ∥f∥2 +∞tν([1,∞)) < ∞. +We develop the first degree Taylor polynomial of f(Z− +As +x) to obtain +f ′(Z− +As)x = f(Z− +As +x)− f(Z− +As)−R(x), +x ∈ (0,1), +where the remainder R satisfies |R(x)| ≤ 1 +2∥f ′′∥∞x2. Therefore +E +�� t +0 +� +(0,1) +� +f(Z− +As +x)− f(Z− +As)− f ′(Z− +As)x +� +ν(dx)ds +� +≤ 1 +2∥f ′′∥∞tE +�� +(0,1) x2 ν(dx) +� +< ∞. +Theorem 5.2.1 from [App09] ensures that M is a (Ft)-local martingale and Lemma 2 implies that MC +is a ( � +Ft)-local martingale. Furthermore, for t ≥ 0 it holds that +E +� +sup +s≤t +|MCs| +� +≤ E +� +sup +s≤t +|Ms| +� +≤ +� +2∥f∥∞ + 1 +2∥f ′′∥2 +∞ +� +t +� +(0,∞)(1∧x2)ν(dx) < ∞. +It follows from Theorem 51 from [Pro04, Chapter I] that MC is a true martingale. +Gathering all the expressions involved in Itˆo’s formula (14), we get the semimartingale decomposi- +tion +f(Zt)− f(z) =Mt + +� t +0 bf ′(Z− +s )(1−I(Zs = 0))ds+ +� t +0 γ f ′(0+)I(Zs = 0)ds ++ 1 +2 +� t +0 σ 2 f ′′(Z− +s )(1−I(Zs = 0))ds+MCt ++ +� Ct +0 +� +(0,∞) +� +f(Z− +As +x)− f(Z− +As)− f ′(Z− +As)xI(x ∈ (0,1)) +� +ν(dx)ds. +Recall that the extended generator of X (as in [RY99, Ch. VII]) is given by +L f(z) = bf ′(z)+ σ 2 +2 f ′′(z)+ +� +R+ +� +f(z+x)− f(z)− f ′(z)xI(x ∈ (0,1)) +� +ν(dx) + +The Sticky L´evy Process as a solution to a Time Change Equation +14 +on C2,b functions and that the extended generator of X0 is given by L f on C2,b functions f on [0,∞) +which vanish (together with its derivatives) at 0 and ∞. Note that L f(z) is bounded. Define +˜ +L f(0) by +˜ +L f(0) = (b−γ) f ′(0+)+ σ 2 +2 f ′′(0+)+ +� +R+ +� +f(x)− f(0+)− f ′(0+)xI(x ∈ (0,1)) +� +ν(dx). +Given that +˜ +L f(0) = L f(0+)−γ f ′(0+), we can write the martingale M +MC as +M +MCt = f(Zt)− f(z)− +� t +0 L f(Z− +s )ds+ +� t +0 +˜ +L f(0)I(Zs = 0)ds. +We deduce that if a function f ∈ C2[0,∞) satisfies the boundary condition +˜ +L f(0) = 0 or equivalently +γ f ′(0+) = L f(0+), then f(Zt) − f(z) − +� t +0 L f(Zs)ds is a martingale. By hypothesis, the last term +is bounded by a linear function of t, so that E[f(Zt)] is differentiable at zero and the derivative equals +L f(z). +□ +We conclude this section with the proof of Lemma 2. +Proof. (Lemma 2) Let (βn,n ≥ 1) be localizing sequence for X, then βn → ∞ as n → ∞ and for each +n ≥ 1, the process XβnI(βn > 0) is a uniformly integrable martingale. Keeping the notation A for the +inverse of C, we will prove that (A(βn),n ≥ 1) is a sequence of ( � +Ft)-stopping times that localizes to XC. +The property of being (Ft)-stopping time is deduced by observing that {βn ≤ Ct} ∈ Fβn ∩FCt ⊂ � +Ft, +which implies that +{A(βn) ≤ t}∩{Ct ≤ s} = {βn ≤ Ct}∩{Ct ≤ s} ∈ Fs. +Since C ◦A = Id, then +(Z ◦C)A(βn) +t += ZCt∧βn = Zβn +Ct . +Given that Zβn is a (Ft)-martingale, Optional Stopping Theorem guarantees that +E +� +Zβn +Ct +���FCs +� += Zβn +Cs , +0 ≤ s ≤ t. +Hence (Z ◦C)A(βn) is a ( � +Ft)-martingale. 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Appl. 52 (1994), no. 1, 135–164. ↑3 + diff --git a/-tAyT4oBgHgl3EQfRPa5/content/tmp_files/load_file.txt b/-tAyT4oBgHgl3EQfRPa5/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..254970da46679d69e95cd658d0a908e8623c78e3 --- /dev/null +++ b/-tAyT4oBgHgl3EQfRPa5/content/tmp_files/load_file.txt @@ -0,0 +1,564 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf,len=563 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content='00063v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content='PR] 30 Dec 2022 The Sticky L´evy Process as a solution to a Time Change Equation Miriam Ram´ırez & Ger´onimo Uribe Bravo Instituto de Matem´aticas Universidad Nacional Aut´onoma de M´exico ABSTRACT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Stochastic Differential Equations (SDEs) were originally devised by Itˆo to provide a path- wise construction of diffusion processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' A less explored approach to represent them is through Time Change Equations (TCEs) as put forth by Doeblin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' TCEs are a generalization of Ordinary Differential Equations driven by random functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' We present a simple example where TCEs have some advantage over SDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' We represent sticky L´evy processes as the unique solution to a TCE driven by a L´evy process with no negative jumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' The solution is adapted to the time-changed filtration of the L´evy process driving the equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' This is in contrast to the SDE describing sticky Brownian motion, which is known to have no adapted solutions as first proved by Chitashvili.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' A known consequence of such non-adaptability for SDEs is that certain natural approximations to the solution of the corresponding SDE do not converge in probability, even though they do converge weakly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Instead, we provide strong approximation schemes for the solution of our TCE (by adapting Euler’s method for ODEs), whenever the driving L´evy process is strongly approximated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' INTRODUCTION AND STATEMENT OF THE RESULTS Feller’s discovery of sticky boundary behavior for Brownian motion on [0,∞) (in [Fel52, Fel54]) is, undoubtedly, a remarkable achievement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' The discovery is inscribed in the problem of describing every diffusion processes on [0,∞) that behaves as a Brownian motion up to the time the former first hits 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' See [EP14] for a historical account and [IM63] for probabilistic intuitions and constructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' We now consider a definition for sticky L´evy processes associated L´evy processes which only jump upwards (also known as Spectrally Positive L´evy process and abbreviated SPLP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' General information on SPLPs can be consulted in [Ber96, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' VII].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Let X be a SPLP and X0 stand for X killed upon reaching zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' An extension of X0 will be c`adl`ag a strong Markov process Z with values in [0,∞) such that X and Z have the same law if killed upon reaching 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' We say that Z is a L´evy process with sticky boundary at 0 based on X (or a sticky L´evy process for short) if Z is an extension of X0 for which 0 is regular and instantaneous and which spends positive time at zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' In other words, if Z0 = 0 then 0 = inf{t > 0 : Zt = 0} = inf{t > 0 : Zt ̸= 0} and � ∞ 0 I(Zs = 0)ds > 0 almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' It is well known that sticky Brownian motion satisfies a stochastic differential equation (SDE) of the form (1) Zt = z+ � t 0 I(Zs > 0)dBs +γ � t 0 I(Zs = 0)ds, t ≥ 0, 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' 60G51, 60G17, 34F05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Research supported by UNAM-DGAPA-PAPIIT grant IN114720.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' 1 The Sticky L´evy Process as a solution to a Time Change Equation 2 where B is a standard Brownian motion, the stickiness parameter γ is strictly positive and I denotes the indicator function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' This equation has no strong solutions, which means that any process satisfying (1) involves some extra randomness to that of Brownian motion B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' This result was conjectured by Skorohod and initially proved by R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Chitashvili in [Chi89] (later published as [Chi97]) and [War97].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' More recent proofs can be found in [EP14, Bas14] and [HCA17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' In contrast to the representation of the sticky Brownian motion as a solution to an SDE, we propose a representation of any SPLP with a sticky boundary as a solution to a TCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' The particularity of our representation is that it does not require any extra randomness to that generated by the L´evy process driving the equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' In the L´evy process case, a fundamental hypothesis to construct sticky L´evy processes will be that the sample paths have unbounded variation on any interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Equivalently, we can assume that either there is a Gaussian component or the sum of jumps is absolutely divergent (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' ∑s≤t |Xs −Xs−| = ∞ almost surely for some t > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Let X be a SPLP adapted to a right-continuous and complete filtration (Ft,t ≥ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Assume that the sample paths of X have unbounded variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Given a parameter γ > 0 and a point z ≥ 0, there exists a unique pair of stochastic processes Z = (Zt,t ≥ 0) and C = (Ct,t ≥ 0) satisfying (2) Zt = z+XCt +γ � t 0 I(Zs = 0)ds, where Ct = � t 0 I(Zs > 0)ds, for every t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' For the unique pair (Z,C) verifying Equation (2), it holds that C is a (Ft)-time change and that Z is adapted to the time-changed filtration ( � Ft,t ≥ 0) given by � Ft = FCt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Furthermore, Z is a sticky L´evy process based on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' This result attempts to honor the memory of Wolfgang Doeblin, the pioneer of TCEs, because for historical reasons that can be consulted in [BY02], the representation of diffusion processes suggested by Doeblin using TCEs is less known than the one given by Kiyosi Itˆo via SDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' In particular, the region of applicability of TCEs has not been as carefully delineated as the one for SDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Note, however, that TCEs a priori do not even need the notion of a stochastic integral to be stated and, as showed in [CPGUB17, CPGUB13], TCEs have much better stability properties than SDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' To explain the unbounded variation assumption, it implies that the Dini derivatives of X are infinite (as proved originally in [Rog68];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' see [AHUB20] for an extension and further applications).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' In other words, at any given stopping time T (such as the hitting time of zero), we have −liminf h→0+ XT+h −XT h = limsup h→0+ XT+h −XT h = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' This will aid in proving that 0 is regular and instantaneous for Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' The following (counter)example also indirectly shows its relevance: the equation h(t) = β � t 0 I(h(s) > 0)ds+γ � t 0 I(h(s) = 0)ds does not admit solutions if β < 0 < γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' The difficulty with a time-change equation such as (2) is the discontinuity of the indicator functions of (0,∞) and of {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' The success in its analysis follows from an explicit description of a solution in terms of reflection in the sense of Skorohod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' This is done for a deterministic version of (2) in Proposition 3 of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Sticky L´evy processes are a one parameter family of processes built from the trajectories of X and are part of the notion of recurrent extensions of X0 analyzed in [RUB22] in terms of three non-negative constants and a measure on (0,∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Such processes are called SPLP (with values) in [0,∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' As in Feller’s result, these parameters describe the domain of the infinitesimal generator L of the corresponding recurrent extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' A possible boundary condition describing such a domain is given by f ′(0+) = γ−1L f(0+) The Sticky L´evy Process as a solution to a Time Change Equation 3 for some constant γ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' In the Brownian case, this condition corresponds to the so-called sticky Brownian motion with stickiness parameter γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Generalizing the Brownian case, we will compute the boundary condition for the generator of the sticky L´evy process of Theorem 1 in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Gen- erator considerations are also relevant to explain the assumption on X having no negative jumps: The generator L of such a L´evy process acts on functions defined on R, but immediately makes sense on functions only defined on [0,∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' This last assertion is not true for the generator of a L´evy process with jumps of both signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Our second main result exposes a positive consequence of the adaptability of the solution to the TCE (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' In [Bas14], an equivalent system to the SDE (1) is studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' In particular, it is showed that the non- existence of strong solutions prevent the convergence in probability of certain natural approximations to the solutions of the corresponding SDE, even though they converge weakly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' In contrast, we present a simple (albeit strong!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=') approximation scheme for the solution to the TCE (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' To establish such a convergence result, we start from an approximation to the L´evy process X which drives the TCE (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Let X be a SPLP with unbounded variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Let (Z,C) denote the unique solution to the TCE (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Consider (Xn,n ≥ 1) a sequence of processes with c`adl`ag paths, such that each Xn is the piecewise constant extension of some discrete-time process defined on N/n and starts at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Suppose that Xn → X in the Skorohod topology, either weakly or almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Let (zn,n ≥ 1) be a sequence of non-negative real numbers converging to a point z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Consider the processes Cn and Zn defined by Cn(0) = Cn(0−) = 0, Cn(t) = Cn(⌊nt⌋/n−)+(t −⌊nt⌋/n)I(Zn(t) > 0) (3) and Zn(t) = (zn +Xn −γ Id)(Cn(⌊nt⌋/n))+γ⌊nt⌋/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' (4) Then Cn →C uniformly on compact sets and Zn → Z in the Skorohod topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' The type of convergence will be weak or almost sure, depending on the type of convergence of (Xn,n ≥ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Observe that the above procedure corresponds to an Euler-type approximation for the solution to the TCE (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' If we consider the same equation but now driven by a process for which we could not guarantee the existence of a solution, our approximation scheme might converge but the limit might not be solution, as shown in the following simple but illustrative example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Let X = −Id, z = 0 and γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Then the approximations proposed in (3) and (4) reduce to Cn �2k −1 n � = Cn �2k n � = k n and Zn �k n � = � 0 if k is even − 1 n if k is odd for each k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' These sequences converge to C∗(t) = t/2 and Z∗ = 0, but clearly such processes do not satisfy TCE (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' In general, TCEs are very robust under approximations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' the failure to converge is related to the fact that the equation that we just considered actually admits no solutions, as commented in a previous paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Weak approximation results for sticky Brownian motion or of L´evy processes of the sticky type have been given in [Yam94] and [HL81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' In the latter reference, reflecting Brownian motion is used, while in the former, an SDE representation is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' In [BRHC20], the reader will find an approximation of sticky Brownian motions by discrete space Markov chains and by diffusions in deep-well potentials as well a numerical study and many references regarding applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' In particular, we find there the following phrase which highlights why Theorem 2 is surprising: .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' there are currently no methods to simulate a sticky diffusion directly: there is no practical way to extend existing methods for discretizing SDEs based on choosing discrete time steps, such as Euler-Maruyama or its variants .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' to sticky processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' It is argued that the Markov chain approximation can be extended to multiple sticky Brownian motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' In the setting of multiple sticky Brownian motions, one can consult [BR20] and [RS15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' We are only The Sticky L´evy Process as a solution to a Time Change Equation 4 aware of a strong approximation of sticky Brownian motion, in terms of time-changed embedded simple and symmetric random walks, in [Ami91].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' The rest of this paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' We split the proof of Theorem 1 into several parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' In Section 2 we explore a deterministic version of the TCE (2), which is applied in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content='1 to show a monotonicity property, the essential ingredient to show uniqueness and convergence of the proposed approximation scheme (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' In Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content='2, we obtain conditions for the existence of the unique solution to the deterministic version of the TCE (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' The purpose of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content='1 is to apply the deterministic analysis to prove existence and uniqueness of the solution to the TCE (2) and the approximation Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Then in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content='2, we verify that the unique process satisfying the TCE (2) is is measurable with respect to the time-changed filtration and that it is a sticky L´evy process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Finally in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content='3, using stochastic calculus instead of Theorem 2 from [RUB22], we analyze the boundary behavior of the solution to the proposed TCE to describe the infinitesimal generator of a sticky L´evy process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' DETERMINISTIC ANALYSIS Following the ideas from [CPGUB13] and [CPGUB17], we start by considering a deterministic version of the TCE (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' We will prove that every solution to the corresponding equation satisfies a monotonicity property, which will be the key in the proof of uniqueness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Assume that Z solves almost surely the TCE (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Hence, its paths satisfy an equation of the type (5) h(t) = f(c(t))+g(t), c(t) = � t 0 I(h(s) > 0)ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' where f : [0,∞) → R is a c`adl`ag function without negative jumps starting at some non-negative value and g is an non-decreasing c`adl`ag function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' (Indeed, we can take as f a typical sample path of t �→ z + Xt − γt and g(t) = γt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=') Recall that, f being c`adl`ag , we can define the jump of f at t, denoted ∆ f(t), as f(t) − f(t−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' By a solution to (5), we might refer either to the function h (from which c is immediately constructed), or to the pair (h,c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' We first verify the non-negativity of the function h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Let f and g be c`adl`ag and assume that ∆ f ≥ 0, g is non-decreasing and f(0)+g(0) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Then, every solution h to the TCE (5) is non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Furthermore, if g is strictly increasing, the function c given by c(t) = � t 0 I(h(s) > 0)ds is also strictly increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Let h be a solution to (5) and suppose that it takes negative values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Note that h(0) = f(0) + g(0) ≥ 0 and that h is c`adl`ag without negative jumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Hence, h reaches (−∞,0) continuously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' The right continuity of f (and then of h) ensures the existence of some non-degenerate interval on which h is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Fix ε > 0 small enough to ensure that τ defined by τ = inf{t ≥ 0 : h < 0 on (t,t +ε)} is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' (Note that, with this definition and the fact that f decreases continuously, we have that h(τ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' ) Given that h is negative on a right neighborhood of τ, then � τ 0 I(h(s) > 0)ds = � τ+ε 0 I(h(s) > 0)ds, which leads us to a contradiction because 0 = h(τ) = f �� τ 0 I(h(s) > 0)ds � +g(τ) ≤ f �� τ+ε 0 I(h(s) > 0)ds � +g(τ +ε) = h(τ +ε) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Hence, h is non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' The Sticky L´evy Process as a solution to a Time Change Equation 5 Assume now that g is strictly increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' By definition, c is non-decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' We prove that c is strictly increasing by contradiction: assume that c(t) = c(s) for some s < t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Then, h = 0 on (s,t) and, by working on a smaler interval, we can assume that h(s) = h(t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' However, we then get 0 = h(s) = f ◦c(s)+g(s) < f ◦c(s)+g(t) = f ◦c(t)+g(t) = h(t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' The contradiction implies that c is strictly increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' □ If f−(t) = f(t−), note that the above result and (a slight modification of) its proof also holds for solutions to the inequality � t s I(h(r) > 0)dr ≤ c(t)−c(s) ≤ � t s I(h(r) ≥ 0)dr where h(r) = f− ◦c(r)+g−(r) and f and g satisfy the hypotheses of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' These inequalities are natural when studying the stability of solutions to (5) and will come up in the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Monotonicity and Uniqueness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' The following comparison result for the solutions to Equation (5) will be the key idea in the uniqueness proof of Theorem (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Moreover, we pick up it in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content='3, where it also plays an essential role in the approximation of sticky L´evy processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Let ( f 1,g1) and ( f 2,g2) be pairs of functions satisfying that f i and gi are c`adl`ag , ∆ f i ≥ 0, gi is strictly increasing and f i(0)+gi(0) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Suppose that f 1 ≤ f 2 and g1 ≤ g2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' If h1 and h2 satisfy hi(t) = f i(ci(t))+gi(t), ci(t) = � t 0 I(hi(s) > 0)ds, for i = 1,2, then we have the inequality c1 ≤ c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' In particular, Equation (5) admits has at most one solution when g is strictly increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Fix ε > 0 and define cε(t) = c2(ε +t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Set τ = inf{t > 0 : c1(t) > cε(t)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' To get a contradiction, suppose that τ < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' The continuity of c1 and cε guarantees that c1(τ) = cε(τ) and c1 is bigger than cε at some point t of every right neighborhood of τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' At such points, the inequality cε(t)−cε(τ) < c1(t)−c1(τ) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Applying a change of variable, this is equivalent to (6) � t τ I(h2(ε +s) > 0)ds < � t τ I(h1(s) > 0)ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' The assumpions about g1 and g2 imply that g1(τ) < g2(ε +τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Therefore 0 ≤ h1(τ) = f 1(c1(τ))+g1(τ) < f 2(cε(τ))+g2(ε +τ) = h2(ε +τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Thanks to the right continuity of h2, we can choose t close enough to τ such that h2(ε +s) > 0 for every s ∈ [τ,t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Going back to the inequality (6), we see that t −τ = � t τ I(h2(ε +s) > 0)ds < � t τ I(h1(s) > 0)ds ≤ t −τ, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Therefore τ = ∞ and we conclude the announced result by letting ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' In particular, if (h1,c1) and (h2,c2) are two solutions to (5) (driven by the same functions f and g), then the above monotonicity result (applied twice) implies c1 = c2 and therefore h1 = f ◦c1 +g = f ◦c2 +g = h2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' □ The Sticky L´evy Process as a solution to a Time Change Equation 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Existence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' The following variant of a well-known result of Skorohod (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' [RY99, Chapter VI, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content='1]) will be helpful to verify the existence of the unique solution to the TCE (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Let f : [0,∞) → R be a c`adl`ag function with non-negative jumps and f(0) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Then there exists a unique pair of functions (r,l) defined on [0,∞) which satisfies: r = f + l, r is non-negative, l is a non-decreasing continuous function that increases only on the set {s : r(s) = 0} and such that l(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Moreover, the function l is given by l(t) = sup s≤t (− f(s)∨0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Note that the lack of negative jumps of f is fundamental to obtain a continuous process l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' With the above Lemma, we can give a deterministic existence result for equation (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Assume that f is c`adl`ag , ∆ f ≥ 0 and f(0) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Let (r,l) be the pair of processes of Lemma 1 applied to f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' If {t ≥ 0 : r(t) = 0} has Lebesgue measure zero, then, for every γ > 0 there exists a solution h to (7) h = f �� t 0 I(h(s) > 0)ds � +γ � t 0 I(h(s) = 0)ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Equivalently, in terms of Equation (5), the function h satisfies (8) h = f γ ◦c+γ Id, c(t) = � t 0 I(h(s) > 0)ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' where f γ(t) = f(t)−γt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Applying Lemma 1 to f, we deduce the existence of a unique pair of processes (r,l) satisfying r(t) = f(t) + l(t) with r is a non-negative function and l a continuous function with non-decreasing paths such that l(0) = 0 and (9) � t 0 I(r(s) > 0)l(ds) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' To construct the solution to the deterministic TCE (7), let us consider the continuous and strictly in- creasing function a defined by a(t) = t + l(t)/γ for every t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Denote its inverse by c and consider the composition h = r ◦c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' The hypothesis on f implies that � t 0 I(r(s) = 0)ds = 0 for all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Therefore, since r is non-negative, then t = � t 0 I(r(s) > 0)ds = � t 0 I(r(s) > 0)(ds+γ−1l(ds)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Substituting the deterministic time t for c(t) in the previous expression and using that c is the inverse of a, we have c(t) = � c(t) 0 I(r(s) > 0)a(ds) = � t 0 I(h(s) > 0)ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Finally, the definition of a and its continuity imply l(t) = γ(a(t)−t), so that l(c(t)) = γ(t −c(t)) = γ � t 0 I(h(s) = 0)ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Hence, the identity h(t) = r(c(t)) can be written as h(t) = f �� t 0 I(h(s) > 0)ds � +γ � t 0 I(h(s) = 0)ds, as we wanted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' □ The Sticky L´evy Process as a solution to a Time Change Equation 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' It is our purpose now to discuss a simple method to approximate the solution to the TCE (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Among the large number of existing discretization schemes, we choose a widely used method, an adaptation of that of Euler’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Again, the key to the proof relies deeply on our monotonicity result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Let f be c`adl`ag and satisfy ∆ f ≥ 0, and f(0) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Assume that Equation (7), or equivalently (8), admits a unique solution denoted by (h,c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Let ˜f n be a sequence of c`adl`ag functions which converge to f and let f n = ˜f n −γ⌊n·⌋/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Let cn and hn be given by cn(0) = cn(0−) = 0, cn(t) = cn(⌊nt⌋/n−)+(t −⌊nt⌋/n)I(hn(t) > 0) (10) and hn(t) = f n(cn(⌊nt⌋/n))+γ⌊nt⌋/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' (11) Then hn → h in the Skorohod J1 topology and cn → c uniformly on compact sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Note that Propositions 2 and 3 give us conditions for the existence of a unique solution, which is the main assumption in the above proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Also, hn is piecewise on [(k −1)/n,k/n) and, therefore, cn is piecewise linear on [(k − 1)/n,k/n] and, at the endpoints of this interval, cn takes values in N/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Hence, cn(⌊tn⌋/n) = ⌊ncn(t)/n⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' The proof of Proposition 4 is structured as follows: we prove that the sequence (cn,n ≥ 1) is relatively compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Given (cnj, j ≥ 1) a subsequence that converges to certain limit c∗, we see that ((cnj,hnj), j ≥ 1) also converges and its limit is given by (c∗,h∗), where h∗ = f γ ◦c∗ +γ Id and we re- call that f γ = f −γ Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' A slight modification of the proof of Proposition 2 implies that the limit (c∗,h∗) does not depend on the choice of the subsequence (nj, j ≥ 1) and consequently the whole sequence ((cn,hn),n ≥ 1) converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Since γ Id is continuous, then our hypothesiss ˜f n → f implies that f n → f − γ Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' (Since addition is not a continuous operation on Skorohod space as in [Bil99, Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content='2], we need to use Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content='1 in [Whi80] or Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content='3 in [Whi02].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=') Fix t0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Note that Equation (10) can be written as cn(t) = � t 0 I(hn(s) > 0)ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' This guarantees that the functions cn are Lipschitz continuous with Lipschitz constant equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Hence they are non-decreasing, equicontinuous and uniformly bounded on [0,t0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' It follows from Arzel`a- Ascoli Theorem that (cn,n ≥ 1) is relatively compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Let (cnj, j ≥ 1) be a subsequence which con- verges uniformly in the space of continuous function on [0,t0], let us call c∗ to the limit, which is non-decreasing and continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Actually, c∗ is 1-Lipschitz continuous, so that c∗(t)−c∗(s) ≤ t −s for s ≤ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' This is a fundamental fact which will be relevant to proving that c = c∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Since cnj(⌊njt⌋/nj) = ⌊njcnj(t)⌋/nj for every t ≥ 0, we can write hnj = f nj ◦cnj +γ⌊nj·⌋/nj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' We now prove that: as j → ∞: (cnj, f nj ◦cnj) → (c∗, f γ ◦c∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Indeed, the convergence f n → f γ implies that liminfn→∞ f n(tn) ≥ f γ −(t) whenever tn → t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' (If a proof is needed, note that Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content='5 in [EK86] tells us that the accumu- lation points of f n(tn) belong to {f γ −(t), f γ(t)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=') Then, I( f γ − ◦c∗(s)+γs > 0) ≤ liminf j I( f nj ◦cnj(s)+γ⌊ns⌋/n > 0), so that, by Fatou’s lemma, � t s I( f γ − ◦c∗(r)+γr > 0)dr ≤ c∗(t)−c∗(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' But now, arguing as in Proposition 1, we see that f γ − ◦c∗ +γ Id is non-negative and that c∗ is strictly in- creasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Since c∗ is continuous and stricly increasing, Theorem 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content='2 in [Whi80, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' 430] implies that The Sticky L´evy Process as a solution to a Time Change Equation 8 the composition operation is continuous at ( f γ,c∗), so that f nj ◦cnj → f γ ◦c∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Since γ Id is continuous, we see that hnj → h∗ := f γ ◦c∗ +γ Id, as asserted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Another application of Fatou’s lemma gives � t s I( f γ ◦c∗(r)+γr > 0)dr ≤ c∗(t)−c∗(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Now, arguing as in the monotonicity result of Proposition 2, we get c ≤ c∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Let us obtain the converse inequality c∗ ≤ c by a small adaptation of the proof of the aforementioned proposition, which then finishes the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Let ε > 0, define ˜c(t) = c(ε + t) and let τ = inf{t ≥ 0 : c∗(t) > ˜c(t)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' If τ < ∞, note that c∗(τ) = ˜c(τ) and, in every right neighborhood of τ, there exists t such that c∗(t) > ˜c(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' At τ, observe that 0 ≤ h∗(τ) = f γ ◦c∗(τ)+γτ < f γ ◦ ˜c(τ)+γ(τ +ε) = h(τ +ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Thanks to the right continuity of the right hand side, there exists a right neighborhood of τ on which h(· + ε) is strictly positive and on which, by definition of c, ˜c grows linearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Let t belong to that right-neighborhood and satisfy c∗(t) > ˜c(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Since c∗ is 1-Lipschitz continuous, we then obtain the contradiction: (t −τ) = � t τ I(h(ε +r) > 0)dr = ˜c(t)− ˜c(τ) < c∗(t)−c∗(τ) ≤ t −τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Hence, τ = ∞ and therefore c∗ ≤ ˜c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Since this inequality holds for any ε > 0, we deduce that c∗ ≤ c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' The above implies that c∗ = c and consequently h∗ = h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' In other words, the limits c∗ and h∗ do not depend on the subsequence (nj, j ≥ 1) and then we conclude the convergence of the whole sequence ((cn,hn),n ≥ 1) to the unique solution to the TCE (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' APPLICATION TO STICKY L´EVY PROCESSES The aim of this section is to apply the deterministic analysis of the preceeding section to prove Theorems 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' The easy part is to obtain existence, uniqueness and approximation, while the Markov property and the fact that the solution Z to Equation (2) is a sticky L´evy process require some extra (probabilistic) work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' We tackle the existence and uniqueness assertions in Theorem 1 and prove Theorem 2 in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Then, we prove the strong Markov property of solutions to Equation 2 in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' This allows us to prove that solutions are sticky L´evy processes, thus finishing the proof of Theorem 1, but leaves open the precise computation of the stickiness parameter (or, equivalently, the boundary condition for its infinitesimal generator).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' We finally obtain the boundary condition in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' We could use the excursion analysis of [RUB22] to obtain the boundary condition but decided to also include a different proof via stochastic analysis to make the two works independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Existence, Uniqueness and Approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' We now turn to the proof of the existence and uniqueness assertions in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Proof of Theorem 1, Existence and Uniqueness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Note that uniqueness of Equation (2) is immediate from Proposition 2 by replacing the c`adl`ag function f by the paths of x+X −γ Id and taking g = γ Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' To get existence, note that applying Lemma 1 to the paths of X, we deduce the existence of a unique pair of processes (R,L) satisfying Rt = z + Xt + Lt with R a non-negative process and L a continuous process with non-decreasing paths such that L0 = 0 and � t 0 I(Rs > 0)dLs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' In fact, we have an explicit representation of L as (12) Lt = sup s≤t ((−z−Xs)∨0) = −inf s≤t((z+Xs)∧0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Note that R corresponds to the process X reflected at its infimum which has been widely studied as a part of the fluctuation theory of L´evy processes (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' [Ber96, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' VI, VII], [Bin75] and [Kyp14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' The Sticky L´evy Process as a solution to a Time Change Equation 9 From the explicit description of the process L given in (12), it follows that P(Rt = 0) = P(Xt = Xt), where Xt = infs≤t(Xs ∧ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Similarly, we denote Xt = sups≤t(Xs ∨ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Proposition 3 from [Ber96, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' VI] ensures that the pairs of variables (Xt −Xt,−Xt) and (Xt,Xt −Xt) have the same distribution under P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Consequently P(Xt = Xt) = P((Xt −Xt,−Xt) ∈ {0}×[0,∞)) = P((Xt,Xt −Xt) ∈ {0}×[0,∞)) ≤ P(Xt = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' The unbounded variation of X guarantees that 0 is regular for (−∞,0) and for (0,∞) (as mentioned, this result can be found in [Rog68] and has been extended in [AHUB20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Hence, for any t > 0, Xt > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' We decude that P(Xt = 0) = 1−P(Xs > 0 for some s ≤ t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Thus, E �� ∞ 0 I(Rt = 0)dt � = � ∞ 0 P(Xt = Xt)dt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Therefore, we can apply Proposition 3 to deduce the existence of solutions to Equation (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' □ Let us now pass to the proof of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' As we have stated in Theorem 2, we allow the convergence Xn → X to be weak or almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Using Skorohod’s representation Theorem, we may assume that it is satisfied almost surely in some suitable probability space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' The desired result follows immediately from Proposition 4 by considering the paths of f = z+X −γ Id and f n = zn +Xn −γ⌊n·⌋/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Measurability details and the strong Markov property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' In order to complete the proof of The- orem 1, it remains to verify the adaptability of the unique solution to the TCE (2) to the time changed filtration ( � Ft,t ≥ 0) and that such a solution is, in fact, a sticky L´evy process based on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' This is the objective of the current section, which ends the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' By construction the mapping t �→ Ct is continuous and strictly increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Furthermore, given that C is the inverse of the map t �→ t +Lt/γ, we can write {Ct ≤ s} = {γ(t −s) ≤ Ls} ∈ Fs, for every t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' In other words, the random time Ct is a (Fs)-stopping time, since the filtration is right-continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Therefore the process C is a (Fs)-time change and Z is adapted to the time-changed filtration ( � Ft,t ≥ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' In this sense we say that Z exhibits no extra randomness to that of the original L´evy process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' This contrasts with the SDE describing sticky Brownian motion (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' [War97, Theorem 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Let us verify that the unique solution Z to (2) is an extension of the killed process X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' By construc- tion, we see that if Z0 = z > 0, then Z equals X until they both reach zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Hence Z and X have the same law if killed upon reaching zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Let now Z be the unique solution of (2) with Z0 = z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' The concrete construction which proves existence to (2) of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content='2 shows that γ � t 0 I(Zs = 0)ds = L◦C where Ct = � t 0 I(Zs > 0)ds, Lt = −infs≤t Xs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' We have already argued that the unbounded variation hypothesis implies that Lt > 0 for any t > 0 and therefore L∞ > 0 almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' As above, recalling that C is the inverse of Id+L/γ, we see that C∞ = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' We conclude that L◦C∞ > 0 almost surely, so that Z spends positive time at zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' We will now use the unbounded variation of X to guarantee the regular and instantaneous character of 0 for Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' By construction, the unique solution Z to the TCE (2) is the process X reflected at its infimum by applying a continuous strictly increasing time change C to it, that is Z = R◦C where R = X −X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Consequently P(inf{s > 0 : Zs = 0} = 0) = P(inf{s > 0 : X ◦Cs = X ◦Cs} = 0) = P(inf{s > 0 : Xs = Xs} = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' The Sticky L´evy Process as a solution to a Time Change Equation 10 Since 0 is regular for (−∞,0) thanks to the unbounded variation hypothesis (meaning that X visits (−∞,0) immediatly upon reaching 0), we conclude the regularity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Similarly, given the regularity of 0 for (0,∞) for X, we have P(inf{s > 0 : Zs > 0} = 0) = P(inf{s > 0 : Xs > Xs} = 0) ≥ P(inf{s > 0 : Xs > 0} = 0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Thus, 0 is an instantaneous point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' To conclude the proof of Theorem 1, it now remains to prove the strong Markov property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' From the construction of the unique solution to the TCE (2), we deduce the existence of a measurable mapping Fs that maps the paths of the L´evy process X and the initial condition z to the unique solution to the TCE (2) evaluated at time s, that is, Zs = Fs(X,z) for s ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Let T be a ( � Ft)-stopping time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Approximating T by a decreasing sequence of ( � Ft)-stopping times (T n,n ≥ 1) taking only finitely many values, we see that CT is an (Ft)-stopping time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' From the TCE (2), we deduce that ZT+s = ZT +(XC(T+s) −XC(T))+γ � s 0 I(ZT+r = 0)dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Consider the processes ˜C, ˜X and ˜Z given by ˜Cs =C(T +s)−C(T), ˜Xs = XC(T)+s −XC(T) and ˜Zs = ZT+s respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' We can write the last equation as (13) ˜Zs = ZT + ˜X ˜C(s) +γ � s 0 I( ˜Zr = 0)dr, and ˜C satisfies ˜Cs = � s 0 I( ˜Zr > 0)dr for s ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' In other words, ˜Z is solution to the TCE (2) driven by ˜X with initial condition ZT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Consequently ˜Zs = Fs( ˜X,ZT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Note that ˜X has the same distribution as X and it is independent of � FT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Hence, the conditional law of ˜Z given � FT is that of F(·,ZT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' (One could make appeal to Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content='7 in [Kal21, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' 169] if needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=') This allows us to conclude that Z is a strong Markov process and concludes the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Stickiness and martingales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' In this section we aim at describing the boundary condition of the infinitesimal generator of the sticky L´evy process Z of Theorem 1 by proving the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Let X be a L´evy process of unbounded variation and no negative jumps and let L be its infinitesimal generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' For a given z ≥ 0, let Z be the unique (strong Markov) process satisfying the time-change equation (2): Zt = z+X� t 0 I(Zs>0)ds +γ � t 0 I(Zs = 0)ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Then, for every f : [0,∞) → R which is of class C2,b and which satisfies the boundary condition γ f ′(0+) = L f(0+), the process M defined by Mt = f(Zt)− � t 0 L f(Zs)ds is a martingale and ∂ ∂t ���� t=0 E( f(Zt)) = L f(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Theorem 2 from [RUB22] describes the domain of the infinitesimal generator of any recurrent exten- sion of X0 (which is proved to be a Feller process) by means of three non-negative constants pc, pd, pκ and a measure µ on (0,∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' To describe such parameters we note a couple of important facts about the unique solution to (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' By construction we can see that it leaves 0 continuously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Indeed, if we consider the left endpoint g of some excursion interval of Z, then Cg is the left endpoint of some excursion inter- val of the process reflected at its infimum R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Thanks to Proposition 2 from [RUB22], such excursions start at 0, so Z leaves 0 continuously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Thus, from [RUB22], pc > 0 and µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Note also that Z has infinite lifetime because R has it and C is bounded by the identity function, so pκ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Finally, since Z The Sticky L´evy Process as a solution to a Time Change Equation 11 spends positive time at 0, then pd > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Theorem 2 from [RUB22] ensures that every function f in the domain of the infinitesimal generator of Z satisfies f ′(0+) = pd pc L f(0+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Our proof of Proposition 5 does not require the results from [RUB22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' The main intention is to give an application of stochastic calculus, since we recall that a classical computation of the infinitesimal generator for L´evy processes is based on Fourier analysis (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' [Ber96]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Regarding the generator L , recall that it can be applied to C2,b functions such as f and that L f is continuous (an explicit expression is forthcoming).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' The lack of negative jumps implies that L f is defined even if f is only defined and C2,b on an open set containing [0,∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Let Z be the unique solution to the TCE (2) driven by the SPLP X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Itˆo’s formula for semimartingales [Pro04, Chapter II, Theorem 32] guarantees that for every function f ∈ C2 0[0,∞): f(Zt) = f(z)+ � t 0 f ′(Z− s )dXCs + � t 0 γ f ′(Z− s )I(Z− s = 0)ds+ 1 2 � t 0 f ′′(Z− s )d[Z,Z]c s +∑ s≤t (∆ f(Zs)− f ′(Z− s )∆Zs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' (14) In order to analyze this expression, we recall the so-called L´evy-Itˆo decomposition, which describes the structure of any L´evy process in terms of three independent auxiliary L´evy processes, each with a different type of path behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Consider the Poisson point process N of the jumps of X given by Nt = ∑ s≤t δ(s,∆Xs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Denote by ν the characteristic measure of N, which is called the L´evy measure of X and fulfills the integrability condition � (0,∞)(1 ∧ x2)ν(dx) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Then, we write the L´evy-Itˆo decomposition as X = X(1) + X(2) + X(3), where X(1) = bt + σBt is a Brownian motion independent of N, with diffusion coefficient σ 2 ≥ 0 and drift b = E[X1 − � (0,1] � [1,∞) xN(ds,dx)], X(2) = � (0,t] � [1,∞) xN(ds,dx) is a compound Poisson process consisting of the sum of the large jumps of X and finally X(3) = � (0,t] � (0,1) x(N(ds,dx)−ν(dx)ds) is a square-integrable martingale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Assuming the L´evy-Itˆo decomposition of X and using the next result, whose proof is postponed, we will see that � t 0 f ′(Z− s )dXCs is a semimartingale of the form (15) Mt + � t 0 bf ′(Z− s )(1−I(Zs = 0))ds+ � t 0 f ′(Z− s )dX(2) Cs , for some square-integrable martingale M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Let C be a (Ft)-time change whose paths are continuous and locally bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Let X be a right-continuous local martingale with respect to (Ft,t ≥ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Then the time-changed process XC is a right-continuous local martingale with respect to the time-changed filtration ( � Ft,t ≥ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Lemma 2 ensures that the time-changed process (σB + X(3)) ◦C remains a local martingale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Ac- cording to Theorem 20 from [Pro04, Chapter II], square-integrable local martingales are preserved The Sticky L´evy Process as a solution to a Time Change Equation 12 under stochastic integration provided that the integrand process is adapted and has c`adl`ag paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Con- sequently the stochastic integral1 M = f ′(Z−) · (σBC + X(3) C ) is a ( � Ft)-local martingale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Thanks to Corollary 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content='3 from [Pro04, Chapter II], we know that a necessary and sufficient condition for a local martingale to be a square-integrable martingale is that its quadratic variation is integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Let us verify that E[[M,M]t] < ∞ for every t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content='17 from [Jac79] implies the quadratic variation of the time-changed process coincides with the time change of the quadratic variation � σBC +X(3) C ,σBC +X(3) C � t = � σB+X(3),σB+X(3)� Ct , t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Given that the Brownian motion B is independent of X(3), the quadratic variation is σ 2Ct + � X(3),X(3)� Ct, which is bounded by σ 2t + � X(3),X(3)� t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Thus E[[M,M]t] ≤ ∥f ′∥2 ∞E �� σB+X(3),σB+X(3)� Ct � ≤ ∥f ′∥2 ∞ � σ 2t +t � (−1,1) x2 ν(dx) � < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' This verifies the decomposition (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Later we will deal with the last term of this decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Coming back to Itˆo’s formula (14), we need to calculate the term corresponding to the integral with respect to the continuous part of the quadratic variation of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' First, we decompose the variation as [Z,Z]s = [XC,XC]s +2[XC,γ(Id−C)]s +γ2[Id−C,Id−C]s, for every s ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' The first term is [X,X]Cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Given the finite variation of γ(Id−C) and the continuity of C, Theorem 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content='6 from [Kal02] implies that almost surely the other two terms are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Thereby [Z,Z]s = [X,X]Cs for every s ≥ 0 and 1 2 � t 0 f ′′(Z− s )d[Z,Z]c s = 1 2 � t 0 σ 2 f ′′(Z− s )(1−I(Zs = 0))ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Now we analyze the last term on the right-hand side from (14), which corresponds to the jump part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Let us note that the discontinuities of f ◦Z derive from the discontinuities of Z, which are caused by the jumps of X ◦C, in other words {s ≤ t : |∆ f(Zs)| > 0}⊆{s ≤ t : ∆Zs > 0} = {s ≤ t : ∆(X ◦C)s > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Making the change of variable r = Cs, the sum of the jumps in (14) can be written as (16) ∑ r≤Ct (∆ f(Z ◦Ar)− f ′(Z− ◦Ar)∆(Z ◦Ar)), where A denotes the inverse of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' We claim that A is a ( � Ft)-time change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Indeed, splitting in the cases r < t and r ≥ t, we see that {At ≤ s}∩{Cs ≤ r} = {t ≤Cs ≤ r} ∈ Fr for any r ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Exercise 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content='12 from [RY99, Chapter V] ensures that the time-changed filtration ( � FAt,t ≥ 0) is in fact (Ft,t ≥ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Thus, for any continuous function g, the process (g(Z− At),t ≥ 0) is (Ft)-predictable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' We return to (15) to put together the sum of the jumps in (16) and the stochastic integral ( f ′ ◦Z−)· (X(2) ◦C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' For this purpose, it is convenient to rewrite the last integral as ( f ′ ◦Z− ◦A ◦C) · (X(2) ◦C) and apply Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content='18 from [Jac79] to deduce that ( f ′ ◦Z−) · (X(2) ◦C) = (( f ′ ◦Z− ◦A) · X(2)) ◦C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' 1We use both notations � Hs dXs and H ·X to refer to the stochastic integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' The Sticky L´evy Process as a solution to a Time Change Equation 13 Consequently � t 0 f ′(Z− s )dX(2) Cs + ∑ s≤Ct (∆ f(Z ◦As)− f ′(Z− ◦As)∆(Z ◦As)) = � Ct 0 � (0,∞) � f(Z− As +x)− f(Z− As)− f ′(Z− As)xI(x ∈ (0,1)) � (N(ds,dx)−ν(dx)ds) + � Ct 0 � (0,∞) � f(Z− As +x)− f(Z− As)− f ′(Z− As)xI(x ∈ (0,1)) � ν(dx)ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' (17) Define the process M by Mt =− � t 0 � [1,∞) � f(Z− As +x)− f(Z− As) � ν(dx)ds + � t 0 � [1,∞) � f(Z− As +x)− f(Z− As) � N(ds,dx) + � t 0 � (0,1) � f(Z− As +x)− f(Z− As)− f ′(Z− As)x � (N(ds,dx)− ds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Since ν is a L´evy measure, then E �� t 0 � [1,∞) �� f(Z− As +x)− f(Z− As) �� ds � ≤ ∥f∥2 ∞tν([1,∞)) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' We develop the first degree Taylor polynomial of f(Z− As +x) to obtain f ′(Z− As)x = f(Z− As +x)− f(Z− As)−R(x), x ∈ (0,1), where the remainder R satisfies |R(x)| ≤ 1 2∥f ′′∥∞x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Therefore E �� t 0 � (0,1) � f(Z− As +x)− f(Z− As)− f ′(Z− As)x � ν(dx)ds � ≤ 1 2∥f ′′∥∞tE �� (0,1) x2 ν(dx) � < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content='1 from [App09] ensures that M is a (Ft)-local martingale and Lemma 2 implies that MC is a ( � Ft)-local martingale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Furthermore, for t ≥ 0 it holds that E � sup s≤t |MCs| � ≤ E � sup s≤t |Ms| � ≤ � 2∥f∥∞ + 1 2∥f ′′∥2 ∞ � t � (0,∞)(1∧x2)ν(dx) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' It follows from Theorem 51 from [Pro04, Chapter I] that MC is a true martingale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Gathering all the expressions involved in Itˆo’s formula (14), we get the semimartingale decomposi- tion f(Zt)− f(z) =Mt + � t 0 bf ′(Z− s )(1−I(Zs = 0))ds+ � t 0 γ f ′(0+)I(Zs = 0)ds + 1 2 � t 0 σ 2 f ′′(Z− s )(1−I(Zs = 0))ds+MCt + � Ct 0 � (0,∞) � f(Z− As +x)− f(Z− As)− f ′(Z− As)xI(x ∈ (0,1)) � ν(dx)ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Recall that the extended generator of X (as in [RY99, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' VII]) is given by L f(z) = bf ′(z)+ σ 2 2 f ′′(z)+ � R+ � f(z+x)− f(z)− f ′(z)xI(x ∈ (0,1)) � ν(dx) The Sticky L´evy Process as a solution to a Time Change Equation 14 on C2,b functions and that the extended generator of X0 is given by L f on C2,b functions f on [0,∞) which vanish (together with its derivatives) at 0 and ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Note that L f(z) is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Define ˜ L f(0) by ˜ L f(0) = (b−γ) f ′(0+)+ σ 2 2 f ′′(0+)+ � R+ � f(x)− f(0+)− f ′(0+)xI(x ∈ (0,1)) � ν(dx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Given that ˜ L f(0) = L f(0+)−γ f ′(0+), we can write the martingale M +MC as M +MCt = f(Zt)− f(z)− � t 0 L f(Z− s )ds+ � t 0 ˜ L f(0)I(Zs = 0)ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' We deduce that if a function f ∈ C2[0,∞) satisfies the boundary condition ˜ L f(0) = 0 or equivalently γ f ′(0+) = L f(0+), then f(Zt) − f(z) − � t 0 L f(Zs)ds is a martingale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' By hypothesis, the last term is bounded by a linear function of t, so that E[f(Zt)] is differentiable at zero and the derivative equals L f(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' □ We conclude this section with the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' (Lemma 2) Let (βn,n ≥ 1) be localizing sequence for X, then βn → ∞ as n → ∞ and for each n ≥ 1, the process XβnI(βn > 0) is a uniformly integrable martingale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Keeping the notation A for the inverse of C, we will prove that (A(βn),n ≥ 1) is a sequence of ( � Ft)-stopping times that localizes to XC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' The property of being (Ft)-stopping time is deduced by observing that {βn ≤ Ct} ∈ Fβn ∩FCt ⊂ � Ft, which implies that {A(βn) ≤ t}∩{Ct ≤ s} = {βn ≤ Ct}∩{Ct ≤ s} ∈ Fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Since C ◦A = Id, then (Z ◦C)A(βn) t = ZCt∧βn = Zβn Ct .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Given that Zβn is a (Ft)-martingale, Optional Stopping Theorem guarantees that E � Zβn Ct ���FCs � = Zβn Cs , 0 ≤ s ≤ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Hence (Z ◦C)A(βn) is a ( � Ft)-martingale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Moreover A(βn) → ∞ as n → ∞ since C ≤ Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' □ REFERENCES [AHUB20] Osvaldo Angtuncio Hern´andez and Ger´onimo Uribe Bravo, Dini derivatives and regularity for exchangeable increment processes, Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' B 7 (2020), 24–45.' metadata={'source': 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+page_content=' i Primenen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' 13 (1968), 507–512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' MR 0242261 ↑2, ↑9 [RS15] Mikl´os Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' R´acz and Mykhaylo Shkolnikov, Multidimensional sticky Brownian motions as limits of exclusion processes, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} 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Research, Springer-Verlag, New York, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' ↑7 [Yam94] Keigo Yamada, Reflecting or sticky Markov processes with L´evy generators as the limit of storage processes, Stochastic Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' 52 (1994), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' 1, 135–164.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAyT4oBgHgl3EQfRPa5/content/2301.00063v1.pdf'} +page_content=' ↑3' metadata={'source': 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of input DNA. Standard mathematical solutions to this back-calculation prob- +lem do not take adequate account of such noise and are error-prone. Here, we develop a +parsimonious mathematical model of the stochastic mapping of input DNA onto experi- +mental outputs that accounts, in a natural way, for amplification noise. We use the model +to derive the probability density of the quantification cycle, a frequently reported exper- +imental output, which can be fit to data to estimate input DNA. Strikingly, the model +predicts that a sample with only one input DNA molecule has a <4% chance of testing +positive, which is >25-fold lower than assumed by a standard method of interpreting PCR +data. We provide formulae for calculating both the limit of detection and the limit of quan- +tification, two important operating characteristics of DNA quantification methods that are +frequently assessed by using ad-hoc mathematical techniques. Our results provide a math- +ematical foundation for the rigorous analysis of DNA quantification. +2 +Introduction +The quantification of genomic targets is of interest in a large variety of applications in +biology, biotechnology and medicine, from determining an individual’s disease status to +detecting minute changes in gene expression profiles occurring across space and time (eg. +[1, 2]). This is typically achieved by converting non-DNA genomic targets into DNA, which +is then amplified to enable its quantification. In principle, this allows even small numbers +∗Address for correspondence: wndifon@aims.ac.za +1 +arXiv:2301.02149v1 [q-bio.QM] 5 Jan 2023 + +of genomic targets to be accurately measured. However, in practice, the DNA amplifica- +tion process, being stochastic, generates outputs that contain noise. Accurate measure- +ment, therefore, requires an adequate, quantitative understanding of this noise. Thus far, +this has proved challenging to achieve. +A specific and very popular instance of a DNA quantification method is the real-time +polymerase chain reaction (PCR) [3, 4]. In PCR, DNA molecules are repeatedly amplified +in a cyclic manner. As they are amplified, fluorescently labeled nucleotides are incorpo- +rated into the newly formed DNA molecules, increasing the overall fluorescence emitted. +The resulting fluorescence profile is used to determine the quantification cycle (denoted +Cq or Ct value), at which the number of molecules exceeds a defined threshold, called +the quantification threshold. A PCR reaction is considered to be positive if its Ct value +is less than or equal to the maximum possible cycle. Despite the fact that the Ct value is +only an indirect readout of the number of input DNA molecules, it is often the only re- +ported output of PCR experiments. A variant of conventional PCR, called digital PCR [5], +uses the fraction of positive reactions to estimate the number of input DNA molecules. To +this end, it assumes that a reaction is positive if and only if it contains at least one target +molecule. It is unclear under what conditions this assumption is valid, and when it must +be discarded in favor of a more realistic alternative. +Here we describe a parsimonious mathematical model that is useful for analysing the +DNA quantification process, and for guiding the interpretation of experimental outputs. +We use PCR as an example, although our analysis is applicable to other methods such as +loop-mediated isothermal amplification of DNA [6]. Experiments indicate that the PCR +process exhibits different phases, characterized by different efficiencies of DNA amplifi- +cation. Therefore, we construct a mathematical model of a PCR process with an arbitrary +number of phases, each with its own amplification efficiency. We use this model to obtain +the following results: +• We derive the generating function for the probability distribution of the number of +molecules found in a PCR experiment at an arbitrary time t. We also derive the +probability density function (pdf), mean, variance, and cumulative density function +(cdf) of the Ct values produced by such an experiment. Either the pdf or the cdf can +be fit to PCR data to estimate the number of input DNA molecules. +• In the simplest instance of our model – a single-phase PCR model that accounts for +amplification noise but not for (upstream) DNA sampling noise – the mean Ct value, +given by (ψ(x +1)−ψ(n))/r, is well approximated by ln(x/n)/r [7] when n ≫ 1, where +n is the number of input molecules, r (defined on a base-e scale) is the amplification +efficiency, x is the quantification threshold, and ψ(·) denotes the digamma function. +• We provide a formula for calculating the limit of detection (LoD) of a PCR experi- +2 + +ment, that is, the smallest number of input molecules that can be detected with a +failure rate not exceeding α. Using a single-phase PCR model, we find that when +α = 0.05, the LoD increases from 3, the value determined while accounting for sam- +pling noise only, to ≈10 when both sampling noise and amplification noise (with r +set to 95% of the maximum possible efficiency, m.p.e.) are considered. The LoD +increases as r decreases, doubling to ≈20 at 90% m.p.e. This illustrates the under- +appreciated, dramatic effect that amplification efficiency has on the LoD. +• We provide a formula for calculating the limit of quantification (LoQ) of a PCR ex- +periment, that is, the smallest number of molecules that can be quantified with a +defined level of precision and a given maximum failure rate α. Counter-intuitively, +the single-phase PCR model predicts that the LoQ does not depend on amplifica- +tion efficiency. When α = 0.05, the LoQ increases from 10, obtained when up to a +two-fold deviation from the expected number of input molecules is allowed, to 820, +when at most a 10% deviation is allowed. This indicates that 10 or fewer molecules +cannot be measured with a better than 2-fold error more than 95% of the time. +• The model indicates that a key assumption commonly used when interpreting digital +PCR data – that a PCR experiment with only one input molecule will always produce +a positive outcome – is invalid under a wide range of conditions. Even when the +amplification efficiency is set to a high value of 95% m.p.e, the probability that such +an experiment will yield a positive outcome is predicted to be <4%. We describe +two different approaches by which accurate estimates of the number of input DNA +molecules may be obtained from digital PCR data. +It should be noted that there have been previous attempts to improve the interpretation +of PCR data through mathematical modeling. The classical approach to estimating the +amount of DNA found in a focal sample involves comparing data generated by that sample +versus data obtained from a reference sample containing either a known or an unknown +amount of DNA [8]. The need for a reference sample with a known amount of DNA, the +determination of which is itself subject to experimental error, makes accurate absolute +quantification of DNA found in the focal sample challenging. An alternative approach +involves fitting mathematical models, mostly phenomenological in their construction, to +PCR data generated by the focal sample alone [4, 8, 9, 10, 11]. See [12] for a comparison +of various methods based on this approach. None of these methods provides an adequate +accounting of how amplification noise shapes PCR data. +The remainder of this paper is organized as follows: We provide an overview of the +model’s structure in Section 3.1 and present our main mathematical results in Sections +3.2 and 3.3. We apply these results to compute the LoD and LoQ in Section 3.4, and +we investigate how amplification noise complicates the accurate interpretation of digital +3 + +PCR data in Section 3.5. We summarize the results and discuss other applications of our +methods in Section 4. To improve readability, we only present mathematical proofs and +detailed calculations in the Appendix (Section 5.1). +3 +Results +3.1 +Preliminaries +We model the PCR process as a continuous-time, discrete-state Markov jump process [13] +evolving up to time T . This representation of the PCR process is based on the facts that +(1) the primary products of PCR reactions, DNA molecules, are countable, and (2) what +happens in the next cycle of the reaction is conditionally independent of what happened +in the past given the present state of the reaction. Our decision to make time continuous +(rather than discrete) is based on the fact that experimentally measured Ct values are +positive real numbers. As a consequence, reaction rates are defined in base e instead of +base 2 (expected for a discrete-time PCR process), but it is straightforward to convert +between these two bases. +We divide the time interval [0,T ] of the PCR process into p non-overlapping subin- +tervals Ii, each one corresponding to a distinct phase of the process and associated with +the probabilistic state transition rate ri, i = 1,2,...,p. These transition rates govern the ef- +ficiency of DNA amplification. We derive the probability generating function [14] for the +number of target molecules found at an arbitrary time t. We use this generating func- +tion to derive the corresponding probability distribution and, importantly, the probability +density function (pdf) of the Ct value. We derive the pdf in two different cases, namely +1. when the initial state of the PCR process is deterministic, and the PCR phase lengths +and amplification efficiencies are given; and +2. when the initial state is Poisson-distributed, and the phase lengths and amplification +efficiencies are given. +To illustrate the mathematical ideas, we will report calculations and simulations based +on a single-phase model. We argue that this simpler instance of our model is sufficient +for analysing a large variety of real-world PCR experiments. In principle, each PCR ex- +periment can be divided into the following three amplification rate-dependent phases: a +pre-exponential phase, in which the amplification rate is sub-exponential; an exponential +phase; and a post-exponential phase where the rate slows down as DNA molecules saturate +the reagents required for their further amplification. However, in practice, the usual out- +put of PCR experiments – the Ct value – is determined as soon as the PCR process enters +the exponential phase, meaning that dynamics occurring in the pre-exponential phase pri- +marily determine this particular outcome. Therefore, for the purposes of understanding +4 + +the factors that shape the Ct value and its statistics, and evaluating related operating char- +acteristics of PCR, a single-phase model appears sufficient. Accordingly, when applicable, +we highlight the forms taken by our mathematical equations in the case of a single-phase +model. In addition, we estimate the LoD and LoQ using a single-phase model (Section +3.4), which we also apply to critique the standard method of interpreting digital PCR data +(Section 3.5). +3.2 +Case 1: A PCR process with a deterministic initial state +3.2.1 +Probability generating function for the number of molecules +Theorem 1. Let {X(t),t ∈ R} be a continuous-time Markov process with p phases, a countable +state space S ⊂ N+, phase-specific transition rates ri, i ∈ 1,2,...,p, and state transition proba- +bility given by +P (X(t′ + ∆t) = x|X(t′) = x′) = δ(x′ − x + 1) +p +� +i=1 +ri1Ii(t′), +(1) +where 1 denotes the indicator function and δ(.) denotes the Kronecker delta function. If the pro- +cess starts with n molecules, then the probability generating function for the number of molecules +present at time t ∈ Ik, k ≤ p, is given by +G(n,⃗r,t,⃗τ;s) = +� +se−z +1 − s(1 − e−z) +�n +, +(2) +where +z = rkt + +k−1 +� +i=1 +(ri − rk)τi, +(3) +Ii denotes the i’th phase and τi = |Ii|, i < k, is its length. +The proof of this theorem is given in Section 5.1.1. We will now use the theorem to +derive the probability distribution of the number of molecules found at time t. +3.2.2 +Probability distribution of the number of molecules +Corollary 1. The probability that there are x molecules at time t ∈ Ik in the PCR process de- +scribed in Theorem 1 is given by the following negative binomial distribution: +P(x|n,⃗r,t,⃗τ) = +�x − 1 +n − 1 +� +e−nz × (1 − e−z)x−n , +(4) +where z is given by (3). +The proof of this corollary is given in Section 5.1.2. We will now use this corollary to +derive the pdf, mean, variance and cdf of the Ct value. +5 + +3.2.3 +pdf, mean and variance of the Ct value +Let t be the Ct value of the PCR process described in Theorem 1. By definition, t is the time +at which the number of molecules reaches the quantification threshold, which we denote +by x. Let t ∈ Ik. In the Appendix [Section 5.1.5], we show that, given n,⃗r = (r1,r2,...,rk−1), +and ⃗τ = (τ1,τ2,...,τk−1), the pdf of t has the following form: +P(t|n,⃗r,⃗τ,x) += +rke−nz (1 − e−z)x−n +Bθ(n,x − n + 1) , +(5) +where Bθ(n,x − n + 1) is the incomplete Beta function, z is given by (3), and +θ = e−�k−1 +i=1 riτi. +(6) +For the single-phase PCR process, θ = 1, so the pdf is given by +P(t|n,r1,x) += +r1e−nz (1 − e−z)x−n +B(n,x − n + 1) +. +(7) +The mean Ct value is given by (see Section 5.1.5) +E(t) += +k−1 +� +i=1 +τi + Γ(n)2θn 3 ˜F2(n,n,n − x;n + 1,n + 1;θ) +rkBθ(n,x − n + 1) +, +(8) +where 3 ˜F2(n,n,n − x;n + 1,n + 1;θ) is the regularized generalized hypergeometric function. +For the single-phase process, the mean is given by +E(t) += +ψ(x + 1) − ψ(n) +r1 +. +(9) +Observe that when n ≫ 1, the right-hand-side of (9) is well-approximated by ln(x/n)/r1. +The latter expression is commonly used to approximate the mean Ct value. For example, +it was used in [7] to estimate PCR amplification efficiency from data. +The variance of the Ct value is given by E(t2) − E(t)2, where E(t2) is given by (85). For +the single-phase process, the variance is given by +Var(t) = ψ1(n) − ψ1(x + 1) +r2 +1 +, +(10) +where ψ1(·) is the second polygamma function (also called the trigamma function). +Finally, the cdf of the Ct value is given by [see Section 5.1.5] +F(t|n,⃗r,⃗τ,x) += +1 − Be−z(n,x − n + 1) +Bθ(n,x − n + 1) . +(11) +6 + +For the single-phase process, the cdf is given by +F(t|n,r1,x) += +1 − Ie−r1t(n,x − n + 1), +(12) +where Ie−r1t(n,x − n + 1) = Be−rt (n,x − n + 1)/B(n,x − n + 1) is the regularized incomplete Beta +function. Sampling from this cdf is relatively straightforward: A random Ct value t is +obtained as follows: +t += +−lnI−1 +1−u(n,x − n + 1) +r1 +, +(13) +where u is sampled uniformly at random from the interval (0,1) and I−1 +1−u is the inverse of +the regularized incomplete Beta function. To find a Ct value that corresponds to a quantile +q ∈ (0,1), q is substituted for u. +To estimate n, either the pdf or the cdf of t can be fit to data. Alternatively, the posterior +density of n conditioned on t can be computed. In Section 5.1.5, we show that, for the +single-phase process, it is given by +P (n|r1,t,x) += +e−(n−1)r1t(1 − e−r1t)x−n +xB(n,x − n + 1) +. +(14) +In Figure 1, we illustrate the shape of the single-phase pdf for different numbers of +input molecules and different amplification efficiencies. To this end, we set T = 35 (a com- +mon upper-bound for the duration of real-world PCR experiments) and x = 2T , which is +equal to the number of molecules expected after T cycles under perfect amplification con- +ditions (a sample that contains only one, perfectly amplified input molecule is expected +to reach the quantification threshold, x, at time t ≤ T). We vary the efficiency from 60% +m.p.e. (equivalent to setting r1 = 0.6 × ln2) to 100% m.p.e. The pdf has a bell shape, the +location and width of which are governed by both the efficiency and the number of input +molecules (Figure 1, left panel). Higher efficiencies or larger numbers of input molecules +produce smaller mean Ct values, smaller variances, and narrower pdfs. In contrast, lower +efficiencies or smaller numbers of input molecules produce larger mean Ct values, larger +variances, and wider pdfs (Figure 1, left panel). In fact, Equation (5) predicts that in the +limit as the efficiency goes to 0, the pdf will become flat as it will map every Ct value to 0. +3.3 +Case 2: A PCR process with a Poisson-distributed initial state +3.3.1 +Probability generating function for the number of molecules +Theorem 2. Let {X(t),t ∈ R} be the continuous-time Markov process described in Theorem 1. +If, instead of starting with a precisely known number of input DNA molecules, the initial state +of the process is Poisson-distributed with mean λ, then the probability generating function for +7 + +Figure 1: pdf of the quantification cycle for the single-phase process. Processes with +either a deterministic (left panel) or a Poisson-distributed (right panel) initial state were +considered. The pdf was calculated using Equation (5) for the former case, and Equation +(18) for the latter case. The quantification threshold, x, was set to 235. The mean µ and +variance σ2 corresponding to different amplification efficiences r are shown. For ease of +comprehension, r, which in our model is defined on a base-e scale, is shown as a percentage +of its maximum possible value of ln(2). +8 + +n =1 +入 =1 +r=1.0,μ=35.83,α2=3.424 +r=1.0,μ=35.12,o2=3.257 +0.0010- +r=0.9,μ=39.81,α2=4.227 +r=0.9,μ=39.02,2=4.021 +0.2- +=0.8,μ=44.79,2=5.35 +r=0.8,μ=43.9,g2=5.089 +0.7,μ=51.19,02=6.987 +r=0.7,50.17,o2=6.647 +0.6,μ59/72,2=9.51 +=0.6，=58.54,2=9.048 +0.0005- +0.1- +0.0000 +0.0 +30 +40 +50 +60 +70 +30 +40 +50 +60 +70 +n=10 +入=10 +0.6- +0.004 +r=1.0,μ=31.75,2=0.219 +r=1.0,μ=31.84,2=0.54 +Probability density +r=0.9.μ=35.28,g2=0.27 +0.5- +r=0.9,μ=35.38,o2=0.666 +0.003- +r=0.8,μ=39.69,g2=0.342 +r=0.8,μ=39.8,α2=0.843 +0.4- +r=0.7，μ=45.36,?元0.447 +r=6|7,μ=45.49.@2=1.101 +0.002 +r=06/μ=52.92,=0.608 +0.3- +r=0.6，μ=53.07/=1.499 +0.2- +0.001 +0.1- +0.000 +0.0- +30 +40 +50 +30 +40 +50 +n=100 +入=100 +2.0- +r=1.0,μ=28.36,2=0.021 +r=1.0,μ=28.37o2=0.042 +r=0.9,μ=31.51,2=0.026 +r=0.9,μ=31.52,2=0.052 +1.5 +0.010 +r=0.8,μ=35.45,2=0.033 +r=0.8,μ=35.46,o2=0.066 +r=0.7 +μ=40.52,2=0.@43 +r=0.7μ=40.53,2=0.087 +r=0.6. +μ=47.27, 2=0.958 +1.0- +r=0.6,μ=47.28,o2=q.118 +0.005 +0.5- +0.000- +0.0 +30 +35 +40 +45 +30 +35 +40 +45 +Cycles +Cyclesthe state of the process at a future time t ∈ Ik is given by +G(λ,⃗r,t,⃗τ;s) = e +λ(s−1) +1−s(1−e−z) , +(15) +where z is given by (3). +The proof of Theorem 2 is given in Section 5.1.3. We now use Theorem 2 to derive the +probability distribution of the number of molecules found in the PCR process at time t. +3.3.2 +Probability distribution of the number of molecules +Corollary 2. The probability that there are x molecules at cycle t ∈ Ik in the PCR process +described in Theorem 2 is given by: +P(x|λ,⃗r,t,⃗τ) = e−λ (1 − e−z)x +x +� +i=1 +�x−1 +i−1 +� +i! +� λe−z +1 − e−z +�i +, +(16) +where z is given by (3). +The proof of this corollary is provided in Section 5.1.4. It is interesting to note that +from the proof emerged the following combinatorial triangle, which is related to the well- +known Narayana triangle [15]: +x +1 +1 +2 +1 +2 +3 +1 +6 +6 +4 +1 +12 +36 +24 +5 +1 +20 +120 +240 +120 +1 +2 +3 +4 +5 +k +. +The entries of this triangle, given by +T(x,k) = +� x +k − 1 +��x − 1 +k − 1 +� +(k − 1)!, x ∈ Z+,k = 1,2,...,x, +(17) +count the number of ways of obtaining x molecules by replicating a randomly selected +subset of k molecules. T (x,k) is related to the Narayana numbers N(x,k) by +T (x,k) = k! N(x,k). +We will now use this corollary to derive the pdf of the Ct value together with the mean, +variance and cdf. +9 + +3.3.3 +pdf, mean and variance of the Ct value +Let t be the Ct value of the PCR process described in Theorem 2. As noted earlier, the Ct +value t is the time at which the number of DNA molecules found in the process reaches +the quantification threshold, which we denote by x. In the Appendix [Section 5.1.6], we +show that the pdf of t is given by +P(t|λ,⃗r,⃗τ,x) += +rkλe−z(1 − e−z)x−1 1F1 +� +1 − x;2; −λe−z +1−e−z +� +�x +j=1 +(x−1 +j−1)λj +j! +Bθ(j,x − j + 1) +, +(18) +where θ is given by (6), z is given by (3), 1F1(a;b;c) is the hypergeometric function (also +called the Kummer confluent hypergeometric function of the first kind). +For the single-phase process, the pdf is given by [see Section 5.1.6] +P(t|λ,r1,x) = +r1xλe−r1t(1 − e−r1t)x−1 1F1 +� +1 − x;2; −λe−r1t +1−e−r1t +� +eλ − 1 +. +(19) +The mean Ct value is given by [see Section 5.1.6] +E(t) += +�x +j=1 +(x−1 +j−1)λj +j! +� +rkBθ(j,x − j + 1)�k−1 +i=1 τi + Γ(j)2θj 3 ˜F2(j,j,j − x;j + 1,j + 1;θ) +� +rk +�x +j=1 +(x−1 +j−1)λj +j! +Bθ(j,x − j + 1) +, (20) +while the second moment is given by (119). +For the single-phase process, the mean and variance are, respectively, given by [see +Section 5.1.6] +E(t) += +ψ(x + 1) +r1 +− +�x +j=1 +λj +j! ψ(j) +r1 +� +eλ − 1 +� +and +(21) +Var(t) = +� +eλ − 1 +��x +j=1 +λj +j! +� +ψ1(j) + ψ(j)2 +� +− +��x +j=1 +λj +j! ψ(j) +�2 +� +r1(eλ − 1) +�2 +− ψ1(x + 1) +r2 +1 +. +(22) +The cdf of the Ct value is given by [see Section 5.1.6] +F(t|λ,⃗r,⃗τ,x) += +1 − +�x +j=1 +(x−1 +j−1)λj +j! +Be−z(j,x − j + 1) +�x +j=1 +(x−1 +j−1)λj +j! +Bθ(j,x − j + 1) +. +(23) +10 + +For the single-phase process, the cdf is given by +F(t|λ,r1,x) += +1 − +x�x +j=1 +(x−1 +j−1)λj +j! +Be−r1t(j,x − j + 1) +eλ − 1 +. +(24) +To estimate λ, either the pdf or the cdf of t can be fit to data. Alternatively, the posterior +density of λ conditioned on t can be computed. In Section 5.1.6, we show that, for the +single-phase process, it is given by +P(λ|r1,t,x) = +λw 1F1(1 − x;2; −λw +1−w ) +(eλ − 1)(1 − w)�x +j=1 +�x−1 +j−1 +�� w +1−w +�j ζ(j + 1) +, +(25) +where w = e−r1t and ζ(·) denotes the Riemann zeta function. +Note that, because they result from calculating expectations over the Poisson distribu- +tion, the summations found in Equations (18) - (25) can be truncated at any value of j ≫ λ +without a loss of accuracy. +In Figure 1, we illustrate the shape of the single-phase pdf for different values of λ +and different amplification efficiencies. As was the case for the PCR process with a deter- +ministic initial state (Figure 1, left panel), the pdf also has a bell shape (Figure 1, right +panel). Its location and width are governed by both the efficiency and λ. Consistent with +expectations, higher efficiencies or larger values of λ produce smaller mean Ct values, +smaller variances, and narrower pdfs (Figure 1, right panel). In contrast, lower efficiencies +or smaller values of λ produce larger mean Ct values, larger variances, and wider pdfs. +3.4 +Limit of detection and limit of quantification +We will now demonstrate theoretically how the mathematical framework we have devel- +oped can be applied to achieve certain operationally important objectives. In particular, +it is often of interest to quantify the limit of detection (LoD) of a particular instance of +the PCR method (henceforth referred to as “PCR protocol”). The LoD of a PCR protocol +is the smallest number of molecules that it can detect with a failure rate not exceeding a +defined threshold α (α is also called the significance level). Protocols with smaller LoDs +are in general preferred to those with larger LoDs. Ideally, the LoD should be either equal +to or smaller than the number of input DNA molecules expected in the considered sample. +Another important operational objective is to determine a PCR protocol’s limit of quan- +tification (LoQ) – i.e. the smallest number of molecules that it can estimate with a given +level of precision (measured here using the parameter β) and a given maximum failure +rate α. When a ≥ β-fold change in the number of target molecules needs to be detected, +the protocol should have an LoQ with precision ≤ β. The methods available for estimat- +ing LoD and LoQ are laborious [16] and frequently rely on certain ad-hoc mathematical +11 + +approximations [16, 17], which we would like to circumvent by developing and executing +mathematically precise statements of the estimation problem. +We begin with the LoD estimation problem. For the PCR process with a deterministic +initial state, the LoD can be expressed as follows: +LoD = +min n +s.t. F(T |n,⃗r,⃗τ,x) > 1 − α, +(26) +where F(T|n,⃗r,⃗τ,x) is given by (11) and T is the maximum practical duration of the PCR +process. For the process with a Poisson-distributed initial state, n is replaced by λ. +In Supplementary Figure 5.1, we show how the LoD varies with amplification effi- +ciency in a single-phase process with either a deterministic or a Poisson-distributed initial +state. In the former case, the process contains amplification noise but no sampling noise +while in the latter case it contains both sampling noise and amplification noise. For com- +parison, we also show the LoD in a process with sampling noise, modeled by using the +Poisson distribution, but without amplification noise. The LoD is lowest in a process with +sampling noise alone (LoD = 3 molecules) and it is highest when both sampling noise and +amplification noise are present (LoD ranges from 6 molecules, at 100% of maximum pos- +sible efficiency or m.p.e, to 157 molecules, at 80% m.p.e). A process with amplification +noise but without sampling noise has an intermediate LoD. In these computational exam- +ples, the parameters of the equation used to estimate the LoQ are perfectly known, and +this makes it possible to obtain perfect knowledge of the LoD. In real-world applications, +the parameter values will be associated with uncertainty, which will, in a quantifiable way, +make uncertain the LoD estimates. +We now turn our attention to the problem of estimating the LoQ in a PCR process +with a deterministic initial state. Suppose that a Ct value t is generated by such a process +and then used to obtain an estimate, denoted ˆn, of the number of input molecules. Let n +be the actual number of input molecules. We want to calculate the probability that, for +any data t generated by the same process, ˆn will not differ from n by more than a factor +β,β ≥ 1. We define the LoQ as the smallest value of n for which this probability exceeds +1 − α. Specifically, the LoQ is given by +LoQ = +min n +s.t. P +��n/β� ≤ ˆn ≤ �βn� | n,⃗r,⃗τ,x +� +> 1 − α, +(27) +where ⌊v⌋ (respectively ⌈v⌉) denotes the largest (respectively smallest) integer less than +(respectively greater than) or equal to v. +Focusing on the single-phase process, to obtain P (�n/β� ≤ ˆn ≤ �βn� | n,r1,x), we marginal- +ize the right-hand-side of (14) with respect to t and then take the sum from �n/β� to �βn�, +12 + +yielding [see Section 5.1.5] +P (�n/β� ≤ ˆn ≤ �βn� | n,r1,x) += +⌈βn⌉ +� +ˆn=⌊n/β⌋ +B( ˆn + n − 1,2x − ˆn − n + 1) +xB( ˆn,x − ˆn + 1)B(n,x − n + 1) += +P (�n/β� ≤ ˆn ≤ �βn� | n,x). +(28) +Strikingly, while Equation (28) depends on both n and x, it does not depend on the ampli- +fication efficiency r1. In the Appendix [see Equation (130)], we follow a similar procedure +to obtain P +��λ/β� ≤ ˆλ ≤ �βλ� | λ,r1,x +� +, the probability that, for any data t generated by a +single-phase PCR process with a Poisson-distributed initial state, the estimated value ofλ, +denoted ˆλ, will not differ from the actual value by more than a factor β. +Setting α = 0.05 and allowing at most a 10% deviation of ˆn from n (corresponding to +setting β = 1.1) results in an LoQ of 820 molecules. The LoQ decreases to 146 molecules +when a deviation of up to 25% from expectation is allowed (corresponding to β = 1.25), +and to 43 molecules when the allowable deviation increases to 50% (corresponding to β = +1.5). The analysis suggests that at the considered 5% failure rate, 10 input molecules can be +detected with an error of at least ≈200%, that is, a ≈ 2 fold deviation from n (corresponding +to β = 2). Because these calculations do not account for sampling noise, they provide only +a lower-bound for the LoQ that is achievable at the considered failure rate and level of +precision (β). As noted earlier, in real-world applications the parameters of the equation +used to estimate LoQ will be imperfectly known, and this will determine the amount of +uncertainty associated with the estimated LoQ. +3.5 +Amplification noise determines digital PCR outcomes +As noted earlier, digital PCR is a variant of conventional PCR that was developed to im- +prove the quantification of DNA. In digital PCR, a sample master mix (containing an un- +known number of input DNA molecules together with all the reagents required for DNA +replication) is uniformly distributed into hundreds (and sometimes thousands) of phys- +ical partitions, which may take the form of droplets or microwells [5]. Each partition is +expected to receive zero, one, or more DNA molecules following a Poisson distribution +with mean λ = CV /D, where C is the concentration of the DNA in the original sample, V +is the partition volume, and D is the (known) dilution factor applied to the sample during +preparation of the mastermix. PCR reactions are independently and simultaneously run +inside each partition, and positive partitions are identified. The resulting data – i.e. the +positive or negative outcome of each PCR reaction – are thus digital. The standard method +of interpreting these data assumes that partitions that receive at least one target molecule +will test positive, and their fraction, ˆf , is approximated by ˆf ≈ 1 − e−λ, from which λ is +estimated as ˆλ = −ln(1 − ˆf ) and then used to estimate C. +13 + +Figure 2: Under-estimation of λ by the standard method of interpreting digital PCR +data. We varied both the amplification efficiency and the amount of input DNA λ and +used Equation (29) to estimate λ. The resulting estimate, denoted ˆλ, was plotted against +efficiency, expressed as a percentage of the maximum possible efficiency. At efficiencies +lower than 95%, only a very small amount of the input DNA is detected. At 95% efficiency, +the amount detected ranges from 12%, when λ = 1, to 69%, when λ = 100. The amount +detected increases to ≈80% when the efficiency equals 100%. +However, according to our model, the assumption that the fraction of positive parti- +tions equals the Poisson probability that a partition receives one or more target molecules +is untenable due to the effects of PCR amplification noise. Indeed, setting the amplifi- +cation efficiency to a reasonably high value of 95% m.p.e (i.e. r1 = 0.95 ln2) and us- +ing T = 35,x = 2T in Equation (24), we predict that <4% of partitions that contain only +one molecule will test positive, which is >25-fold smaller than assumed by the Poisson +method. In Supplementary Figure 5.2, we compare the fraction of positive partitions cal- +culated using our model [Equation(24)] versus the positive fraction calculated by the Pois- +son method, for different values of λ and different amplification efficiencies. We find that +the Poisson method over-estimates the fraction of positive partitions for small values of λ, +including the value (1.61) at which the method is expected [18] to produce its most precise +estimates of λ. Only for a relatively large value of λ (10) do we find the Poisson method’s +estimate of the fraction of positive partitions to agree with the noise-adjusted expectation +calculated using our model (Supplementary Figure 5.2). +We use our model to investigate how this over-estimation of the fraction of positive +partitions affects the accuracy of the estimate of λ (denoted ˆλ) produced by the Poisson +14 + +入=1 +入 =1.61 +0.8- +1.0- +0.6 +0.4- +0.5 +0.2- +0.0 +0.0 +80 +85 +90 +95 +100 +80 +85 +90 +95 +100 +> +入=10 +入=100 +6- +80 +5 +60 +4 +3- +40 +2- +20 +1 +0 +0. +80 +85 +90 +95 +100 +80 +85 +90 +95 +100 +Amplificationefficiency,r(%)method. Setting t = T in Equation (24), we find that +ˆλ += +−ln +� +1 − ˆf +� += +ln +� +eλ − 1 +�x +j=1 +(x +j) +(j−1)!λjBe−r1T (j,x − j + 1) +� +. +(29) +According to Equation (29), ˆλ depends strongly on the amplification efficiency r1. ˆλ equals +0 in the limit as r1 tends to 0. As r1 increases to its maximum possible value of ln(2), ˆλ +also increases, approaching λ. In Figure 2, we illustrate the relationship between ˆλ/λ and +amplification efficiency. Strikingly, for efficiencies lower than 95% m.p.e, ˆλ/λ is very small, +indicating that λ is markedly under-estimated by the Poisson method. When the efficiency +equals 95% m.p.e, ˆλ/λ increases from ≈ 12% (at λ = 1) to ≈69% (at λ = 100). Increasing the +efficiency to the maximum possible value of 100% m.p.e causes ˆλ/λ to increase to ≈80% +(at λ = 100). These results indicate that the Poisson method is expected to under-estimate +λ because it does not account for amplification noise. Indeed, experimental data show a +strong tendency by the method to under-estimate the number of input DNA molecules +(eg. see Supplementary Table 6 in [19]). +4 +Discussion +The outputs of DNA quantification experiments, including those based on the polymerase +chain reaction (PCR), tend to vary within and across different experimental instances, +making the results difficult to interpret and limiting their utility beyond the particular +contexts in which they are generated. Indeed, various factors are known to contribute +to the variability of PCR outputs [20, 21, 22] including the varying complexity of DNA +templates and the random distribution of target molecules in the reaction environment; +the type of PCR machine and buffer components used; the durations and temperatures of +the three thermal cycles of PCR; the binding kinetics of oligonucleotide primers to target +DNA; and the stability of DNA polymerase and other PCR reagents. Taylor et. al [22] +reviewed the sources of variability in PCR experiments and proposed a stepwise process +to minimize such variability in practice. +A common output of a PCR experiment is the quantification cycle (denoted Ct or Cq +value), the PCR cycle at which the number of DNA molecules exceeds a defined threshold, +called the quantification threshold. The Ct value varies with both the number of input +DNA molecules and the PCR amplification efficiency, which in turn varies with the afore- +mentioned experimental variables. It is desirable to deconvolute such variable outputs to +estimate the number of input DNA molecules, which is of greatest interest in experiments, +by applying mathematical methods that account for the stochasticity that is inherent in the +15 + +underlying generative process. +We have developed a mathematical approach to modeling DNA quantification that +takes into account the underlying stochasticity. We used PCR, the most widely used class +of DNA quantification process, to illustrate our mathematical ideas, which are also ap- +plicable to a broader class of such processes (eg. [6]). Using the model, we derived the +probability generating function for the number of molecules found in a PCR process with +either a deterministic or a Poisson-distributed number of input molecules as well as the +probability density function (pdf), mean, variance and cumulative density function (cdf) +of the Ct value produced by such a process. In contrast to the deterministic case, in which +PCR outputs are contaminated only by amplification noise, in the latter case the outputs +also contain sampling noise. The equations we derived for these important statistical prop- +erties of the PCR process revealed functional relationships between the Ct value and un- +derlying variables that could previously only be accessed by empirical means. +To illustrate our mathematical ideas, we focused on the single-phase PCR process, for +which our modeling results take relatively simple mathematical forms. We found that +the common assumption that the mean Ct value is a simple logarithmic function of the +number of input DNA molecules n is correct when n is large. For small n, corrections +are required. An exact mean Ct value is given by (ψ(x + 1) − ψ(n))/r, where x is the quan- +tification threshold, r is the amplification efficiency and ψ(·) denotes the first polygamma +function. The variance has the elegant form (ψ1(n) − ψ1(x + 1))/r2, where ψ1(·) denotes the +second polygamma function. Therefore, the variance is strongly dependent on amplifica- +tion efficiency. This effect is illustrated in Figure 1, which shows that the pdf of the Ct +value becomes wider as the amplification efficiency decreases. Interestingly, in this simple +case, the coefficient of variation of the Ct value (i.e. the ratio of the standard deviation to +the mean) does not depend on amplification efficiency. +Two important numbers that characterize the performance of a PCR process are the +limit of detection (LoD) and the limit of quantification (LoQ). The LoD is the smallest +number of molecules that can be detected with a failure rate not exceeding a threshold +α, while LoQ is the smallest number of molecules that can be quantified with a given +level of precision (i.e. allowing a defined maximum fold deviation from the true value) +and a given maximum failure rate α. We provided mathematical formulae for calculating +both LoD and LoQ. Close examination of these formulae in the context of a single-phase +PCR process revealed that a small reduction of the amplification efficiency may cause a +large increase of LoD. For example, when α = 5%, reducing the efficiency from 95% of the +maximum possible efficiency (m.p.e) to 90% m.p.e. caused the LoD to double, from ≈10 +input molecules to ≈20 molecules. In contrast to LoD, we found that LoQ is independent +of efficiency. Allowing up to a 2-fold difference between n and its estimate results in an +LoQ of ≈10 molecules. Reducing the allowed fold difference to 10% increases LoQ to 820 +16 + +molecules. Our methods may be used to improve significantly the current approaches to +estimating LoD and LoQ, which are laborious [16] and frequently rely on certain crude +mathematical approximations [16, 17] that can be avoided by using our methods. +Furthermore, we applied our methods to shed light on the effects that amplification +noise has on estimates of the expected number of input DNA molecules λ obtained by the +standard method of interpreting digital PCR data. A key assumption of this method is +that a PCR reaction will be positive if it contains at least one input DNA molecule. We +showed that this assumption is in general invalid because of the stochastic nature of PCR +amplification. Stochastic effects are particularly large when λ is small, which is the regime +in which digital PCR preferentially operates. At a high amplification efficiency of 95% +m.p.e, we find that the ratio of the fraction of positive digital PCR reactions calculated by +the standard method versus the value obtained after accounting for amplification noise is +only 18.3% when λ = 1 and it increases to ≈100% when λ = 10 (Figure 5.2). Accordingly, +stochastic effects were found to cause a significant under-estimation of λ by the standard +method. Indeed, at a high efficiency of 95% m.p.e., the standard method is predicted +to under-estimate λ by factors of ≈8.1, ≈3.1, and ≈1.4 when λ equals 1, 10, and 100, +respectively (Figure 2). This is in the same range as empirically observed (eg. [19]). +Using our mathematical methods, the following two different approaches may be used +to obtain much more accurate estimates of λ. Firstly, if Ct values are available from pos- +itive digital PCR reactions, then Equation (19) can be fit to those Ct values, using either +a likelihood-based or a Bayesian statistical approach, to estimate the most probable value +of λ together with a confidence (or credible) interval for it. Secondly, if only binary (ie. +positive or negative) outcomes are available from individual reactions, then the variability +of such outcomes can still be exploited to estimate λ. Specifically, suppose there are N +different reactions. These can be randomly distributed into groups of N′ reactions each. +Assuming a binomial distribution of the number of positive reactions found in each group, +their first and second moments are given by F(T )N′ and F(T)N′ (1 + F(T )(N′ − 1)), respec- +tively, where F(T ) is calculated using (24). These moments contain information about the +two free parameters of F(T ) (i.e. λ and r1), which can be readily extracted to estimate λ +together with a confidence (or credible) interval. +The Ct value is estimated from the fluorescence profiles produced by DNA molecules +as they are amplified during PCR. Our mathematical analysis can be straightforwardly ex- +tended to obtain a time-dependent probability density of the fluorescence intensity Pt(y), +which can then be fit to fluorescence profiles as an alternative approach to estimating the +number of input DNA molecules. Using standard results from probability theory (eg. see +[23]), Pt(y) can be derived from both the cumulative distribution function of the number +of molecules found in the PCR process at time t, Ft(x) [calculated based on Equation (4) +or (16)] and the linear relation expected [24] between y and x. Specifically, Pt(y) can be +17 + +expressed as +Pt(y) = 1 +α ht(g−1(y)), +(30) +where +ht(x) = d +dxFt(x), +(31) +y = αx + β = g(x), and α,β > 0. We will explore in detail this alternative approach to +estimating the number of input DNA molecules in a future paper. +5 +Supporting Information +5.1 +Appendix +This section contains mathematical proofs and detailed calculations supporting the results +presented in Section 3. +5.1.1 +Proof of Theorem 1 +Proof. We will prove Theorem 1 by mathematical induction on k. +• k = 1: +The Chapman-Kolmogorov forward equation corresponding to the single-phase pro- +cess is given by: +∂P(X = x,t|X = x′,t′) +∂t += r1(x−1)P(X = x−1,t|X = x′,t′)−r1xP(X = x,t|X = x′,t′), (32) +where we have set t = t′ +∆t, and r1 is the amplification efficiency associated with the +process. To simplify our notation, we will abbreviate P(X = x,t|X = x′,t′) by P(x,t). +We will solve (32) by using a powerful combinatorial device called the probability +generating function (pgf) [14]. Recall that the pgf of P(x,t) is defined as: +G(s,t) = +∞ +� +x=0 +sxP(x,t), +where s is a book-keeping variable. +18 + +Multiplying both sides of (32) by sx and summing over all possible values of x yields: +∞ +� +x=0 +sx ∂P(x,t) +∂t += +r1 +∞ +� +x=0 +(x − 1)sxP(x − 1,t) − r1 +∞ +� +x=0 +xsxP(x,t) += +r1s2 +∞ +� +x=0 +(x − 1)sx−2P(x − 1,t) − r1s +∞ +� +x=0 +xsx−1P(x,t) += += r1s +� +������s +∞ +� +x=0 +(x − 1)sx−2P(x − 1,t) − +∞ +� +x=0 +xsx−1P(x,t) +� +������. +(33) +Using +∂G(s,t) +∂s += +∞ +� +x=0 +xsx−1P(x,t) and +∂G(s,t) +∂t += +∞ +� +x=0 +sx ∂P(x,t) +∂t +, +(34) +we simplify (33) to obtain +∂G(s,t) +∂t += r1s(s − 1)∂G(s,t) +∂s +, +(35) +which is a partial differential equation (pde) in G(s,t). +We will solve (35) by the method of characteristics. To this end, we define new +variables +u = u(s,t) and v = v(s,t), +which will transform (35) into the simpler equation +∂W(u,v) +∂u ++ H(u,v)W(u,v) = F(u,v), +(36) +which has the solution +W(u,v) = e− +� +H(u,v)du +�� +F(u,v)e +� +H(u,v)du + Ψ(v) +� +, +where +W(u,v) = G(s(u,v),t(u,v)). +This requires that v(s,t) = c, where c is an arbitrary constant. The resulting charac- +teristic equation is given by +ds +dt = −r1s(s − 1), +19 + +which has the solution +s − 1 +s +er1t = c = v(s,t). +Setting u(s,t) = t, we obtain +∂G +∂t += +∂W +∂t = ∂W +∂u +∂u +∂t + ∂W +∂v +∂v +∂t += +∂W +∂u + r1(s − 1) +s +er1t ∂W +∂v +(37) +and +∂G +∂s += +∂W +∂u +∂u +∂s + ∂W +∂v +∂v +∂s += +1 +s2 er1t ∂W +∂v . +(38) +Substituting (37) and (38) into (35) gives +∂W +∂u = 0, +(39) +which has the same form as (36). The solution to (39) is given by +W(u,v) += Ψ(v) +=⇒ +G(s,t) += Ψ +�s − 1 +s +er1t� +. +(40) +If there are n molecules at the start of the process (t = 0), then p(x,0) = 1 if x = n and +p(x,0) = 0 otherwise. Therefore, +G(s,0) = Ψ +�s − 1 +s +� += +∞ +� +x=0 +sxP(x,0) = sn. +(41) +In (41), the argument y of Ψ (y) maps onto ( 1 +1−y )n, implying that +G(s,t) = Ψ +�s − 1 +s +er1t� += +� +����� +1 +1 − s−1 +s er1t +� +����� +n += +sne−nr1t +[1 − s(1 − e−r1t)]n . +(42) +Equation (42) matches (2) when k = 1. +Corollary 3. Equation (42) solves (35). +20 + +Proof. The right-hand-side of (35) is +∂G +∂s += +nsn−1e−nr1t � +1 − s(1 − e−r1t) +�−n + nsne−nr1t � +1 − s(1 − e−r1t) +�−(n+1) � +1 − e−r1t� += +nsn−1e−nr1t +[1 − s(1 − e−r1t)]n +� +1 + +s(1 − e−r1t) +1 − s(1 − e−r1t) +� += +nsn−1e−nr1t +[1 − s(1 − e−r1t)](n+1) +� +1 − s(1 − e−r1t) + s(1 − e−r1t) +� += +nsn−1e−nr1t +[1 − s(1 − e−r1t)](n+1) , +(43) +and the left hand-side is +∂G +∂t += +−nr1sne−nr1t � +1 − s(1 − e−r1t) +�−n + nsn+1r1e−(n+1)r1t � +1 − s(1 − e−r1t) +�−(n+1) += +nr1sne−nr1t +[1 − s(1 − e−r1t)]n +� +se−r1t +1 − s(1 − e−r1t) − 1 +� += +nr1sne−nr1t +[1 − s(1 − e−r1t)](n+1) +� +se−r1t − 1 + s − se−r1t� += +nr1sne−nr1t(s − 1) +[1 − s(1 − e−r1t)](n+1) += +r1s(s − 1) +this matches (43) +������������������������������������������������ +� +����� +nsn−1e−nr1t +[1 − s(1 − e−r1t)](n+1) +� +����� = r1s(s − 1)∂G +∂s . +(44) +• k = 2: +There are two amplification phases with rates r1 and r2, respectively. The first one +runs from time t = 0 to t = τ1, and the second one runs from t = τ1 to t = τ1 + +τ2. In the second phase, the probability generating function takes exactly the same +general functional form as in the first phase, albeit with a different initial condition. +Specifically, we have +G(s,t) = Ψ +�s − 1 +s +er2(t−τ1)� +, +with the initial condition (at time t = τ1) +G(s,τ1) = Ψ +�s − 1 +s +� += +sne−nr1τ1 +[1 − s(1 − e−r1τ1)]n . +21 + +Using the same procedure as in the case when k = 1, we obtain +G(s,t) += +Ψ +�s − 1 +s +er2(t−τ1)� += +� +1 +1− s−1 +s er2(t−τ1) +�n +e−nr1τ1 +� +1 − +� +1 +1− s−1 +s er2(t−τ1) +� +(1 − e−r1τ1) +�n += +sne−n[r2t+(r1−r2)τ1] +� +1 − s +� +1 − e−[r2t+(r1−r2)τ1]��n . +(45) +The right side of (45) equals (35) when k = 2, as expected. +• We assume the statement is true for t ∈ Ik, that is +G(s,t) = +� +se−z +1 − s(1 − e−z) +�n +, +where z = rkt + �k−1 +i=1(ri − rk)τi, and we prove it for t ∈ Ik+1. As before, in phase k + 1, +the generating function has the functional form +G(s,t) = Ψ +�s − 1 +s +erk+1(t−�k +i=1 τi)� +. +At time t = �k +i=1 τi, by the induction step, we have +G(s,t) = Ψ +�s − 1 +s +� += +� +se−z +1 − s(1 − e−z) +�n +. +Using the same arguments as before, we find that, for t ∈ Ik+1, +G(s,t) += +Ψ +�s − 1 +s +erk+1(t−�k +i=1 τi)� += +� +��������� +1 +1− s−1 +s erk+1(t−�k +i=1 τi ) e−z +1 − +1 +1− s−1 +s erk+1(t−�k +i=1 τi ) (1 − e−z) +� +��������� +n += +sne−n[rk+1t+�k +i=1(ri−rk)τi] +� +1 − s +� +1 − e−[rk+1t+�k +i=1(ri−rk)τi] +��n , +(46) +and this ends the proof of Theorem 1. +22 + +5.1.2 +Proof of Corollary 1 +Proof. From Theorem 1, we know that the generating function for P(x|n,⃗r,t,⃗τ) is given by +G(s,t) = +� +se−z +1 − s(1 − e−z) +�n +. +Let p = e−z and q = 1 − p. Then, +G(s,t) = +� +sp +1 − sq +�n += (sp)n [1 − sq]−n . +P(x|n,⃗r,t,⃗τ) is the coefficient of sx in the power series expansion of G(s,t), given by +G(s,t) += +(sp)n [1 − sq]−n = (sp)n � +i=0 +�−n +i +� +(−sq)i = (sp)n � +i=0 +�−n +i +� +(−1)i(sq)i. +(47) +But +�−n +i +� += +−n(−n − 1)(−n − 2)(−n − 3)...(−n − (i − 2))(−n − (i − 1)) +1.2.3....(i − 1)i += +(−1)i n(n + 1)(n + 2)(n + 3)...(n + (i − 2))(n + (i − 1)) +i! += +(−1)i +�n + i − 1 +i +� +=⇒ +(−1)i +�−n +i +� += +�n + i − 1 +i +� +. +(48) +Substituting (48) into (47) gives +G(s,t) += +(sp)n � +i=0 +�−n +i +� +(−1)i(sq)i = (sp)n � +i=0 +�n + i − 1 +i +� +(sq)i. +(49) +Let x = n + i. Then, +G(s,t) += +(sp)n +∞ +� +x=n +�x − 1 +x − n +� +(sq)x−n = +∞ +� +x=n +�x − 1 +x − n +� +pnqx−nsx. +(50) +The probability of having x molecules at time t, P(x,t), is therefore given by +P(x,t) = +�x − 1 +x − n +� +pnqx−n = +�x − 1 +n − 1 +� +e−nz (1 − e−z)x−n . +(51) +Corollary 4. The probability distribution given in Theorem 1 solves the Chapman-Kolmogorov +23 + +equation given by (32). +Proof. +∂P(x,t) +∂t += �x−1 +x−n +�� +−nrke−z (1 − e−z)x−n + rk(x − n)e−2z (1 − e−z)x−n−1� += rk +�x−1 +x−n +�e−z (1 − e−z)x−n � +−n + (x − n) e−z +1−e−z +� += rk +�x−1 +x−n +�e−z (1 − e−z)x−n � +−n + (x − n) e−z +1−e−z − x−n +1−e−z + x−n +1−e−z +� += rk +�x−1 +x−n +�e−z (1 − e−z)x−n � x−n +1−e−z − n + (x−n) +1−e−z (e−z − 1) +� += rk +�x−1 +x−n +�e−z (1 − e−z)x−n � x−n +1−e−z − x +� += rk +� +(x − n)�x−1 +x−n +�� +e−z (1 − e−z)x−n−1 − rkx�x−1 +x−n +�e−z (1 − e−z)x−n += rk(x − 1) +�� x−2 +x−n−1 +�e−z (1 − e−z)x−n−1� +− rkx +��x−1 +x−n +�e−z (1 − e−z)x−n� += rk(x − 1)P(x − 1,t) − rkxP(x,t), +(52) +which equals the right-hand side of (32). +5.1.3 +Proof of Theorem 2 +Proof. We prove Theorem (2) by mathematical induction on k. +• k = 1: +There is only one phase with amplification efficiency r1. Recall that the Chapman- +Kolmogorov forward equation for the dynamics of P (x,t) is given by (32), with the +initial condition +P (x,0) = e−λλx +x! +. +Using the same arguments as above, we can write the generating function for P (x,t) +as +G(s,t) = Ψ +�s − 1 +s +er1t� +, +(53) +with the initial condition +G(s,0) += +Ψ +�s − 1 +s +� += +∞ +� +x=0 +sxP(x,0) = eλ(s−1). +(54) +Therefore, +G(s,t) += +Ψ +�s − 1 +s +er1t� += +e +λ +� +1 +1− s−1 +s +er1t −1 +� += +e +� +λ(s−1) +1−s(1−e−r1t) +� +. +(55) +24 + +Equation (55) matches (15) when k = 1. +Corollary 5. The generating function given by (55) solves Equation (35). +Proof. Differentiate the right hand side of (55) with respect to s. +• k = 2: +There are two phases with amplification efficiencies r1 and r2, respectively. The first +phase runs from time t = 0 to t = τ1, and the second one runs from t = τ1 to t = τ1+τ2. +As before, for t ∈ I2 the probability generating function has the form +G(s,t) = Ψ +�s − 1 +s +er2(t−τ1)� +, +with the initial condition (at time t = t1) +G(s,τ1) = Ψ +�s − 1 +s +� += e +� +λ(s−1) +1−s(1−e−r1t) +� +. +Therefore, +G(s,t) += +Ψ +�s − 1 +s +er2(t−τ1)� += +e +� +������� +λ( +1 +1− s−1 +s +er2(t−τ1) −1) +1− +1 +1− s−1 +s +er2(t−τ1) (1−e−r1t) +� +������� += +e +� +λ(s−1) +1−s(1−e−(r2t+(r1−r2)τ1)) +� +. +(56) +The right side of (56) equals (15) for the case k = 2. +• We assume the statement is true for t ∈ Ik, that is +G(s,t) = e +� +λ(s−1) +1−s(1−e−z) +� +, +where z = rkt + �k−1 +i=1(ri − rk)τi, and we prove it for t ∈ Ik+1. +In phase k + 1, the probability generating function has the form +G(s,t) = Ψ +�s − 1 +s +erk+1(t−�k +i=1 τi)� +. +By the induction step, the initial condition (at time t = �k +i=1 τi) is given by +G(s,t) = Ψ +�s − 1 +s +� += e +� +λ(s−1) +1−s(1−e−z) +� +. +25 + +Therefore, for t ∈ Ik+1, we have +G(s,t) += +Ψ +�s − 1 +s +erk+1(t−�k +i=1 τi)� += +e +� +���������� +λ( +1 +1− s−1 +s +erk+1(t−�k +i=1 τi ) +−1) +1− +1 +1− s−1 +s +erk+1(t−�k +i=1 τi ) +(1−e−z) +� +���������� += +e +� +λ(s−1) +1−s(1−e−z′ ) +� +, +(57) +where z′ = rk+1t + �k +i=1(ri − rk+1)τi, and this ends the proof of Theorem 2. +5.1.4 +Proof of Corollary 2 +Proof. Recall that P(x|λ,⃗r,t,⃗τ) is the coefficient of sx in the power series expansion of the +probability generating function given in Theorem 2, that is +P(x|λ,⃗r,t,⃗τ) += +1 +x! +∂xG(s,t) +∂sx +���s=0. +(58) +Now, let us compute the partial derivatives of G(s,t) with respect to s. For brevity, we set +a = e−z,v = (1 − e−z) = (1 − a),u = λe−z = aλ, and Q(s,t) = 1 − s(1 − e−z) = 1 − sv. +Observe that +G(s,t) = e +λ(s−1) +Q ,Q(0,t) = 1,G(0,t) = e−λ, ∂Q(s,t) +∂s += ∂Q(s,t) +∂s +���s=0 = −v +26 + +and +∂ +∂sG(s,t) += +u G(s,t) +Q2 +=⇒ ∂ +∂sG(s,t) +���s=0 = ue−λ, +∂2 +∂s2 G(s,t) += +ue−λ +Q4 +�Q2uG(s,t) +Q2 ++ 2vQG(s,t) +� += u [u + 2vQ] G(s,t) +Q4 +=⇒ +∂2 +∂s2 G(s,t) +���s=0 = e−λ � +u2 + 2uv +� +, +∂3 +∂s3 G(s,t) += +u [u + 2v] +Q8 +� +uG(s,t)(u + 2vQ)Q2 − 2v2G(s,t)Q4 + 4vG(s,t)(u + 2vQ)Q3� += +u G(s,t) +Q6 +� +u2 + 6uvQ + 6v2Q2� +=⇒ +∂3 +∂s3 G(s,t) +���s=0 = e−λ(u3 + 6u2v + 6uv2), +∂4 +∂s4 G(s,t) += +u G(s,t) +Q8 +� +u3 + 12u2vQ + 36uv2Q2 + 24v3Q3� +=⇒ +∂4 +∂s4 G(s,t) +���s=0 = e−λ � +u4 + 12u3v + 36u2v2 + 24uv3� +∂5 +∂s5 G(s,t) += +u G(s,t) +Q8 +� +u4 + 20u3vQ + 120u2v2Q2 + 240uv3Q3 + 120v4Q4� +, +=⇒ +∂4 +∂s5 G(s,t) +���s=0 = e−λ � +u5 + 20u4v + 120u3v2 + 240u2v3 + 120uv4� +. +(59) +By closely examining the coefficients of powers of the terms u,v and uv in (59), the follow- +ing combinatorial triangle emerges +x +1 +1 +2 +1 +2 +3 +1 +6 +6 +4 +1 +12 +36 +24 +5 +1 +20 +120 +240 +120 +1 +2 +3 +4 +5 +i +In particular, the (x,i)’th entry of the triangle is given by +T (x,i) = +� x +i − 1 +��x − 1 +i − 1 +� +(i − 1)!, for i = 1,2,...x. +(60) +27 + +Thus +P(x|λ,⃗r,t,⃗τ) += +1 +x! +∂x +∂sx G(s,t) +����s=0 = e−λ +x! +x +� +i=1 +T (x,i) += +e−λ +x! +x +� +i=1 +� x +i − 1 +��x − 1 +i − 1 +� +(i − 1)!(λe−z)x−i+1 (1 − e−z)i−1 += +e−λ +x +� +i=1 +1 +(x − i + 1)! +�x − 1 +i − 1 +� +(λe−z)x−i+1 (1 − e−z)i−1 . +(61) +Setting k = x − i + 1 in (61) gives the desired result. +Corollary 6. The probability distribution given in Theorem 2 solves (32). +Proof. Differentiate the right-hand-side of Equation (61) with respect to t. +We will now derive the probability density function (pdf), mean, variance, and cumu- +lative density function (cdf) of the Ct value. We will consider two different cases, namely: +1. when the initial state of the PCR process is deterministic, and the PCR phase lengths +and amplification efficiencies are given; and +2. when the initial state is Poisson-distributed, and the phase lengths and amplification +efficiencies are given. +5.1.5 +Case 1: The initial state is deterministic, and the phase lengths and amplifica- +tion efficiencies are given +• General form of the pdf +Consider the PCR process described in Theorem 1. The process begins with n cDNA +molecules, which are amplified across up to p successive phases {Ii} of lengths ⃗τ = +(τ1,τ2,...,τp) at the corresponding amplification efficiencies ⃗r = (r1,r2,...,rp). By def- +inition, the Ct value t is the time at which the number of molecules reaches the quan- +tification threshold, which we denote by x. By Bayes’ theorem, a general expression +for the pdf of t is given by +P(t|n,⃗r,⃗τ,x) += +P(n,⃗r,⃗τ,x|t)P(t) +P(n,⃗r,⃗τ,x) +. +(62) +Since n is independent of ⃗r, ⃗τ, and t, and ⃗r is also independent of t and of the values +taken by the entries of ⃗τ, we re-write the numerator of the right-hand-side of (62) as +28 + +follows: +P(n,⃗r,⃗τ,x|t)P(t) += +P(x|n,⃗r,⃗τ,t)P(n,⃗r,⃗τ|t)P(t) += +P(x|n,⃗r,⃗τ,t)P(n|⃗r,⃗τ,t)P(⃗r,⃗τ|t)P(t) += +P(x|n,⃗r,⃗τ,t)P(n)P(⃗r|⃗τ,t)P(⃗τ|t)P(t) += +P(x|n,⃗r,⃗τ,t)P(n)P(⃗r)P(t|⃗τ)P(⃗τ). +(63) +Similarly, the denominator of the right-hand-side of (62) can be simplified to +P(n,⃗r,⃗τ,x) += +� ∞ +�k−1 +i=1 τi +P(n,⃗r,⃗τ,x|t)P(t)dt += +P(n)P(⃗r)P(⃗τ) +� ∞ +�k−1 +i=1 τi +P(x|n,⃗r,⃗τ,t)P(t|⃗τ)dt. +(64) +Therefore, (62) can be re-written as +P(t|n,⃗r,⃗τ,x) += +P(x|n,⃗r,⃗τ,t)P(t|⃗τ) +� ∞ +�k−1 +i=1 τi P(x|n,⃗r,⃗τ,t)P(t|⃗τ)dt +. +(65) +We will derive the pdf, mean, variance, and cdf of the Ct value for a PCR process +with an arbitrary number of phases p. Without loss of generality, we suppose that +t ∈ Ik,k ≤ p. We will assume a uniform prior density for t given ⃗τ. As we will +demonstrate later, this assumption produces very similar results to those we obtain +by assuming a Jeffreys prior [25]. We will state results for the case of a single-phase +PCR process whenever these cannot be readily gleaned from the general results. +We will first consider the case when the lengths of the intermediate phases, recorded +in the vector ⃗τ, are given. This is useful, for example, when it is of interest to estimate +the lengths of such phases from data. We will then show how to marginalize ⃗τ out +of the pdf. +• pdf +Using the posterior density given in Equation (65) and the likelihood function given +in Corollary 1, we obtain the following functional form for the pdf: +P(t|n,⃗r,⃗τ,x) +∝ +e−nz (1 − e−z)x−n , +(66) +where +z = rkt + +k−1 +� +i=1 +(ri − rk)τi, +(67) +⃗r = (r1,r2,...,rk) is a vector of amplification efficiencies, ⃗τ = (τ1,τ2,...,τk) is a vector of +29 + +phase lengths, and we have used a uniform prior for t. +The normalizing constant is given by +C = +� ∞ +�k−1 +i=1 τi +e−nz(1 − e−z)x−ndt. +(68) +Let w = e−z. Then, +C += +1 +rk +� θ +0 +wn−1(1 − w)x−ndw += +Bθ(n,x − n + 1) +rk +, +(69) +where +θ = e−�k−1 +i=1 riτi. +(70) +Therefore, the pdf is given by +P(t|n,⃗r,⃗τ,x) += +rke−nz (1 − e−z)x−n +Bθ(n,x − n + 1) . +(71) +For the single-phase process, θ = 1, so the pdf simplifies to +P(t|n,r1,x) += +r1e−nr1t � +1 − e−r1t�x−n +B(n,x − n + 1) +. +(72) +Note that in some cases (eg. when knowledge of the lengths of individual PCR ampli- +fication phases is not of interest), it may be useful to marginalize ⃗τ out of P(t|n,⃗r,⃗τ,x). +This can be achieved by using the fact that +P(t|n,⃗r,x) += +� +Ω +P(t,⃗τ|n,⃗r,x)d⃗τ += +� +Ω +P(t|n,⃗r,⃗τ,x)P(⃗τ|n,⃗r,x)d⃗τ, +(73) +where Ω is the domain of ⃗τ. +In addition, note that an alternative formulation of the prior for t, based on an ap- +proach proposed by Jeffreys [25] for generating priors that are invariant to reparametriza- +tion, is the following: +p(t|⃗τ) ∝ +� +|I(t|⃗τ)|, +(74) +30 + +where I(t|⃗τ) is the Fisher information of the likelihood function and is given by +I(t|⃗τ) += +EX +�� ∂ +∂t lnP(x|n,⃗r,⃗τ,x) +�2 � += +EX +�r2 +k +� +w2x2 − 2nwx + n2� +(1 − w)2 +� += +r2 +k w2 +(1 − w)2 EX +� +x2� +− 2nr2 +k w +(1 − w)2 EX +� +x +� ++ +n2r2 +k +(1 − w)2 += +r2 +k +(1 − w)2 +� +�����w2 +∞ +� +j=1 +x2 +�x − 1 +j − 1 +� +wn(1 − w)x−n − 2nw +∞ +� +j=1 +x +�x − 1 +j − 1 +� +wn(1 − w)x−n + n2 +� +�����, +(75) +where w = e−z. +Observe that +∞ +� +x=1 +x +�x − 1 +n − 1 +� +wn(1 − w)x−n += +∞ +� +x=1 +x! +(x − n)!(n − 1)!wn(1 − w)x−n += +∞ +� +y=2 +(y − 1)! +(y − m)!(m − 2)!wm−1(1 − w)y−m += +m − 1 +w +∞ +� +y=1 +(y − 1)! +(y − m)!(m − 1)!wm(1 − w)y−m += +m − 1 +w += +n +w +(76) +31 + +and +∞ +� +x=1 +x2 +�x − 1 +n − 1 +� +wn(1 − w)x−n += +∞ +� +x=1 +x!x +(x − n)!(n − 1)!wn(1 − w)x−n += +∞ +� +y=2 +(y − 1)!(y − 1) +(y − m)!(m − 2)!wm−1(1 − w)y−m += +∞ +� +y=1 +(y − 1)!y +(y − m)!(m − 2)!wm−1(1 − w)y−m − m − 1 +w += +∞ +� +y′=2 +(y′ − 1)! +(y′ − m′)!(m′ − 3)!wm′−2(1 − w)y′−m′ − m − 1 +w += +(m′ − 1)(m′ − 2) +w2 +∞ +� +y′=1 +(y′ − 1)! +(y′ − m′)!(m′ − 1)!wm′(1 − w)y′−m′ − m − 1 +w += +(m′ − 1)(m′ − 2) +w2 +− m − 1 +w += +n(n + 1) +w2 +− n +w , +(77) +where m = n + 1,m′ = m + 1,y = x + 1,y′ = y + 1. +Plugging (76) and (77) into (75), we obtain +I(t|⃗τ) += +nr2 +k +1 − w +=⇒ +p(t|⃗τ) ∝ +1 +√ +1 − w += +1 +√ +1 − e−z . +(78) +Using this prior, and following the steps we used earlier to derive (71), we find that +the pdf is given by +P(t|n,⃗r,⃗τ,x) +∝ +e−nz (1 − e−z)x−n−1/2 +=⇒ P(t|n,⃗r,⃗τ,x) += +rke−nz (1 − e−z)x−n−1/2 +Bθ(n,x − n + 1/2) +, +(79) +which has a similar form as (71). +For simplicity, we will continue to use a uniform prior for t. +• Mean +The mean Ct value is given by +E(t) += +rk +� ∞ +�k−1 +i +τi te−nz(1 − e−z)x−ndt +Bθ(n,x − n + 1) +, +(80) +32 + +where z is given by (67). +Let w = e−z. Then, +E(t) += +rk +� 0 +θ +� +− (lnw−lnθ′) +rk +� +wn(1 − w)x−n +� +− dw +rkw +� +Bθ(n,x − n + 1) += +� 0 +θ (lnw − lnθ′)wn−1(1 − w)x−ndw +rkBθ(n,x − n + 1) += +lnθ′ � θ +0 wn−1(1 − w)x−ndw − +� θ +0 lnw wn−1(1 − w)x−ndw +rkBθ(n,x − n + 1) += +lnθ′ +rk +− +� +∂ +∂n + ∂ +∂x +� +Bθ(n,x − n + 1) +rkBθ(n,x − n + 1) += +ln θ′ +θ +rk ++ Γ(n)2θn 3 ˜F2(n,n,n − x;n + 1,n + 1;θ) +rkBθ(n,x − n + 1) += +k−1 +� +i=1 +τi + Γ(n)2θn 3 ˜F2(n,n,n − x;n + 1,n + 1;θ) +rkBθ(n,x − n + 1) +, +(81) +where θ is given by (70), ψ(·) is the first polygamma function (also called the digamma +function), and +θ′ = θerk +�k−1 +i=1 τi. +(82) +Note that for the single-phase process, θ = θ′ = 1. In this case, using +3 ˜F2(n,n,n − x;n + 1,n + 1;1) = nΓ(x − n + 1)[ψ(x + 1) − ψ(n)] +Γ(n + 1)Γ(x + 1) +, +we find that the mean Ct value is given by +E(t) = ψ(x + 1) − ψ(n) +r1 +. +(83) +• Variance +The variance of the Ct value is given by E(t2) − E(t)2, where +E(t2) += +rk +� ∞ +�k−1 +i=1 τi t2e−nz(1 − e−z)x−ndt +Bθ(n,x − n + 1) +, +(84) +and z is given by (67). +33 + +Let w = e−z. Then, +E(t2) += +rk +� 0 +θ +� +lnw−lnθ′ +rk +�2 +wn(1 − w)x−n +� +− dw +rkw +� +Bθ(n,x − n + 1) += +� θ +0 (lnw − lnθ′)2 wn−1(1 − w)x−ndw +rk2Bθ(n,x − n + 1) += +� θ +0 (lnw)2 wn−1(1 − w)x−ndw − 2lnθ′ � θ +0 lnw wn−1(1 − w)x−ndw +rk2Bθ(n,x − n + 1) ++ +(lnθ′)2 � θ +0 wn−1(1 − w)x−ndw +rk2Bθ(n,x − n + 1) += +� +∂2 +∂n2 + 2 ∂2 +∂n∂x + ∂2 +∂x2 − 2lnθ′ +� +∂ +∂n + ∂ +∂x +�� +Bθ(n,x − n + 1) +rk2Bθ(n,x − n + 1) ++ (lnθ′)2 +rk2 += +� +∂2 +∂n2 + 2 ∂2 +∂n∂x + ∂2 +∂x2 +� +Bθ(n,x − n + 1) +rk2Bθ(n,x − n + 1) ++ 2lnθ′Γ(n)2θn 3 ˜F2(n,n,n − x;n + 1,n + 1;θ) +r2 +k Bθ(n,x − n + 1) ++ +lnθ′ ln θ′ +θ2 +r2 +k += +� +∂2 +∂n2 + 2 ∂2 +∂n∂x + ∂2 +∂x2 +� +Bθ(n,x − n + 1) +rk2Bθ(n,x − n + 1) ++ 2lnθ′Γ(n)2θn 3 ˜F2(n,n,n − x;n + 1,n + 1;θ) +r2 +k Bθ(n,x − n + 1) ++ +� +������ +k−1 +� +i=1 +τi +� +������ +2 +− +� +������ +k−1 +� +i=1 +riτi +rk +� +������ +2 +, +(85) +where θ is given by (70) and θ′ is given by (82). +For the single-phase process, the second moment of the Ct value is given by +E(t2) += +� +∂2 +∂n2 + 2 ∂2 +∂n∂x + ∂2 +∂x2 +� +B(n,x − n + 1) +rk2B(n,x − n + 1) += +ψ1(n) − ψ1(x + 1) + [ψ(x + 1) − ψ(n)]2 +rk2 +, +(86) +where ψ1(·) is the second polygamma function (also called the trigamma function). +Therefore, the variance is +Var(t) = ψ1(n) − ψ1(x + 1) +rk2 +. +(87) +• cdf +34 + +The cdf of the Ct value is given by +F(t|n,⃗r,⃗τ,x) += +rk +Bθ(n,x − n + 1) +� t +�k−1 +i=1 τi +e−nz′(1 − e−z′)x−nds, +(88) +where z′ = rks + �k−1 +i=1(ri − rk)τi +Let w = e−z′. Then, +F(t|n,⃗r,⃗τ,x) += +� θ +e−z wn−1(1 − w)x−ndw +Bθ(n,x − n + 1) += +Bθ(n,x − n + 1) − Be−z(n,x − n + 1) +Bθ(n,x − n + 1) += +1 − Be−z(n,x − n + 1) +Bθ(n,x − n + 1) , +(89) +where θ is given by 70. +For the single-phase process, the cdf is given by +F(t|n,r1,x) += +1 − Ie−r1t(n,x − n + 1), +(90) +where Ie−r1t(n,x−n+1) = Be−rt (n,x−n+1) +B(n,x−n+1) +is the regularized incomplete Beta function. Be- +cause the cdf is in closed analytical form, we can apply the efficient inverse transform +method to generate random samples of Ct values as follows: +t += +−lnI−1 +1−u(n,x − n + 1) +r1 +, +(91) +where u is a real number sampled uniformly at random from the interval (0,1) and +I−1 +1−u is the inverse of the regularized incomplete Beta function. To find a Ct value +that corresponds to a quantile q ∈ (0,1), simply replace u in Equation (91) by q. +• Probability distribution of n +We conclude by deriving the probability distribution of n, denoted P(n|r1,t,x), for the +single-phase PCR process. This distribution can be used to estimate n from measured +Ct values. It can also be used to calculate the LoD and LoQ of a PCR assay, as we +demonstrated in the main text. The steps described below can also be used to derive +P(n|⃗r,t,⃗τ), for a PCR process with an arbitrary number of phases, although this will +not yield a closed-form result like we will obtain in the single-phase case. +By Bayes’ Theorem, we have +P(n|r1,t,x) +∝ +wn(1 − w)x−n +B(n,x − n + 1), +(92) +35 + +where +w = e−r1t. +(93) +The normalizing constant is given by +C += +x +� +n=1 +wn(1 − w)x−n +B(n,x − n + 1) += +x +� +n=1 +x! wn(1 − w)x−n +(n − 1)! (x − n)! . +(94) +Let m = n − 1. Then, +C += +x−1 +� +m=0 +x! wm+1(1 − w)x−1−m +m! (x − 1 − m)! += +xw +x−1 +� +m=0 +�x − 1 +m +� +wm(1 − w)x−1−m += +xw. +(95) +Therefore, we have +P (n|r1,t,x) += +wn−1(1 − w)x−n +xB(n,x − n + 1). +(96) +Suppose that t is a Ct value generated by a PCR process with n input molecules. If +we replace n by ˆn in Equation (96), then the equation will give the probability that +ˆn will be obtained as the estimate of n based on the data t. It is useful – eg. for the +purpose of determining the LoQ – to calculate the probability that ˆn will be obtained +as the estimate of n based on any data t that can be generated by a PCR process with +n input molecules. This probability is given by +P ( ˆn|n,r1,x) += +� ∞ +0 +P ( ˆn,t|n,r1,x)dt += +� ∞ +0 +P ( ˆn|n,r1,t,x)P (t|n,r1,x)dt += +� ∞ +0 +P ( ˆn|r1,t,x)P (t|n,r1,x)dt += +r1 +� ∞ +0 +w ˆn−1(1 − w)x− ˆn +xB( ˆn,x − ˆn + 1) +wn(1 − w)x−n +B(n,x − n + 1)dt += +� 1 +0 +w ˆn+n−2(1 − w)2x−m−n +xB( ˆn,x − ˆn + 1)B(n,x − n + 1)dw += +B( ˆn + n − 1,2x − ˆn − n + 1) +xB( ˆn,x − ˆn + 1)B(n,x − n + 1) = P( ˆn|n,x), +(97) +36 + +where, using (96), we have assumed that ˆn is conditionally independent of n given t. +Strikingly, (97) does not depend on r1. +5.1.6 +Case 2: The initial state is Poisson-distributed, and the phase lengths and am- +plification efficiencies are given +• General form of the pdf +Let t be the Ct value of the PCR process described in Theorem 2. The process begins +with a Poisson-distributed number of input DNA molecules, with mean λ, which +are replicated across up to p distinct phases with lengths ⃗τ = (τ1,τ2,...,τp) and am- +plification efficiencies ⃗r = (r1,r2,...,rp). As noted earlier, t is the time at which the +number of molecules reaches the quantification threshold, which we denote by x. +Let us denote the pdf of t by P(t|λ,⃗r,⃗τ,x). By Bayes’ theorem, we have +P(t|λ,⃗r,⃗τ,x) += +P(λ,⃗r,⃗τ,x|t)P(t) +P(λ,⃗r,⃗τ,x) +(98) +However, λ is independent of ⃗r, ⃗τ, and t, while ⃗r is also independent of t and of the +precise values taken by the entries of ⃗τ. Therefore, by following the same steps we +used earlier to derive (65), we find that +P(t|λ,⃗r,⃗τ,x) += +P(x|λ,⃗r,t,⃗τ)P(t|⃗τ) +� ∞ +�k−1 +i=1 τi P(x|λ,⃗r,t,⃗τ)P(t|⃗τ)dt +. +(99) +We will derive the pdf, mean, variance, and cdf by assuming, without loss of gener- +ality, that t ∈ Ik, and then we will specify the functional forms taken by the results +in the instructive case when t ∈ I1. As before, for simplicity, we will use a uniform +prior density for t. +• pdf +We derive the pdf of the Ct value t by using the general expression given in Equation +(99), with the probability distribution of the number of molecules given in Theorem +37 + +2 serving as the likelihood. Specifically, +P(t|λ,⃗r,⃗τ,x) +∝ +(1 − e−z)x +x +� +j=1 +�x−1 +j−1 +� +j! +� λe−z +1 − e−z +�j +(100) += +(1 − e−z)x +x−1 +� +j=0 +�x−1 +j +� +(j + 1)! +� λe−z +1 − e−z +�j+1 +(101) +since (x +j)=0 for j>x += +(1 − e−z)x +∞ +� +j=0 +�x−1 +j +� +(j + 1)! +� λe−z +1 − e−z +�j+1 +(102) += +λe−z(1 − e−z)x−1 +∞ +� +j=0 +(x − 1)(x − 2)···(x − j) +(j + 1)! j! +� λe−z +1 − e−z +�j +(103) += +λe−z(1 − e−z)x−1 +∞ +� +j=0 +(1 − x)(2 − x)···(j − x) +(j + 1)! j! +� −λe−z +1 − e−z +�j +(104) += +λe−z(1 − e−z)x−1 +∞ +� +j=0 +(1 − x)j +(2)j +�−λe−z +1−e−z +�j +j! +(105) += +λe−z(1 − e−z)x−1 +1F1 +� +1 − x;2; −λe−z +1 − e−z +� +, +(106) +where z is given by (67), 1F1 is the hypergeometric function (also called the Kummer +confluent hypergeometric function of the first kind), defined as +1F1 +� +1 − x;2; −λe−r1t +1 − e−r1t +� += +∞ +� +j=0 +(1 − x)j +(2)j +� −λe−r1t +1−e−r1t +�j +j! +, +and (α)j denotes the rising factorial, i.e. (α)j = α(α+1)(α+2)...(α+j−1) with (α)0 = 1. +The normalizing constant is given by +C += +x +� +j=1 +�x−1 +j−1 +�λj +j! +� ∞ +�k−1 +i=1 τi +e−jz(1 − e−z)x−jdt. +(107) +Let w = e−z. Then, +C += +1 +rk +x +� +j=1 +�x−1 +j−1 +�λj +j! +� θ +0 +wj−1(1 − w)x−jdw += +1 +rk +x +� +j=1 +�x−1 +j−1 +�λj +j! +Bθ(j,x − j + 1), +(108) +where θ is given by (70). +38 + +Therefore, the pdf is given by +P(t|λ,⃗r,⃗τ,x) += +rk(1 − e−z)x �x +j=1 +(x−1 +j−1) +j! +� λe−z +1−e−z +�j +�x +j=1 +(x−1 +j−1)λj +j! +Bθ(j,x − j + 1) += +rkλe−z(1 − e−z)x−1 1F1 +� +1 − x,2, −λe−z +1−e−z +� +�x +j=1 +(x−1 +j−1)λj +j! +Bθ(j,x − j + 1) +, +(109) +Recall that for the single-phase process, θ = 1, so we have +x +� +j=1 +�x−1 +j−1 +�λj +j! +B(j,x − j + 1) += +∞ +� +j=1 +�x−1 +j−1 +�λj +j! +B(j,x − j + 1) += +∞ +� +j=0 +�x−1 +j +�λj+1 +(j + 1)! B(j + 1,x − j) += +∞ +� +j=0 +λj+1 +(j + 1)! += +eλ − 1 +x +, +(110) +implying that the pdf is given by +P(t|λ,r1,x) += +r1xλe−r1t(1 − e−r1t)x−1 1F1 +� +1 − x,2, −λe−r1t +1−e−r1t +� +eλ − 1 +. +(111) +Note that in some cases (eg. when knowledge of the lengths of individual PCR ampli- +fication phases is not of interest), it may be useful to marginalize ⃗τ out of P(t|λ,⃗r,⃗τ,x). +This can be achieved by using the fact that +P(t|λ,⃗r,x) += +� +Ω +P(t,⃗τ|λ,⃗r,x)d⃗τ += +� +Ω +P(t|λ,⃗r,⃗τ,x)P(⃗τ|λ,⃗r,x)d⃗τ, +(112) +where Ω is the domain of ⃗τ. +• Mean +39 + +The mean Ct value is given by +E(t) += +rk +�x +j=1 +(x−1 +j−1)λj +j! +Bθ(j,x − j + 1) +x +� +j=1 +�x−1 +j−1 +�λj +j! +(D) +�������������������������������������������������������� +� ∞ +�k−1 +i=1 τi +te−jz(1 − e−z)x−jdt, +(113) +where z is given by (67). +Let w = e−z. Then, +D +see (81) += +Bθ(j,x − j + 1)�k−1 +i=1 τi +rk ++ Γ(j)2θj 3 ˜F2(j,j,j − x;j + 1,j + 1;θ) +r2 +k +. +(114) +Therefore +E(t) += +�x +j=1 +(x−1 +j−1)λj +j! +� +rkBθ(j,x − j + 1)�k−1 +i=1 τi + Γ(j)2θj 3 ˜F2(j,j,j − x;j + 1,j + 1;θ) +� +rk +�x +j=1 +(x−1 +j−1)λj +j! +Bθ(j,x − j + 1) +. +(115) +Recall that for the single-phase process, θ = θ′ = 1, so +E(t) += +ψ(x + 1) +r1 +− +�x +j=1 +λj +j! ψ(j) +r1 +� +eλ − 1 +� . +(116) +• Variance +The variance is given by E(t2) − E(t)2, where +E(t2) += +rk +�x +j=1 +(x−1 +j−1)λj +j! +Bθ(j,x − j + 1) +x +� +j=1 +�x−1 +j−1 +�λj +j! +(D) +������������������������������������������������������������ +� ∞ +�k−1 +i=1 τi +t2e−jz(1 − e−z)x−jdt, +(117) +where z is given by (67). +40 + +Let w = e−z. Then, +D +see (85) += +� +∂2 +∂n2 + 2 ∂2 +∂n∂x + ∂2 +∂x2 +� +Bθ(n,x − n + 1) +rk3 ++ 2lnθ′Γ(n)2θn 3 ˜F2(n,n,n − x;n + 1,n + 1;θ) +r3 +k ++ +Bθ(j,x − j + 1) +� +�������� +��k−1 +i=1 τi +�2 − +��k−1 +i=1 +riτi +rk +�2 +rk +� +�������� +(118) +where θ′ is given by (82). +Plugging (118) into (117), we obtain +E(t2) += +1 +r2 +k +�x +j=1 +(x−1 +j−1)λj +j! +Bθ(j,x − j + 1) +x +� +j=1 +�x−1 +j−1 +�λj +j! +� +����� +� ∂2 +∂n2 + 2 ∂2 +∂n∂x + ∂2 +∂x2 +� +Bθ(n,x − n + 1) + +2lnθ′Γ(n)2θn +3 ˜F2(n,n,n − x;n + 1,n + 1;θ) + r2 +k Bθ(j,x − j + 1) +� +�������� +� +������ +k−1 +� +i=1 +τi +� +������ +2 +− +� +������ +k−1 +� +i=1 +riτi +rk +� +������ +2� +�������� +� +�����. +. +(119) +For the single-phase process, the variance is given by +Var(t) += +(eλ − 1)�x +j=1 +λj +j! +� +ψ1(j) + ψ(j)2 +� +− +� +����� +�x +j=1 +λj +j! ψ(j) +� +����� +2 +� +r1(eλ − 1) +�2 +− ψ1(x + 1) +r2 +1 +. +(120) +• cdf +The cdf of the Ct value is given by +F(t|λ,⃗r,⃗τ,x) += +rk +�x +j=1 +(x−1 +j−1)λj +j! +� t�k−1 +i=1 τi e−jz′(1 − e−z′)x−jds +�x +j=1 +(x−1 +j−1)λj +j! +Bθ(j,x − j + 1) +, +(121) +where z′ = rks + �k−1 +i=1(ri − rk)τi. +41 + +Let w = e−z′. Then, +F(t|λ,⃗r,⃗τ,x) += +�x +j=1 +(x−1 +j−1)λj +j! +� θ +e−z wj−1(1 − w)x−jdw +�x +j=1 +(x−1 +j−1)λj +j! +Bθ(j,x − j + 1) += +�x +j=1 +(x−1 +j−1)λj +j! +�� θ +0 wj−1(1 − w)x−jdw − +� e−z +0 +wj−1(1 − w)x−jdw +� +�x +j=1 +(x−1 +j−1)λj +j! +Bθ(j,x − j + 1) += +�x +j=1 +(x−1 +j−1)λj +j! +� +Bθ(j,x − j + 1) − Be−z(j,x − j + 1) +� +�x +j=1 +(x−1 +j−1)λj +j! +Bθ(j,x − j + 1) += +1 − +�x +j=1 +(x−1 +j−1)λj +j! +Be−z(j,x − j + 1) +�x +j=1 +(x−1 +j−1)λj +j! +Bθ(j,x − j + 1) +, +(122) +where θ is given by (70). +For the single-phase process, using (110), we simplify the cdf to obtain +F(t|λ,r1,x) += +1 − +x�x +j=1 +(x−1 +j−1)λj +j! +Be−r1t(j,x − j + 1) +eλ − 1 +. +(123) +• Probability density of λ +We conclude by deriving the probability density of λ, P(λ|r1,t,x), for the single-phase +process. This density can be used to estimate λ from measured Ct values, and for +calculating both the LoD and the LoQ of a PCR process. The steps described below +can also be used to derive P(λ|⃗r,t,⃗τ), for a PCR process with an arbitrary number of +phases. +By Bayes’ Theorem, we have +P(λ|r1,t,x) +∝ +�x +j=1 +(x−1 +j−1) +j! +� λw +1−w +�j +eλ − 1 +, +(124) +where w = e−r1t. +42 + +The normalizing constant is given by +C += +� ∞ +0 +�x +j=1 +(x−1 +j−1) +j! +� λw +1−w +�j +eλ − 1 +dλ += +x +� +j=1 +�x−1 +j−1 +� +j! +� +w +1 − w +�j � ∞ +0 +λj +eλ − 1dλ += +x +� +j=1 +�x−1 +j−1 +� +j! +� +w +1 − w +�j +Γ(j + 1)ζ(j + 1) += +x +� +j=1 +�x − 1 +j − 1 +�� +w +1 − w +�j +ζ(j + 1), +(125) +where ζ(j) is the Riemann zeta function. +Therefore, the probability density of λ is given by +P(λ|r1,t,x) += +�x +j=1 +(x−1 +j−1) +j! +� λw +1−w +�j +(eλ − 1)�x +j=1 +�x−1 +j−1 +�� w +1−w +�j ζ(j + 1) += +λw 1F1(1 − x,2, −λw +1−w ) +(eλ − 1)(1 − w)�x +j=1 +�x−1 +j−1 +�� w +1−w +�j ζ(j + 1) +. +(126) +The probability that λ takes values between a and b is given by +P(a ≤ λ ≤ b | r1,t,x) += +�x +j=1 +(x−1 +j−1) +j! +� w +1−w +�j � b +a +sj +es−1ds +�x +j=1 +�x−1 +j−1 +�� w +1−w +�j ζ(j + 1) += +�x +j=1 +�x−1 +j−1 +�� w +1−w +�j [ζb(j + 1) − ζa(j + 1)] +�x +j=1 +�x−1 +j−1 +�� w +1−w +�j ζ(j + 1) +, +(127) +where ζλ(·) is the incomplete Riemann zeta function. +It follows that the cumulative density function of λ is given by +F(λ | r1,t,x) += +�x +j=1 +�x−1 +j−1 +�� w +1−w +�j ζλ(j + 1) +�x +j=1 +�x−1 +j−1 +�� w +1−w +�j ζ(j + 1) +. +(128) +As discussed earlier in relation to P(n|⃗r,t,⃗τ,x), Equation (126) can be interpreted +as follows: Suppose that a Ct value t is produced by a PCR process with expected +number of input molecules λ. If we replace λ by an estimate ˆλ, then (126) gives the +43 + +likelihood of ˆλ. For practical purposes (eg. to determine the LoQ), it is useful to +calculate the probability that ˆλ will be obtained as the estimate of λ from any data t +that can be produced by a PCR process with expected number of input molecules λ. +This probability is given by +P +� ˆλ|λ,r1,x +� += +� ∞ +�k−1 +i=1 τi +P +� ˆλ,t|λ,r1,x +� +dt += +� ∞ +�k−1 +i=1 τi +P +� ˆλ|λ,r1,t,x +� +P (t|λ,r1,x)dt += +� ∞ +�k−1 +i=1 τi +P +� ˆλ|r1,t,x +� +P (t|λ,r1,x)dt += +r1x +� +eλ − 1 +�� +e ˆλ − 1 +� +� ∞ +�k−1 +i=1 τi +(1 − w)x +��x +j=1 +(x−1 +j−1) +j! +� λw +1−w +�j ���x +j=1 +(x−1 +j−1) +j! +� ˆλw +1−w +�j � +�x +j=1 +�x−1 +j−1 +�� w +1−w +�j ζ(j + 1) +dt += +x +� +eλ − 1 +�� +e ˆλ − 1 +� +� θ +0 +(1 − w)x +��x +j=1 +(x−1 +j−1) +j! +� λw +1−w +�j ���x +j=1 +(x−1 +j−1) +j! +� ˆλw +1−w +�j � +w�x +j=1 +�x−1 +j−1 +�� w +1−w +�j ζ(j + 1) +dw, +(129) +where θ is given by (70) and we have assumed that ˆλ is conditionally independent +of λ given t. +It follows that the t-independent probability that ˆλ will take values between a and b +is given by +P(a ≤ ˆλ ≤ b | λ,r1,x) = +� b +a +P +� ˆλ|λ,r1,x +� +d ˆλ. +(130) +44 + +5.2 +Supplementary Figures +Figure 5.1: Limit of detection of the single-phase process. The LoD was determined +while accounting for either sampling noise alone (solid green line), amplification noise +alone (solid red line), or both sampling noise and amplification noise (solid blue line). +It was then plotted versus amplification efficiency, which is expressed on a base-2 scale +as a percentage. The LoD based on sampling noise alone equals 3, whereas the LoD is +highest when accounting for both types of noise. In the latter case, it ranges from 157, +when the efficiency is only 80%, to 6, when the efficiency is 100%. The plot shows a strong +dependence of the LoD on efficiency. +45 + +150 - +100 +Noise type +LoD +Sampling noise only +Amplification noise only +Sampling + amplification noise +50 +0 - +80 +85 +90 +95 +100 +Amplification efficiency (%)Figure 5.2: Ratio of expected versus estimated fraction of positive partitions in digital +PCR. 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Mathematical and Physical Sciences, +186(1007):453–461, 1946. +49 + diff --git a/4tA0T4oBgHgl3EQfNf_3/content/tmp_files/load_file.txt b/4tA0T4oBgHgl3EQfNf_3/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a2da1f8e88ccefcda0d9190f58458f75e3ec6d06 --- /dev/null +++ b/4tA0T4oBgHgl3EQfNf_3/content/tmp_files/load_file.txt @@ -0,0 +1,1112 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf,len=1111 +page_content='Stochastics of DNA Quantification Abdoelnaser M Degoot and Wilfred Ndifon∗ African Institute for Mathematical Sciences, Next Einstein Initiative, Rwanda January 6, 2023 1 Abstract A common approach to quantifying DNA involves repeated cycles of DNA amplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' This approach, employed by the polymerase chain reaction (PCR), produces outputs that are corrupted by amplification noise, making it challenging to accurately back-calculate the amount of input DNA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Standard mathematical solutions to this back-calculation prob- lem do not take adequate account of such noise and are error-prone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Here, we develop a parsimonious mathematical model of the stochastic mapping of input DNA onto experi- mental outputs that accounts, in a natural way, for amplification noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' We use the model to derive the probability density of the quantification cycle, a frequently reported exper- imental output, which can be fit to data to estimate input DNA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Strikingly, the model predicts that a sample with only one input DNA molecule has a <4% chance of testing positive, which is >25-fold lower than assumed by a standard method of interpreting PCR data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' We provide formulae for calculating both the limit of detection and the limit of quan- tification, two important operating characteristics of DNA quantification methods that are frequently assessed by using ad-hoc mathematical techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Our results provide a math- ematical foundation for the rigorous analysis of DNA quantification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' 2 Introduction The quantification of genomic targets is of interest in a large variety of applications in biology, biotechnology and medicine, from determining an individual’s disease status to detecting minute changes in gene expression profiles occurring across space and time (eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' [1, 2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' This is typically achieved by converting non-DNA genomic targets into DNA, which is then amplified to enable its quantification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' In principle, this allows even small numbers ∗Address for correspondence: wndifon@aims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='za 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='02149v1 [q-bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='QM] 5 Jan 2023 of genomic targets to be accurately measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' However, in practice, the DNA amplifica- tion process, being stochastic, generates outputs that contain noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Accurate measure- ment, therefore, requires an adequate, quantitative understanding of this noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Thus far, this has proved challenging to achieve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' A specific and very popular instance of a DNA quantification method is the real-time polymerase chain reaction (PCR) [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' In PCR, DNA molecules are repeatedly amplified in a cyclic manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' As they are amplified, fluorescently labeled nucleotides are incorpo- rated into the newly formed DNA molecules, increasing the overall fluorescence emitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' The resulting fluorescence profile is used to determine the quantification cycle (denoted Cq or Ct value), at which the number of molecules exceeds a defined threshold, called the quantification threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' A PCR reaction is considered to be positive if its Ct value is less than or equal to the maximum possible cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Despite the fact that the Ct value is only an indirect readout of the number of input DNA molecules, it is often the only re- ported output of PCR experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' A variant of conventional PCR, called digital PCR [5], uses the fraction of positive reactions to estimate the number of input DNA molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' To this end, it assumes that a reaction is positive if and only if it contains at least one target molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' It is unclear under what conditions this assumption is valid, and when it must be discarded in favor of a more realistic alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Here we describe a parsimonious mathematical model that is useful for analysing the DNA quantification process, and for guiding the interpretation of experimental outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' We use PCR as an example, although our analysis is applicable to other methods such as loop-mediated isothermal amplification of DNA [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Experiments indicate that the PCR process exhibits different phases, characterized by different efficiencies of DNA amplifi- cation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Therefore, we construct a mathematical model of a PCR process with an arbitrary number of phases, each with its own amplification efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' We use this model to obtain the following results: We derive the generating function for the probability distribution of the number of molecules found in a PCR experiment at an arbitrary time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' We also derive the probability density function (pdf), mean, variance, and cumulative density function (cdf) of the Ct values produced by such an experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Either the pdf or the cdf can be fit to PCR data to estimate the number of input DNA molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' In the simplest instance of our model – a single-phase PCR model that accounts for amplification noise but not for (upstream) DNA sampling noise – the mean Ct value, given by (ψ(x +1)−ψ(n))/r, is well approximated by ln(x/n)/r [7] when n ≫ 1, where n is the number of input molecules, r (defined on a base-e scale) is the amplification efficiency, x is the quantification threshold, and ψ(·) denotes the digamma function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' We provide a formula for calculating the limit of detection (LoD) of a PCR experi- 2 ment, that is, the smallest number of input molecules that can be detected with a failure rate not exceeding α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Using a single-phase PCR model, we find that when α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='05, the LoD increases from 3, the value determined while accounting for sam- pling noise only, to ≈10 when both sampling noise and amplification noise (with r set to 95% of the maximum possible efficiency, m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=') are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' The LoD increases as r decreases, doubling to ≈20 at 90% m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' This illustrates the under- appreciated, dramatic effect that amplification efficiency has on the LoD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' We provide a formula for calculating the limit of quantification (LoQ) of a PCR ex- periment, that is, the smallest number of molecules that can be quantified with a defined level of precision and a given maximum failure rate α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Counter-intuitively, the single-phase PCR model predicts that the LoQ does not depend on amplifica- tion efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' When α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='05, the LoQ increases from 10, obtained when up to a two-fold deviation from the expected number of input molecules is allowed, to 820, when at most a 10% deviation is allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' This indicates that 10 or fewer molecules cannot be measured with a better than 2-fold error more than 95% of the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' The model indicates that a key assumption commonly used when interpreting digital PCR data – that a PCR experiment with only one input molecule will always produce a positive outcome – is invalid under a wide range of conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Even when the amplification efficiency is set to a high value of 95% m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='e, the probability that such an experiment will yield a positive outcome is predicted to be <4%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' We describe two different approaches by which accurate estimates of the number of input DNA molecules may be obtained from digital PCR data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' It should be noted that there have been previous attempts to improve the interpretation of PCR data through mathematical modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' The classical approach to estimating the amount of DNA found in a focal sample involves comparing data generated by that sample versus data obtained from a reference sample containing either a known or an unknown amount of DNA [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' The need for a reference sample with a known amount of DNA, the determination of which is itself subject to experimental error, makes accurate absolute quantification of DNA found in the focal sample challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' An alternative approach involves fitting mathematical models, mostly phenomenological in their construction, to PCR data generated by the focal sample alone [4, 8, 9, 10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' See [12] for a comparison of various methods based on this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' None of these methods provides an adequate accounting of how amplification noise shapes PCR data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' The remainder of this paper is organized as follows: We provide an overview of the model’s structure in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='1 and present our main mathematical results in Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' We apply these results to compute the LoD and LoQ in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='4, and we investigate how amplification noise complicates the accurate interpretation of digital 3 PCR data in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' We summarize the results and discuss other applications of our methods in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' To improve readability, we only present mathematical proofs and detailed calculations in the Appendix (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' 3 Results 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='1 Preliminaries We model the PCR process as a continuous-time, discrete-state Markov jump process [13] evolving up to time T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' This representation of the PCR process is based on the facts that (1) the primary products of PCR reactions, DNA molecules, are countable, and (2) what happens in the next cycle of the reaction is conditionally independent of what happened in the past given the present state of the reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Our decision to make time continuous (rather than discrete) is based on the fact that experimentally measured Ct values are positive real numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' As a consequence, reaction rates are defined in base e instead of base 2 (expected for a discrete-time PCR process), but it is straightforward to convert between these two bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' We divide the time interval [0,T ] of the PCR process into p non-overlapping subin- tervals Ii, each one corresponding to a distinct phase of the process and associated with the probabilistic state transition rate ri, i = 1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=',p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' These transition rates govern the ef- ficiency of DNA amplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' We derive the probability generating function [14] for the number of target molecules found at an arbitrary time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' We use this generating func- tion to derive the corresponding probability distribution and, importantly, the probability density function (pdf) of the Ct value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' We derive the pdf in two different cases, namely 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' when the initial state of the PCR process is deterministic, and the PCR phase lengths and amplification efficiencies are given;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' when the initial state is Poisson-distributed, and the phase lengths and amplification efficiencies are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' To illustrate the mathematical ideas, we will report calculations and simulations based on a single-phase model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' We argue that this simpler instance of our model is sufficient for analysing a large variety of real-world PCR experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' In principle, each PCR ex- periment can be divided into the following three amplification rate-dependent phases: a pre-exponential phase, in which the amplification rate is sub-exponential;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' an exponential phase;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' and a post-exponential phase where the rate slows down as DNA molecules saturate the reagents required for their further amplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' However, in practice, the usual out- put of PCR experiments – the Ct value – is determined as soon as the PCR process enters the exponential phase, meaning that dynamics occurring in the pre-exponential phase pri- marily determine this particular outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Therefore, for the purposes of understanding 4 the factors that shape the Ct value and its statistics, and evaluating related operating char- acteristics of PCR, a single-phase model appears sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Accordingly, when applicable, we highlight the forms taken by our mathematical equations in the case of a single-phase model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' In addition, we estimate the LoD and LoQ using a single-phase model (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='4), which we also apply to critique the standard method of interpreting digital PCR data (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='2 Case 1: A PCR process with a deterministic initial state 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='1 Probability generating function for the number of molecules Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Let {X(t),t ∈ R} be a continuous-time Markov process with p phases, a countable state space S ⊂ N+, phase-specific transition rates ri, i ∈ 1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=',p, and state transition proba- bility given by P (X(t′ + ∆t) = x|X(t′) = x′) = δ(x′ − x + 1) p � i=1 ri1Ii(t′), (1) where 1 denotes the indicator function and δ(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=') denotes the Kronecker delta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' If the pro- cess starts with n molecules, then the probability generating function for the number of molecules present at time t ∈ Ik, k ≤ p, is given by G(n,⃗r,t,⃗τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='s) = � se−z 1 − s(1 − e−z) �n , (2) where z = rkt + k−1 � i=1 (ri − rk)τi, (3) Ii denotes the i’th phase and τi = |Ii|, i < k, is its length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' The proof of this theorem is given in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' We will now use the theorem to derive the probability distribution of the number of molecules found at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='2 Probability distribution of the number of molecules Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' The probability that there are x molecules at time t ∈ Ik in the PCR process de- scribed in Theorem 1 is given by the following negative binomial distribution: P(x|n,⃗r,t,⃗τ) = �x − 1 n − 1 � e−nz × (1 − e−z)x−n , (4) where z is given by (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' The proof of this corollary is given in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' We will now use this corollary to derive the pdf, mean, variance and cdf of the Ct value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='3 pdf, mean and variance of the Ct value Let t be the Ct value of the PCR process described in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' By definition, t is the time at which the number of molecules reaches the quantification threshold, which we denote by x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Let t ∈ Ik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' In the Appendix [Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='5], we show that, given n,⃗r = (r1,r2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=',rk−1), and ⃗τ = (τ1,τ2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=',τk−1), the pdf of t has the following form: P(t|n,⃗r,⃗τ,x) = rke−nz (1 − e−z)x−n Bθ(n,x − n + 1) , (5) where Bθ(n,x − n + 1) is the incomplete Beta function, z is given by (3), and θ = e−�k−1 i=1 riτi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (6) For the single-phase PCR process, θ = 1, so the pdf is given by P(t|n,r1,x) = r1e−nz (1 − e−z)x−n B(n,x − n + 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (7) The mean Ct value is given by (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='5) E(t) = k−1 � i=1 τi + Γ(n)2θn 3 ˜F2(n,n,n − x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='n + 1,n + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='θ) rkBθ(n,x − n + 1) , (8) where 3 ˜F2(n,n,n − x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='n + 1,n + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='θ) is the regularized generalized hypergeometric function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' For the single-phase process, the mean is given by E(t) = ψ(x + 1) − ψ(n) r1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (9) Observe that when n ≫ 1, the right-hand-side of (9) is well-approximated by ln(x/n)/r1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' The latter expression is commonly used to approximate the mean Ct value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' For example, it was used in [7] to estimate PCR amplification efficiency from data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' The variance of the Ct value is given by E(t2) − E(t)2, where E(t2) is given by (85).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' For the single-phase process, the variance is given by Var(t) = ψ1(n) − ψ1(x + 1) r2 1 , (10) where ψ1(·) is the second polygamma function (also called the trigamma function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Finally, the cdf of the Ct value is given by [see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='5] F(t|n,⃗r,⃗τ,x) = 1 − Be−z(n,x − n + 1) Bθ(n,x − n + 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (11) 6 For the single-phase process, the cdf is given by F(t|n,r1,x) = 1 − Ie−r1t(n,x − n + 1), (12) where Ie−r1t(n,x − n + 1) = Be−rt (n,x − n + 1)/B(n,x − n + 1) is the regularized incomplete Beta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Sampling from this cdf is relatively straightforward: A random Ct value t is obtained as follows: t = −lnI−1 1−u(n,x − n + 1) r1 , (13) where u is sampled uniformly at random from the interval (0,1) and I−1 1−u is the inverse of the regularized incomplete Beta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' To find a Ct value that corresponds to a quantile q ∈ (0,1), q is substituted for u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' To estimate n, either the pdf or the cdf of t can be fit to data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Alternatively, the posterior density of n conditioned on t can be computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' In Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='5, we show that, for the single-phase process, it is given by P (n|r1,t,x) = e−(n−1)r1t(1 − e−r1t)x−n xB(n,x − n + 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (14) In Figure 1, we illustrate the shape of the single-phase pdf for different numbers of input molecules and different amplification efficiencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' To this end, we set T = 35 (a com- mon upper-bound for the duration of real-world PCR experiments) and x = 2T , which is equal to the number of molecules expected after T cycles under perfect amplification con- ditions (a sample that contains only one, perfectly amplified input molecule is expected to reach the quantification threshold, x, at time t ≤ T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' We vary the efficiency from 60% m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (equivalent to setting r1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='6 × ln2) to 100% m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' The pdf has a bell shape, the location and width of which are governed by both the efficiency and the number of input molecules (Figure 1, left panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Higher efficiencies or larger numbers of input molecules produce smaller mean Ct values, smaller variances, and narrower pdfs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' In contrast, lower efficiencies or smaller numbers of input molecules produce larger mean Ct values, larger variances, and wider pdfs (Figure 1, left panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' In fact, Equation (5) predicts that in the limit as the efficiency goes to 0, the pdf will become flat as it will map every Ct value to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='3 Case 2: A PCR process with a Poisson-distributed initial state 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='1 Probability generating function for the number of molecules Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Let {X(t),t ∈ R} be the continuous-time Markov process described in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' If, instead of starting with a precisely known number of input DNA molecules, the initial state of the process is Poisson-distributed with mean λ, then the probability generating function for 7 Figure 1: pdf of the quantification cycle for the single-phase process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Processes with either a deterministic (left panel) or a Poisson-distributed (right panel) initial state were considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' The pdf was calculated using Equation (5) for the former case, and Equation (18) for the latter case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' The quantification threshold, x, was set to 235.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' The mean µ and variance σ2 corresponding to different amplification efficiences r are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' For ease of comprehension, r, which in our model is defined on a base-e scale, is shown as a percentage of its maximum possible value of ln(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' 8 n =1 入 =1 r=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='0,μ=35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='83,α2=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='424 r=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='0,μ=35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='12,o2=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='257 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='0010- r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='9,μ=39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='81,α2=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='227 r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='9,μ=39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='02,2=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='021 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='2- =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='8,μ=44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='79,2=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='35 r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='8,μ=43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='9,g2=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='089 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='7,μ=51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='19,02=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='987 r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='7,50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='17,o2=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='647 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='6,μ59/72,2=9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='51 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='6，=58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='54,2=9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='048 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='0005- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='1- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='0000 0.' metadata={'source': 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+page_content='53,2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='087 r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' μ=47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='27, 2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='958 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='0- r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='6,μ=47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='28,o2=q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='118 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='5- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='000- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='0 30 35 40 45 30 35 40 45 Cycles Cyclesthe state of the process at a future time t ∈ Ik is given by G(λ,⃗r,t,⃗τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='s) = e λ(s−1) 1−s(1−e−z) , (15) where z is given by (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' The proof of Theorem 2 is given in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' We now use Theorem 2 to derive the probability distribution of the number of molecules found in the PCR process at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='2 Probability distribution of the number of molecules Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' The probability that there are x molecules at cycle t ∈ Ik in the PCR process described in Theorem 2 is given by: P(x|λ,⃗r,t,⃗τ) = e−λ (1 − e−z)x x � i=1 �x−1 i−1 � i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' � λe−z 1 − e−z �i , (16) where z is given by (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' The proof of this corollary is provided in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' It is interesting to note that from the proof emerged the following combinatorial triangle, which is related to the well- known Narayana triangle [15]: x 1 1 2 1 2 3 1 6 6 4 1 12 36 24 5 1 20 120 240 120 1 2 3 4 5 k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' The entries of this triangle, given by T(x,k) = � x k − 1 ��x − 1 k − 1 � (k − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=', x ∈ Z+,k = 1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=',x, (17) count the number of ways of obtaining x molecules by replicating a randomly selected subset of k molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' T (x,k) is related to the Narayana numbers N(x,k) by T (x,k) = k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' N(x,k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' We will now use this corollary to derive the pdf of the Ct value together with the mean, variance and cdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='3 pdf, mean and variance of the Ct value Let t be the Ct value of the PCR process described in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' As noted earlier, the Ct value t is the time at which the number of DNA molecules found in the process reaches the quantification threshold, which we denote by x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' In the Appendix [Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='6], we show that the pdf of t is given by P(t|λ,⃗r,⃗τ,x) = rkλe−z(1 − e−z)x−1 1F1 � 1 − x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' −λe−z 1−e−z � �x j=1 (x−1 j−1)λj j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Bθ(j,x − j + 1) , (18) where θ is given by (6), z is given by (3), 1F1(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='c) is the hypergeometric function (also called the Kummer confluent hypergeometric function of the first kind).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' For the single-phase process, the pdf is given by [see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='6] P(t|λ,r1,x) = r1xλe−r1t(1 − e−r1t)x−1 1F1 � 1 − x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' −λe−r1t 1−e−r1t � eλ − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (19) The mean Ct value is given by [see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='6] E(t) = �x j=1 (x−1 j−1)λj j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' � rkBθ(j,x − j + 1)�k−1 i=1 τi + Γ(j)2θj 3 ˜F2(j,j,j − x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='j + 1,j + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='θ) � rk �x j=1 (x−1 j−1)λj j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Bθ(j,x − j + 1) , (20) while the second moment is given by (119).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' For the single-phase process, the mean and variance are, respectively, given by [see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='6] E(t) = ψ(x + 1) r1 − �x j=1 λj j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' ψ(j) r1 � eλ − 1 � and (21) Var(t) = � eλ − 1 ��x j=1 λj j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' � ψ1(j) + ψ(j)2 � − ��x j=1 λj j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' ψ(j) �2 � r1(eλ − 1) �2 − ψ1(x + 1) r2 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (22) The cdf of the Ct value is given by [see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='6] F(t|λ,⃗r,⃗τ,x) = 1 − �x j=1 (x−1 j−1)λj j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Be−z(j,x − j + 1) �x j=1 (x−1 j−1)λj j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Bθ(j,x − j + 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (23) 10 For the single-phase process, the cdf is given by F(t|λ,r1,x) = 1 − x�x j=1 (x−1 j−1)λj j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Be−r1t(j,x − j + 1) eλ − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (24) To estimate λ, either the pdf or the cdf of t can be fit to data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Alternatively, the posterior density of λ conditioned on t can be computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' In Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='6, we show that, for the single-phase process, it is given by P(λ|r1,t,x) = λw 1F1(1 − x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' −λw 1−w ) (eλ − 1)(1 − w)�x j=1 �x−1 j−1 �� w 1−w �j ζ(j + 1) , (25) where w = e−r1t and ζ(·) denotes the Riemann zeta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Note that, because they result from calculating expectations over the Poisson distribu- tion, the summations found in Equations (18) - (25) can be truncated at any value of j ≫ λ without a loss of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' In Figure 1, we illustrate the shape of the single-phase pdf for different values of λ and different amplification efficiencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' As was the case for the PCR process with a deter- ministic initial state (Figure 1, left panel), the pdf also has a bell shape (Figure 1, right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Its location and width are governed by both the efficiency and λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Consistent with expectations, higher efficiencies or larger values of λ produce smaller mean Ct values, smaller variances, and narrower pdfs (Figure 1, right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' In contrast, lower efficiencies or smaller values of λ produce larger mean Ct values, larger variances, and wider pdfs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='4 Limit of detection and limit of quantification We will now demonstrate theoretically how the mathematical framework we have devel- oped can be applied to achieve certain operationally important objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' In particular, it is often of interest to quantify the limit of detection (LoD) of a particular instance of the PCR method (henceforth referred to as “PCR protocol”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' The LoD of a PCR protocol is the smallest number of molecules that it can detect with a failure rate not exceeding a defined threshold α (α is also called the significance level).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Protocols with smaller LoDs are in general preferred to those with larger LoDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Ideally, the LoD should be either equal to or smaller than the number of input DNA molecules expected in the considered sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Another important operational objective is to determine a PCR protocol’s limit of quan- tification (LoQ) – i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' the smallest number of molecules that it can estimate with a given level of precision (measured here using the parameter β) and a given maximum failure rate α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' When a ≥ β-fold change in the number of target molecules needs to be detected, the protocol should have an LoQ with precision ≤ β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' The methods available for estimat- ing LoD and LoQ are laborious [16] and frequently rely on certain ad-hoc mathematical 11 approximations [16, 17], which we would like to circumvent by developing and executing mathematically precise statements of the estimation problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' We begin with the LoD estimation problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' For the PCR process with a deterministic initial state, the LoD can be expressed as follows: LoD = min n s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' F(T |n,⃗r,⃗τ,x) > 1 − α, (26) where F(T|n,⃗r,⃗τ,x) is given by (11) and T is the maximum practical duration of the PCR process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' For the process with a Poisson-distributed initial state, n is replaced by λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' In Supplementary Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='1, we show how the LoD varies with amplification effi- ciency in a single-phase process with either a deterministic or a Poisson-distributed initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' In the former case, the process contains amplification noise but no sampling noise while in the latter case it contains both sampling noise and amplification noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' For com- parison, we also show the LoD in a process with sampling noise, modeled by using the Poisson distribution, but without amplification noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' The LoD is lowest in a process with sampling noise alone (LoD = 3 molecules) and it is highest when both sampling noise and amplification noise are present (LoD ranges from 6 molecules, at 100% of maximum pos- sible efficiency or m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='e, to 157 molecules, at 80% m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' A process with amplification noise but without sampling noise has an intermediate LoD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' In these computational exam- ples, the parameters of the equation used to estimate the LoQ are perfectly known, and this makes it possible to obtain perfect knowledge of the LoD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' In real-world applications, the parameter values will be associated with uncertainty, which will, in a quantifiable way, make uncertain the LoD estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' We now turn our attention to the problem of estimating the LoQ in a PCR process with a deterministic initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Suppose that a Ct value t is generated by such a process and then used to obtain an estimate, denoted ˆn, of the number of input molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Let n be the actual number of input molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' We want to calculate the probability that, for any data t generated by the same process, ˆn will not differ from n by more than a factor β,β ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' We define the LoQ as the smallest value of n for which this probability exceeds 1 − α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Specifically, the LoQ is given by LoQ = min n s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' P ��n/β� ≤ ˆn ≤ �βn� | n,⃗r,⃗τ,x � > 1 − α, (27) where ⌊v⌋ (respectively ⌈v⌉) denotes the largest (respectively smallest) integer less than (respectively greater than) or equal to v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Focusing on the single-phase process, to obtain P (�n/β� ≤ ˆn ≤ �βn� | n,r1,x), we marginal- ize the right-hand-side of (14) with respect to t and then take the sum from �n/β� to �βn�, 12 yielding [see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='5] P (�n/β� ≤ ˆn ≤ �βn� | n,r1,x) = ⌈βn⌉ � ˆn=⌊n/β⌋ B( ˆn + n − 1,2x − ˆn − n + 1) xB( ˆn,x − ˆn + 1)B(n,x − n + 1) = P (�n/β� ≤ ˆn ≤ �βn� | n,x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (28) Strikingly, while Equation (28) depends on both n and x, it does not depend on the ampli- fication efficiency r1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' In the Appendix [see Equation (130)], we follow a similar procedure to obtain P ��λ/β� ≤ ˆλ ≤ �βλ� | λ,r1,x � , the probability that, for any data t generated by a single-phase PCR process with a Poisson-distributed initial state, the estimated value ofλ, denoted ˆλ, will not differ from the actual value by more than a factor β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Setting α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='05 and allowing at most a 10% deviation of ˆn from n (corresponding to setting β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='1) results in an LoQ of 820 molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' The LoQ decreases to 146 molecules when a deviation of up to 25% from expectation is allowed (corresponding to β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='25), and to 43 molecules when the allowable deviation increases to 50% (corresponding to β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' The analysis suggests that at the considered 5% failure rate, 10 input molecules can be detected with an error of at least ≈200%, that is, a ≈ 2 fold deviation from n (corresponding to β = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Because these calculations do not account for sampling noise, they provide only a lower-bound for the LoQ that is achievable at the considered failure rate and level of precision (β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' As noted earlier, in real-world applications the parameters of the equation used to estimate LoQ will be imperfectly known, and this will determine the amount of uncertainty associated with the estimated LoQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='5 Amplification noise determines digital PCR outcomes As noted earlier, digital PCR is a variant of conventional PCR that was developed to im- prove the quantification of DNA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' In digital PCR, a sample master mix (containing an un- known number of input DNA molecules together with all the reagents required for DNA replication) is uniformly distributed into hundreds (and sometimes thousands) of phys- ical partitions, which may take the form of droplets or microwells [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Each partition is expected to receive zero, one, or more DNA molecules following a Poisson distribution with mean λ = CV /D, where C is the concentration of the DNA in the original sample, V is the partition volume, and D is the (known) dilution factor applied to the sample during preparation of the mastermix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' PCR reactions are independently and simultaneously run inside each partition, and positive partitions are identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' The resulting data – i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' the positive or negative outcome of each PCR reaction – are thus digital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' The standard method of interpreting these data assumes that partitions that receive at least one target molecule will test positive, and their fraction, ˆf , is approximated by ˆf ≈ 1 − e−λ, from which λ is estimated as ˆλ = −ln(1 − ˆf ) and then used to estimate C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' 13 Figure 2: Under-estimation of λ by the standard method of interpreting digital PCR data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' We varied both the amplification efficiency and the amount of input DNA λ and used Equation (29) to estimate λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' The resulting estimate, denoted ˆλ, was plotted against efficiency, expressed as a percentage of the maximum possible efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' At efficiencies lower than 95%, only a very small amount of the input DNA is detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' At 95% efficiency, the amount detected ranges from 12%, when λ = 1, to 69%, when λ = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' The amount detected increases to ≈80% when the efficiency equals 100%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' However, according to our model, the assumption that the fraction of positive parti- tions equals the Poisson probability that a partition receives one or more target molecules is untenable due to the effects of PCR amplification noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Indeed, setting the amplifi- cation efficiency to a reasonably high value of 95% m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='e (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' r1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='95 ln2) and us- ing T = 35,x = 2T in Equation (24), we predict that <4% of partitions that contain only one molecule will test positive, which is >25-fold smaller than assumed by the Poisson method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' In Supplementary Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='2, we compare the fraction of positive partitions cal- culated using our model [Equation(24)] versus the positive fraction calculated by the Pois- son method, for different values of λ and different amplification efficiencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' We find that the Poisson method over-estimates the fraction of positive partitions for small values of λ, including the value (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='61) at which the method is expected [18] to produce its most precise estimates of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Only for a relatively large value of λ (10) do we find the Poisson method’s estimate of the fraction of positive partitions to agree with the noise-adjusted expectation calculated using our model (Supplementary Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' We use our model to investigate how this over-estimation of the fraction of positive partitions affects the accuracy of the estimate of λ (denoted ˆλ) produced by the Poisson 14 入=1 入 =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='8- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='0- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='4- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='2- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='0 80 85 90 95 100 80 85 90 95 100 > 入=10 入=100 6- 80 5 60 4 3- 40 2- 20 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' 80 85 90 95 100 80 85 90 95 100 Amplificationefficiency,r(%)method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Setting t = T in Equation (24), we find that ˆλ = −ln � 1 − ˆf � = ln � eλ − 1 �x j=1 (x j) (j−1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='λjBe−r1T (j,x − j + 1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (29) According to Equation (29), ˆλ depends strongly on the amplification efficiency r1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' ˆλ equals 0 in the limit as r1 tends to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' As r1 increases to its maximum possible value of ln(2), ˆλ also increases, approaching λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' In Figure 2, we illustrate the relationship between ˆλ/λ and amplification efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Strikingly, for efficiencies lower than 95% m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='e, ˆλ/λ is very small, indicating that λ is markedly under-estimated by the Poisson method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' When the efficiency equals 95% m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='e, ˆλ/λ increases from ≈ 12% (at λ = 1) to ≈69% (at λ = 100).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Increasing the efficiency to the maximum possible value of 100% m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='e causes ˆλ/λ to increase to ≈80% (at λ = 100).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' These results indicate that the Poisson method is expected to under-estimate λ because it does not account for amplification noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Indeed, experimental data show a strong tendency by the method to under-estimate the number of input DNA molecules (eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' see Supplementary Table 6 in [19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' 4 Discussion The outputs of DNA quantification experiments, including those based on the polymerase chain reaction (PCR), tend to vary within and across different experimental instances, making the results difficult to interpret and limiting their utility beyond the particular contexts in which they are generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Indeed, various factors are known to contribute to the variability of PCR outputs [20, 21, 22] including the varying complexity of DNA templates and the random distribution of target molecules in the reaction environment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' the type of PCR machine and buffer components used;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' the durations and temperatures of the three thermal cycles of PCR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' the binding kinetics of oligonucleotide primers to target DNA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' and the stability of DNA polymerase and other PCR reagents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Taylor et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' al [22] reviewed the sources of variability in PCR experiments and proposed a stepwise process to minimize such variability in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' A common output of a PCR experiment is the quantification cycle (denoted Ct or Cq value), the PCR cycle at which the number of DNA molecules exceeds a defined threshold, called the quantification threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' The Ct value varies with both the number of input DNA molecules and the PCR amplification efficiency, which in turn varies with the afore- mentioned experimental variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' It is desirable to deconvolute such variable outputs to estimate the number of input DNA molecules, which is of greatest interest in experiments, by applying mathematical methods that account for the stochasticity that is inherent in the 15 underlying generative process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' We have developed a mathematical approach to modeling DNA quantification that takes into account the underlying stochasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' We used PCR, the most widely used class of DNA quantification process, to illustrate our mathematical ideas, which are also ap- plicable to a broader class of such processes (eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' [6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Using the model, we derived the probability generating function for the number of molecules found in a PCR process with either a deterministic or a Poisson-distributed number of input molecules as well as the probability density function (pdf), mean, variance and cumulative density function (cdf) of the Ct value produced by such a process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' In contrast to the deterministic case, in which PCR outputs are contaminated only by amplification noise, in the latter case the outputs also contain sampling noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' The equations we derived for these important statistical prop- erties of the PCR process revealed functional relationships between the Ct value and un- derlying variables that could previously only be accessed by empirical means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' To illustrate our mathematical ideas, we focused on the single-phase PCR process, for which our modeling results take relatively simple mathematical forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' We found that the common assumption that the mean Ct value is a simple logarithmic function of the number of input DNA molecules n is correct when n is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' For small n, corrections are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' An exact mean Ct value is given by (ψ(x + 1) − ψ(n))/r, where x is the quan- tification threshold, r is the amplification efficiency and ψ(·) denotes the first polygamma function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' The variance has the elegant form (ψ1(n) − ψ1(x + 1))/r2, where ψ1(·) denotes the second polygamma function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Therefore, the variance is strongly dependent on amplifica- tion efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' This effect is illustrated in Figure 1, which shows that the pdf of the Ct value becomes wider as the amplification efficiency decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Interestingly, in this simple case, the coefficient of variation of the Ct value (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' the ratio of the standard deviation to the mean) does not depend on amplification efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Two important numbers that characterize the performance of a PCR process are the limit of detection (LoD) and the limit of quantification (LoQ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' The LoD is the smallest number of molecules that can be detected with a failure rate not exceeding a threshold α, while LoQ is the smallest number of molecules that can be quantified with a given level of precision (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' allowing a defined maximum fold deviation from the true value) and a given maximum failure rate α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' We provided mathematical formulae for calculating both LoD and LoQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Close examination of these formulae in the context of a single-phase PCR process revealed that a small reduction of the amplification efficiency may cause a large increase of LoD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' For example, when α = 5%, reducing the efficiency from 95% of the maximum possible efficiency (m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='e) to 90% m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' caused the LoD to double, from ≈10 input molecules to ≈20 molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' In contrast to LoD, we found that LoQ is independent of efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Allowing up to a 2-fold difference between n and its estimate results in an LoQ of ≈10 molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Reducing the allowed fold difference to 10% increases LoQ to 820 16 molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Our methods may be used to improve significantly the current approaches to estimating LoD and LoQ, which are laborious [16] and frequently rely on certain crude mathematical approximations [16, 17] that can be avoided by using our methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Furthermore, we applied our methods to shed light on the effects that amplification noise has on estimates of the expected number of input DNA molecules λ obtained by the standard method of interpreting digital PCR data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' A key assumption of this method is that a PCR reaction will be positive if it contains at least one input DNA molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' We showed that this assumption is in general invalid because of the stochastic nature of PCR amplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Stochastic effects are particularly large when λ is small, which is the regime in which digital PCR preferentially operates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' At a high amplification efficiency of 95% m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='e, we find that the ratio of the fraction of positive digital PCR reactions calculated by the standard method versus the value obtained after accounting for amplification noise is only 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='3% when λ = 1 and it increases to ≈100% when λ = 10 (Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Accordingly, stochastic effects were found to cause a significant under-estimation of λ by the standard method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Indeed, at a high efficiency of 95% m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=', the standard method is predicted to under-estimate λ by factors of ≈8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='1, ≈3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='1, and ≈1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='4 when λ equals 1, 10, and 100, respectively (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' This is in the same range as empirically observed (eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' [19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Using our mathematical methods, the following two different approaches may be used to obtain much more accurate estimates of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Firstly, if Ct values are available from pos- itive digital PCR reactions, then Equation (19) can be fit to those Ct values, using either a likelihood-based or a Bayesian statistical approach, to estimate the most probable value of λ together with a confidence (or credible) interval for it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Secondly, if only binary (ie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' positive or negative) outcomes are available from individual reactions, then the variability of such outcomes can still be exploited to estimate λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Specifically, suppose there are N different reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' These can be randomly distributed into groups of N′ reactions each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Assuming a binomial distribution of the number of positive reactions found in each group, their first and second moments are given by F(T )N′ and F(T)N′ (1 + F(T )(N′ − 1)), respec- tively, where F(T ) is calculated using (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' These moments contain information about the two free parameters of F(T ) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' λ and r1), which can be readily extracted to estimate λ together with a confidence (or credible) interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' The Ct value is estimated from the fluorescence profiles produced by DNA molecules as they are amplified during PCR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Our mathematical analysis can be straightforwardly ex- tended to obtain a time-dependent probability density of the fluorescence intensity Pt(y), which can then be fit to fluorescence profiles as an alternative approach to estimating the number of input DNA molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Using standard results from probability theory (eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' see [23]), Pt(y) can be derived from both the cumulative distribution function of the number of molecules found in the PCR process at time t, Ft(x) [calculated based on Equation (4) or (16)] and the linear relation expected [24] between y and x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Specifically, Pt(y) can be 17 expressed as Pt(y) = 1 α ht(g−1(y)), (30) where ht(x) = d dxFt(x), (31) y = αx + β = g(x), and α,β > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' We will explore in detail this alternative approach to estimating the number of input DNA molecules in a future paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' 5 Supporting Information 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='1 Appendix This section contains mathematical proofs and detailed calculations supporting the results presented in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='1 Proof of Theorem 1 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' We will prove Theorem 1 by mathematical induction on k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' k = 1: The Chapman-Kolmogorov forward equation corresponding to the single-phase pro- cess is given by: ∂P(X = x,t|X = x′,t′) ∂t = r1(x−1)P(X = x−1,t|X = x′,t′)−r1xP(X = x,t|X = x′,t′), (32) where we have set t = t′ +∆t, and r1 is the amplification efficiency associated with the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' To simplify our notation, we will abbreviate P(X = x,t|X = x′,t′) by P(x,t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' We will solve (32) by using a powerful combinatorial device called the probability generating function (pgf) [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Recall that the pgf of P(x,t) is defined as: G(s,t) = ∞ � x=0 sxP(x,t), where s is a book-keeping variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' 18 Multiplying both sides of (32) by sx and summing over all possible values of x yields: ∞ � x=0 sx ∂P(x,t) ∂t = r1 ∞ � x=0 (x − 1)sxP(x − 1,t) − r1 ∞ � x=0 xsxP(x,t) = r1s2 ∞ � x=0 (x − 1)sx−2P(x − 1,t) − r1s ∞ � x=0 xsx−1P(x,t) = = r1s � ������s ∞ � x=0 (x − 1)sx−2P(x − 1,t) − ∞ � x=0 xsx−1P(x,t) � ������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (33) Using ∂G(s,t) ∂s = ∞ � x=0 xsx−1P(x,t) and ∂G(s,t) ∂t = ∞ � x=0 sx ∂P(x,t) ∂t , (34) we simplify (33) to obtain ∂G(s,t) ∂t = r1s(s − 1)∂G(s,t) ∂s , (35) which is a partial differential equation (pde) in G(s,t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' We will solve (35) by the method of characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' To this end, we define new variables u = u(s,t) and v = v(s,t), which will transform (35) into the simpler equation ∂W(u,v) ∂u + H(u,v)W(u,v) = F(u,v), (36) which has the solution W(u,v) = e− � H(u,v)du �� F(u,v)e � H(u,v)du + Ψ(v) � , where W(u,v) = G(s(u,v),t(u,v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' This requires that v(s,t) = c, where c is an arbitrary constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' The resulting charac- teristic equation is given by ds dt = −r1s(s − 1), 19 which has the solution s − 1 s er1t = c = v(s,t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Setting u(s,t) = t, we obtain ∂G ∂t = ∂W ∂t = ∂W ∂u ∂u ∂t + ∂W ∂v ∂v ∂t = ∂W ∂u + r1(s − 1) s er1t ∂W ∂v (37) and ∂G ∂s = ∂W ∂u ∂u ∂s + ∂W ∂v ∂v ∂s = 1 s2 er1t ∂W ∂v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (38) Substituting (37) and (38) into (35) gives ∂W ∂u = 0, (39) which has the same form as (36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' The solution to (39) is given by W(u,v) = Ψ(v) =⇒ G(s,t) = Ψ �s − 1 s er1t� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (40) If there are n molecules at the start of the process (t = 0), then p(x,0) = 1 if x = n and p(x,0) = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Therefore, G(s,0) = Ψ �s − 1 s � = ∞ � x=0 sxP(x,0) = sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (41) In (41), the argument y of Ψ (y) maps onto ( 1 1−y )n, implying that G(s,t) = Ψ �s − 1 s er1t� = � ����� 1 1 − s−1 s er1t � ����� n = sne−nr1t [1 − s(1 − e−r1t)]n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (42) Equation (42) matches (2) when k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Equation (42) solves (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' 20 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' The right-hand-side of (35) is ∂G ∂s = nsn−1e−nr1t � 1 − s(1 − e−r1t) �−n + nsne−nr1t � 1 − s(1 − e−r1t) �−(n+1) � 1 − e−r1t� = nsn−1e−nr1t [1 − s(1 − e−r1t)]n � 1 + s(1 − e−r1t) 1 − s(1 − e−r1t) � = nsn−1e−nr1t [1 − s(1 − e−r1t)](n+1) � 1 − s(1 − e−r1t) + s(1 − e−r1t) � = nsn−1e−nr1t [1 − s(1 − e−r1t)](n+1) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (43) and the left hand-side is ∂G ∂t = −nr1sne−nr1t � 1 − s(1 − e−r1t) �−n + nsn+1r1e−(n+1)r1t � 1 − s(1 − e−r1t) �−(n+1) = nr1sne−nr1t [1 − s(1 − e−r1t)]n � se−r1t 1 − s(1 − e−r1t) − 1 � = nr1sne−nr1t [1 − s(1 − e−r1t)](n+1) � se−r1t − 1 + s − se−r1t� = nr1sne−nr1t(s − 1) [1 − s(1 − e−r1t)](n+1) = r1s(s − 1) this matches (43) ������������������������������������������������ � ����� nsn−1e−nr1t [1 − s(1 − e−r1t)](n+1) � ����� = r1s(s − 1)∂G ∂s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (44) k = 2: There are two amplification phases with rates r1 and r2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' The first one runs from time t = 0 to t = τ1, and the second one runs from t = τ1 to t = τ1 + τ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' In the second phase, the probability generating function takes exactly the same general functional form as in the first phase, albeit with a different initial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Specifically, we have G(s,t) = Ψ �s − 1 s er2(t−τ1)� , with the initial condition (at time t = τ1) G(s,τ1) = Ψ �s − 1 s � = sne−nr1τ1 [1 − s(1 − e−r1τ1)]n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' 21 Using the same procedure as in the case when k = 1, we obtain G(s,t) = Ψ �s − 1 s er2(t−τ1)� = � 1 1− s−1 s er2(t−τ1) �n e−nr1τ1 � 1 − � 1 1− s−1 s er2(t−τ1) � (1 − e−r1τ1) �n = sne−n[r2t+(r1−r2)τ1] � 1 − s � 1 − e−[r2t+(r1−r2)τ1]��n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (45) The right side of (45) equals (35) when k = 2, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' We assume the statement is true for t ∈ Ik, that is G(s,t) = � se−z 1 − s(1 − e−z) �n , where z = rkt + �k−1 i=1(ri − rk)τi, and we prove it for t ∈ Ik+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' As before, in phase k + 1, the generating function has the functional form G(s,t) = Ψ �s − 1 s erk+1(t−�k i=1 τi)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' At time t = �k i=1 τi, by the induction step, we have G(s,t) = Ψ �s − 1 s � = � se−z 1 − s(1 − e−z) �n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Using the same arguments as before, we find that, for t ∈ Ik+1, G(s,t) = Ψ �s − 1 s erk+1(t−�k i=1 τi)� = � ��������� 1 1− s−1 s erk+1(t−�k i=1 τi ) e−z 1 − 1 1− s−1 s erk+1(t−�k i=1 τi ) (1 − e−z) � ��������� n = sne−n[rk+1t+�k i=1(ri−rk)τi] � 1 − s � 1 − e−[rk+1t+�k i=1(ri−rk)τi] ��n , (46) and this ends the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' 22 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='2 Proof of Corollary 1 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' From Theorem 1, we know that the generating function for P(x|n,⃗r,t,⃗τ) is given by G(s,t) = � se−z 1 − s(1 − e−z) �n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Let p = e−z and q = 1 − p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Then, G(s,t) = � sp 1 − sq �n = (sp)n [1 − sq]−n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' P(x|n,⃗r,t,⃗τ) is the coefficient of sx in the power series expansion of G(s,t), given by G(s,t) = (sp)n [1 − sq]−n = (sp)n � i=0 �−n i � (−sq)i = (sp)n � i=0 �−n i � (−1)i(sq)i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (47) But �−n i � = −n(−n − 1)(−n − 2)(−n − 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='(−n − (i − 2))(−n − (i − 1)) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='.(i − 1)i = (−1)i n(n + 1)(n + 2)(n + 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='(n + (i − 2))(n + (i − 1)) i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' = (−1)i �n + i − 1 i � =⇒ (−1)i �−n i � = �n + i − 1 i � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (48) Substituting (48) into (47) gives G(s,t) = (sp)n � i=0 �−n i � (−1)i(sq)i = (sp)n � i=0 �n + i − 1 i � (sq)i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (49) Let x = n + i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Then, G(s,t) = (sp)n ∞ � x=n �x − 1 x − n � (sq)x−n = ∞ � x=n �x − 1 x − n � pnqx−nsx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (50) The probability of having x molecules at time t, P(x,t), is therefore given by P(x,t) = �x − 1 x − n � pnqx−n = �x − 1 n − 1 � e−nz (1 − e−z)x−n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (51) Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' The probability distribution given in Theorem 1 solves the Chapman-Kolmogorov 23 equation given by (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' ∂P(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='∂t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='= �x−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='x−n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='−nrke−z (1 − e−z)x−n + rk(x − n)e−2z (1 − e−z)x−n−1� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='= rk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='�x−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='x−n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='�e−z (1 − e−z)x−n � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='−n + (x − n) e−z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='1−e−z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='= rk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='�x−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='x−n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='�e−z (1 − e−z)x−n � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='−n + (x − n) e−z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='1−e−z − x−n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='1−e−z + x−n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='1−e−z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='= rk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='�x−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='x−n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='�e−z (1 − e−z)x−n � x−n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='1−e−z − n + (x−n) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='1−e−z (e−z − 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='= rk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='�x−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='x−n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='�e−z (1 − e−z)x−n � x−n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='1−e−z − x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='= rk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='(x − n)�x−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='x−n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='e−z (1 − e−z)x−n−1 − rkx�x−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='x−n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='�e−z (1 − e−z)x−n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='= rk(x − 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='�� x−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='x−n−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='�e−z (1 − e−z)x−n−1� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='− rkx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='��x−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='x−n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='�e−z (1 − e−z)x−n� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='= rk(x − 1)P(x − 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='t) − rkxP(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (52) which equals the right-hand side of (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='3 Proof of Theorem 2 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' We prove Theorem (2) by mathematical induction on k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' k = 1: There is only one phase with amplification efficiency r1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Recall that the Chapman- Kolmogorov forward equation for the dynamics of P (x,t) is given by (32), with the initial condition P (x,0) = e−λλx x!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Using the same arguments as above, we can write the generating function for P (x,t) as G(s,t) = Ψ �s − 1 s er1t� , (53) with the initial condition G(s,0) = Ψ �s − 1 s � = ∞ � x=0 sxP(x,0) = eλ(s−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (54) Therefore, G(s,t) = Ψ �s − 1 s er1t� = e λ � 1 1− s−1 s er1t −1 � = e � λ(s−1) 1−s(1−e−r1t) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (55) 24 Equation (55) matches (15) when k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' The generating function given by (55) solves Equation (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Differentiate the right hand side of (55) with respect to s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' k = 2: There are two phases with amplification efficiencies r1 and r2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' The first phase runs from time t = 0 to t = τ1, and the second one runs from t = τ1 to t = τ1+τ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' As before, for t ∈ I2 the probability generating function has the form G(s,t) = Ψ �s − 1 s er2(t−τ1)� , with the initial condition (at time t = t1) G(s,τ1) = Ψ �s − 1 s � = e � λ(s−1) 1−s(1−e−r1t) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Therefore, G(s,t) = Ψ �s − 1 s er2(t−τ1)� = e � ������� λ( 1 1− s−1 s er2(t−τ1) −1) 1− 1 1− s−1 s er2(t−τ1) (1−e−r1t) � ������� = e � λ(s−1) 1−s(1−e−(r2t+(r1−r2)τ1)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (56) The right side of (56) equals (15) for the case k = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' We assume the statement is true for t ∈ Ik, that is G(s,t) = e � λ(s−1) 1−s(1−e−z) � , where z = rkt + �k−1 i=1(ri − rk)τi, and we prove it for t ∈ Ik+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' In phase k + 1, the probability generating function has the form G(s,t) = Ψ �s − 1 s erk+1(t−�k i=1 τi)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' By the induction step, the initial condition (at time t = �k i=1 τi) is given by G(s,t) = Ψ �s − 1 s � = e � λ(s−1) 1−s(1−e−z) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' 25 Therefore, for t ∈ Ik+1, we have G(s,t) = Ψ �s − 1 s erk+1(t−�k i=1 τi)� = e � ���������� λ( 1 1− s−1 s erk+1(t−�k i=1 τi ) −1) 1− 1 1− s−1 s erk+1(t−�k i=1 τi ) (1−e−z) � ���������� = e � λ(s−1) 1−s(1−e−z′ ) � , (57) where z′ = rk+1t + �k i=1(ri − rk+1)τi, and this ends the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='4 Proof of Corollary 2 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Recall that P(x|λ,⃗r,t,⃗τ) is the coefficient of sx in the power series expansion of the probability generating function given in Theorem 2, that is P(x|λ,⃗r,t,⃗τ) = 1 x!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' ∂xG(s,t) ∂sx ���s=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (58) Now, let us compute the partial derivatives of G(s,t) with respect to s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' For brevity, we set a = e−z,v = (1 − e−z) = (1 − a),u = λe−z = aλ, and Q(s,t) = 1 − s(1 − e−z) = 1 − sv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Observe that G(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='t) = e λ(s−1) Q ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='Q(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='t) = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='G(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='t) = e−λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' ∂Q(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='t) ∂s = ∂Q(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='t) ∂s ���s=0 = −v 26 and ∂ ∂sG(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='t) = u G(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='t) Q2 =⇒ ∂ ∂sG(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='t) ���s=0 = ue−λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' ∂2 ∂s2 G(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='t) = ue−λ Q4 �Q2uG(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='t) Q2 + 2vQG(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='t) � = u [u + 2vQ] G(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='t) Q4 =⇒ ∂2 ∂s2 G(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='t) ���s=0 = e−λ � u2 + 2uv � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' ∂3 ∂s3 G(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='t) = u [u + 2v] Q8 � uG(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='t)(u + 2vQ)Q2 − 2v2G(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='t)Q4 + 4vG(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='t)(u + 2vQ)Q3� = u G(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='t) Q6 � u2 + 6uvQ + 6v2Q2� =⇒ ∂3 ∂s3 G(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='t) ���s=0 = e−λ(u3 + 6u2v + 6uv2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' ∂4 ∂s4 G(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='t) = u G(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='t) Q8 � u3 + 12u2vQ + 36uv2Q2 + 24v3Q3� =⇒ ∂4 ∂s4 G(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='t) ���s=0 = e−λ � u4 + 12u3v + 36u2v2 + 24uv3� ∂5 ∂s5 G(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='t) = u G(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='t) Q8 � u4 + 20u3vQ + 120u2v2Q2 + 240uv3Q3 + 120v4Q4� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' =⇒ ∂4 ∂s5 G(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='t) ���s=0 = e−λ � u5 + 20u4v + 120u3v2 + 240u2v3 + 120uv4� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (59) By closely examining the coefficients of powers of the terms u,v and uv in (59), the follow- ing combinatorial triangle emerges x 1 1 2 1 2 3 1 6 6 4 1 12 36 24 5 1 20 120 240 120 1 2 3 4 5 i In particular, the (x,i)’th entry of the triangle is given by T (x,i) = � x i − 1 ��x − 1 i − 1 � (i − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=', for i = 1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (60) 27 Thus P(x|λ,⃗r,t,⃗τ) = 1 x!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' ∂x ∂sx G(s,t) ����s=0 = e−λ x!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' x � i=1 T (x,i) = e−λ x!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' x � i=1 � x i − 1 ��x − 1 i − 1 � (i − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (λe−z)x−i+1 (1 − e−z)i−1 = e−λ x � i=1 1 (x − i + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' �x − 1 i − 1 � (λe−z)x−i+1 (1 − e−z)i−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (61) Setting k = x − i + 1 in (61) gives the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' The probability distribution given in Theorem 2 solves (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Differentiate the right-hand-side of Equation (61) with respect to t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' We will now derive the probability density function (pdf), mean, variance, and cumu- lative density function (cdf) of the Ct value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' We will consider two different cases, namely: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' when the initial state of the PCR process is deterministic, and the PCR phase lengths and amplification efficiencies are given;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' when the initial state is Poisson-distributed, and the phase lengths and amplification efficiencies are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='5 Case 1: The initial state is deterministic, and the phase lengths and amplifica- tion efficiencies are given General form of the pdf Consider the PCR process described in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' The process begins with n cDNA molecules, which are amplified across up to p successive phases {Ii} of lengths ⃗τ = (τ1,τ2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=',τp) at the corresponding amplification efficiencies ⃗r = (r1,r2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=',rp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' By def- inition, the Ct value t is the time at which the number of molecules reaches the quan- tification threshold, which we denote by x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' By Bayes’ theorem, a general expression for the pdf of t is given by P(t|n,⃗r,⃗τ,x) = P(n,⃗r,⃗τ,x|t)P(t) P(n,⃗r,⃗τ,x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (62) Since n is independent of ⃗r, ⃗τ, and t, and ⃗r is also independent of t and of the values taken by the entries of ⃗τ, we re-write the numerator of the right-hand-side of (62) as 28 follows: P(n,⃗r,⃗τ,x|t)P(t) = P(x|n,⃗r,⃗τ,t)P(n,⃗r,⃗τ|t)P(t) = P(x|n,⃗r,⃗τ,t)P(n|⃗r,⃗τ,t)P(⃗r,⃗τ|t)P(t) = P(x|n,⃗r,⃗τ,t)P(n)P(⃗r|⃗τ,t)P(⃗τ|t)P(t) = P(x|n,⃗r,⃗τ,t)P(n)P(⃗r)P(t|⃗τ)P(⃗τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (63) Similarly, the denominator of the right-hand-side of (62) can be simplified to P(n,⃗r,⃗τ,x) = � ∞ �k−1 i=1 τi P(n,⃗r,⃗τ,x|t)P(t)dt = P(n)P(⃗r)P(⃗τ) � ∞ �k−1 i=1 τi P(x|n,⃗r,⃗τ,t)P(t|⃗τ)dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (64) Therefore, (62) can be re-written as P(t|n,⃗r,⃗τ,x) = P(x|n,⃗r,⃗τ,t)P(t|⃗τ) � ∞ �k−1 i=1 τi P(x|n,⃗r,⃗τ,t)P(t|⃗τ)dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (65) We will derive the pdf, mean, variance, and cdf of the Ct value for a PCR process with an arbitrary number of phases p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Without loss of generality, we suppose that t ∈ Ik,k ≤ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' We will assume a uniform prior density for t given ⃗τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' As we will demonstrate later, this assumption produces very similar results to those we obtain by assuming a Jeffreys prior [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' We will state results for the case of a single-phase PCR process whenever these cannot be readily gleaned from the general results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' We will first consider the case when the lengths of the intermediate phases, recorded in the vector ⃗τ, are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' This is useful, for example, when it is of interest to estimate the lengths of such phases from data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' We will then show how to marginalize ⃗τ out of the pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' pdf Using the posterior density given in Equation (65) and the likelihood function given in Corollary 1, we obtain the following functional form for the pdf: P(t|n,⃗r,⃗τ,x) ∝ e−nz (1 − e−z)x−n , (66) where z = rkt + k−1 � i=1 (ri − rk)τi, (67) ⃗r = (r1,r2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=',rk) is a vector of amplification efficiencies, ⃗τ = (τ1,τ2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=',τk) is a vector of 29 phase lengths, and we have used a uniform prior for t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' The normalizing constant is given by C = � ∞ �k−1 i=1 τi e−nz(1 − e−z)x−ndt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (68) Let w = e−z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Then, C = 1 rk � θ 0 wn−1(1 − w)x−ndw = Bθ(n,x − n + 1) rk , (69) where θ = e−�k−1 i=1 riτi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (70) Therefore, the pdf is given by P(t|n,⃗r,⃗τ,x) = rke−nz (1 − e−z)x−n Bθ(n,x − n + 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (71) For the single-phase process, θ = 1, so the pdf simplifies to P(t|n,r1,x) = r1e−nr1t � 1 − e−r1t�x−n B(n,x − n + 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (72) Note that in some cases (eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' when knowledge of the lengths of individual PCR ampli- fication phases is not of interest), it may be useful to marginalize ⃗τ out of P(t|n,⃗r,⃗τ,x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' This can be achieved by using the fact that P(t|n,⃗r,x) = � Ω P(t,⃗τ|n,⃗r,x)d⃗τ = � Ω P(t|n,⃗r,⃗τ,x)P(⃗τ|n,⃗r,x)d⃗τ, (73) where Ω is the domain of ⃗τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' In addition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' note that an alternative formulation of the prior for t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' based on an ap- proach proposed by Jeffreys [25] for generating priors that are invariant to reparametriza- tion,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' is the following: p(t|⃗τ) ∝ � |I(t|⃗τ)|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (74) 30 where I(t|⃗τ) is the Fisher information of the likelihood function and is given by I(t|⃗τ) = EX �� ∂ ∂t lnP(x|n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='⃗r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='⃗τ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='x) �2 � = EX �r2 k � w2x2 − 2nwx + n2� (1 − w)2 � = r2 k w2 (1 − w)2 EX � x2� − 2nr2 k w (1 − w)2 EX � x � + n2r2 k (1 − w)2 = r2 k (1 − w)2 � �����w2 ∞ � j=1 x2 �x − 1 j − 1 � wn(1 − w)x−n − 2nw ∞ � j=1 x �x − 1 j − 1 � wn(1 − w)x−n + n2 � �����,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (75) where w = e−z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Observe that ∞ � x=1 x �x − 1 n − 1 � wn(1 − w)x−n = ∞ � x=1 x!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (x − n)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='wn(1 − w)x−n = ∞ � y=2 (y − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (y − m)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (m − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='wm−1(1 − w)y−m = m − 1 w ∞ � y=1 (y − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (y − m)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (m − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='wm(1 − w)y−m = m − 1 w = n w (76) 31 and ∞ � x=1 x2 �x − 1 n − 1 � wn(1 − w)x−n = ∞ � x=1 x!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='x (x − n)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='wn(1 − w)x−n = ∞ � y=2 (y − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (y − 1) (y − m)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (m − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='wm−1(1 − w)y−m = ∞ � y=1 (y − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='y (y − m)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (m − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='wm−1(1 − w)y−m − m − 1 w = ∞ � y′=2 (y′ − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (y′ − m′)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (m′ − 3)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='wm′−2(1 − w)y′−m′ − m − 1 w = (m′ − 1)(m′ − 2) w2 ∞ � y′=1 (y′ − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (y′ − m′)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (m′ − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='wm′(1 − w)y′−m′ − m − 1 w = (m′ − 1)(m′ − 2) w2 − m − 1 w = n(n + 1) w2 − n w , (77) where m = n + 1,m′ = m + 1,y = x + 1,y′ = y + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Plugging (76) and (77) into (75), we obtain I(t|⃗τ) = nr2 k 1 − w =⇒ p(t|⃗τ) ∝ 1 √ 1 − w = 1 √ 1 − e−z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (78) Using this prior, and following the steps we used earlier to derive (71), we find that the pdf is given by P(t|n,⃗r,⃗τ,x) ∝ e−nz (1 − e−z)x−n−1/2 =⇒ P(t|n,⃗r,⃗τ,x) = rke−nz (1 − e−z)x−n−1/2 Bθ(n,x − n + 1/2) , (79) which has a similar form as (71).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' For simplicity, we will continue to use a uniform prior for t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Mean The mean Ct value is given by E(t) = rk � ∞ �k−1 i τi te−nz(1 − e−z)x−ndt Bθ(n,x − n + 1) , (80) 32 where z is given by (67).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Let w = e−z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Then, E(t) = rk � 0 θ � − (lnw−lnθ′) rk � wn(1 − w)x−n � − dw rkw � Bθ(n,x − n + 1) = � 0 θ (lnw − lnθ′)wn−1(1 − w)x−ndw rkBθ(n,x − n + 1) = lnθ′ � θ 0 wn−1(1 − w)x−ndw − � θ 0 lnw wn−1(1 − w)x−ndw rkBθ(n,x − n + 1) = lnθ′ rk − � ∂ ∂n + ∂ ∂x � Bθ(n,x − n + 1) rkBθ(n,x − n + 1) = ln θ′ θ rk + Γ(n)2θn 3 ˜F2(n,n,n − x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='n + 1,n + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='θ) rkBθ(n,x − n + 1) = k−1 � i=1 τi + Γ(n)2θn 3 ˜F2(n,n,n − x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='n + 1,n + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='θ) rkBθ(n,x − n + 1) , (81) where θ is given by (70), ψ(·) is the first polygamma function (also called the digamma function), and θ′ = θerk �k−1 i=1 τi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (82) Note that for the single-phase process, θ = θ′ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' In this case, using 3 ˜F2(n,n,n − x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='n + 1,n + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='1) = nΓ(x − n + 1)[ψ(x + 1) − ψ(n)] Γ(n + 1)Γ(x + 1) , we find that the mean Ct value is given by E(t) = ψ(x + 1) − ψ(n) r1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (83) Variance The variance of the Ct value is given by E(t2) − E(t)2, where E(t2) = rk � ∞ �k−1 i=1 τi t2e−nz(1 − e−z)x−ndt Bθ(n,x − n + 1) , (84) and z is given by (67).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' 33 Let w = e−z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' E(t2) = rk � 0 θ � lnw−lnθ′ rk �2 wn(1 − w)x−n � − dw rkw � Bθ(n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='x − n + 1) = � θ 0 (lnw − lnθ′)2 wn−1(1 − w)x−ndw rk2Bθ(n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='x − n + 1) = � θ 0 (lnw)2 wn−1(1 − w)x−ndw − 2lnθ′ � θ 0 lnw wn−1(1 − w)x−ndw rk2Bθ(n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='x − n + 1) + (lnθ′)2 � θ 0 wn−1(1 − w)x−ndw rk2Bθ(n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='x − n + 1) = � ∂2 ∂n2 + 2 ∂2 ∂n∂x + ∂2 ∂x2 − 2lnθ′ � ∂ ∂n + ∂ ∂x �� Bθ(n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='x − n + 1) rk2Bθ(n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='x − n + 1) + (lnθ′)2 rk2 = � ∂2 ∂n2 + 2 ∂2 ∂n∂x + ∂2 ∂x2 � Bθ(n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='x − n + 1) rk2Bθ(n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='x − n + 1) + 2lnθ′Γ(n)2θn 3 ˜F2(n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='n − x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='n + 1,n + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='θ) r2 k Bθ(n,x − n + 1) + lnθ′ ln θ′ θ2 r2 k = � ∂2 ∂n2 + 2 ∂2 ∂n∂x + ∂2 ∂x2 � Bθ(n,x − n + 1) rk2Bθ(n,x − n + 1) + 2lnθ′Γ(n)2θn 3 ˜F2(n,n,n − x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='n + 1,n + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='θ) r2 k Bθ(n,x − n + 1) + � ������ k−1 � i=1 τi � ������ 2 − � ������ k−1 � i=1 riτi rk � ������ 2 , (85) where θ is given by (70) and θ′ is given by (82).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' For the single-phase process, the second moment of the Ct value is given by E(t2) = � ∂2 ∂n2 + 2 ∂2 ∂n∂x + ∂2 ∂x2 � B(n,x − n + 1) rk2B(n,x − n + 1) = ψ1(n) − ψ1(x + 1) + [ψ(x + 1) − ψ(n)]2 rk2 , (86) where ψ1(·) is the second polygamma function (also called the trigamma function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Therefore, the variance is Var(t) = ψ1(n) − ψ1(x + 1) rk2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (87) cdf 34 The cdf of the Ct value is given by F(t|n,⃗r,⃗τ,x) = rk Bθ(n,x − n + 1) � t �k−1 i=1 τi e−nz′(1 − e−z′)x−nds, (88) where z′ = rks + �k−1 i=1(ri − rk)τi Let w = e−z′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Then, F(t|n,⃗r,⃗τ,x) = � θ e−z wn−1(1 − w)x−ndw Bθ(n,x − n + 1) = Bθ(n,x − n + 1) − Be−z(n,x − n + 1) Bθ(n,x − n + 1) = 1 − Be−z(n,x − n + 1) Bθ(n,x − n + 1) , (89) where θ is given by 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' For the single-phase process, the cdf is given by F(t|n,r1,x) = 1 − Ie−r1t(n,x − n + 1), (90) where Ie−r1t(n,x−n+1) = Be−rt (n,x−n+1) B(n,x−n+1) is the regularized incomplete Beta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Be- cause the cdf is in closed analytical form, we can apply the efficient inverse transform method to generate random samples of Ct values as follows: t = −lnI−1 1−u(n,x − n + 1) r1 , (91) where u is a real number sampled uniformly at random from the interval (0,1) and I−1 1−u is the inverse of the regularized incomplete Beta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' To find a Ct value that corresponds to a quantile q ∈ (0,1), simply replace u in Equation (91) by q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Probability distribution of n We conclude by deriving the probability distribution of n, denoted P(n|r1,t,x), for the single-phase PCR process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' This distribution can be used to estimate n from measured Ct values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' It can also be used to calculate the LoD and LoQ of a PCR assay, as we demonstrated in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' The steps described below can also be used to derive P(n|⃗r,t,⃗τ), for a PCR process with an arbitrary number of phases, although this will not yield a closed-form result like we will obtain in the single-phase case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' By Bayes’ Theorem, we have P(n|r1,t,x) ∝ wn(1 − w)x−n B(n,x − n + 1), (92) 35 where w = e−r1t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (93) The normalizing constant is given by C = x � n=1 wn(1 − w)x−n B(n,x − n + 1) = x � n=1 x!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' wn(1 − w)x−n (n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (x − n)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (94) Let m = n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Then, C = x−1 � m=0 x!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' wm+1(1 − w)x−1−m m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (x − 1 − m)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' = xw x−1 � m=0 �x − 1 m � wm(1 − w)x−1−m = xw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (95) Therefore, we have P (n|r1,t,x) = wn−1(1 − w)x−n xB(n,x − n + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (96) Suppose that t is a Ct value generated by a PCR process with n input molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' If we replace n by ˆn in Equation (96), then the equation will give the probability that ˆn will be obtained as the estimate of n based on the data t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' It is useful – eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' for the purpose of determining the LoQ – to calculate the probability that ˆn will be obtained as the estimate of n based on any data t that can be generated by a PCR process with n input molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' This probability is given by P ( ˆn|n,r1,x) = � ∞ 0 P ( ˆn,t|n,r1,x)dt = � ∞ 0 P ( ˆn|n,r1,t,x)P (t|n,r1,x)dt = � ∞ 0 P ( ˆn|r1,t,x)P (t|n,r1,x)dt = r1 � ∞ 0 w ˆn−1(1 − w)x− ˆn xB( ˆn,x − ˆn + 1) wn(1 − w)x−n B(n,x − n + 1)dt = � 1 0 w ˆn+n−2(1 − w)2x−m−n xB( ˆn,x − ˆn + 1)B(n,x − n + 1)dw = B( ˆn + n − 1,2x − ˆn − n + 1) xB( ˆn,x − ˆn + 1)B(n,x − n + 1) = P( ˆn|n,x), (97) 36 where, using (96), we have assumed that ˆn is conditionally independent of n given t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Strikingly, (97) does not depend on r1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='6 Case 2: The initial state is Poisson-distributed, and the phase lengths and am- plification efficiencies are given General form of the pdf Let t be the Ct value of the PCR process described in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' The process begins with a Poisson-distributed number of input DNA molecules, with mean λ, which are replicated across up to p distinct phases with lengths ⃗τ = (τ1,τ2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=',τp) and am- plification efficiencies ⃗r = (r1,r2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=',rp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' As noted earlier, t is the time at which the number of molecules reaches the quantification threshold, which we denote by x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Let us denote the pdf of t by P(t|λ,⃗r,⃗τ,x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' By Bayes’ theorem, we have P(t|λ,⃗r,⃗τ,x) = P(λ,⃗r,⃗τ,x|t)P(t) P(λ,⃗r,⃗τ,x) (98) However, λ is independent of ⃗r, ⃗τ, and t, while ⃗r is also independent of t and of the precise values taken by the entries of ⃗τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Therefore, by following the same steps we used earlier to derive (65), we find that P(t|λ,⃗r,⃗τ,x) = P(x|λ,⃗r,t,⃗τ)P(t|⃗τ) � ∞ �k−1 i=1 τi P(x|λ,⃗r,t,⃗τ)P(t|⃗τ)dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (99) We will derive the pdf, mean, variance, and cdf by assuming, without loss of gener- ality, that t ∈ Ik, and then we will specify the functional forms taken by the results in the instructive case when t ∈ I1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' As before, for simplicity, we will use a uniform prior density for t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' pdf We derive the pdf of the Ct value t by using the general expression given in Equation (99), with the probability distribution of the number of molecules given in Theorem 37 2 serving as the likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Specifically, P(t|λ,⃗r,⃗τ,x) ∝ (1 − e−z)x x � j=1 �x−1 j−1 � j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' � λe−z 1 − e−z �j (100) = (1 − e−z)x x−1 � j=0 �x−1 j � (j + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' � λe−z 1 − e−z �j+1 (101) since (x j)=0 for j>x = (1 − e−z)x ∞ � j=0 �x−1 j � (j + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' � λe−z 1 − e−z �j+1 (102) = λe−z(1 − e−z)x−1 ∞ � j=0 (x − 1)(x − 2)···(x − j) (j + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' � λe−z 1 − e−z �j (103) = λe−z(1 − e−z)x−1 ∞ � j=0 (1 − x)(2 − x)···(j − x) (j + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' � −λe−z 1 − e−z �j (104) = λe−z(1 − e−z)x−1 ∞ � j=0 (1 − x)j (2)j �−λe−z 1−e−z �j j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (105) = λe−z(1 − e−z)x−1 1F1 � 1 − x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' −λe−z 1 − e−z � , (106) where z is given by (67), 1F1 is the hypergeometric function (also called the Kummer confluent hypergeometric function of the first kind), defined as 1F1 � 1 − x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' −λe−r1t 1 − e−r1t � = ∞ � j=0 (1 − x)j (2)j � −λe−r1t 1−e−r1t �j j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' , and (α)j denotes the rising factorial, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (α)j = α(α+1)(α+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='(α+j−1) with (α)0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' The normalizing constant is given by C = x � j=1 �x−1 j−1 �λj j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' � ∞ �k−1 i=1 τi e−jz(1 − e−z)x−jdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (107) Let w = e−z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Then, C = 1 rk x � j=1 �x−1 j−1 �λj j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' � θ 0 wj−1(1 − w)x−jdw = 1 rk x � j=1 �x−1 j−1 �λj j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Bθ(j,x − j + 1), (108) where θ is given by (70).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' 38 Therefore, the pdf is given by P(t|λ,⃗r,⃗τ,x) = rk(1 − e−z)x �x j=1 (x−1 j−1) j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' � λe−z 1−e−z �j �x j=1 (x−1 j−1)λj j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Bθ(j,x − j + 1) = rkλe−z(1 − e−z)x−1 1F1 � 1 − x,2, −λe−z 1−e−z � �x j=1 (x−1 j−1)λj j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Bθ(j,x − j + 1) , (109) Recall that for the single-phase process, θ = 1, so we have x � j=1 �x−1 j−1 �λj j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' B(j,x − j + 1) = ∞ � j=1 �x−1 j−1 �λj j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' B(j,x − j + 1) = ∞ � j=0 �x−1 j �λj+1 (j + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' B(j + 1,x − j) = ∞ � j=0 λj+1 (j + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' = eλ − 1 x , (110) implying that the pdf is given by P(t|λ,r1,x) = r1xλe−r1t(1 − e−r1t)x−1 1F1 � 1 − x,2, −λe−r1t 1−e−r1t � eλ − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (111) Note that in some cases (eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' when knowledge of the lengths of individual PCR ampli- fication phases is not of interest), it may be useful to marginalize ⃗τ out of P(t|λ,⃗r,⃗τ,x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' This can be achieved by using the fact that P(t|λ,⃗r,x) = � Ω P(t,⃗τ|λ,⃗r,x)d⃗τ = � Ω P(t|λ,⃗r,⃗τ,x)P(⃗τ|λ,⃗r,x)d⃗τ, (112) where Ω is the domain of ⃗τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Mean 39 The mean Ct value is given by E(t) = rk �x j=1 (x−1 j−1)λj j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Bθ(j,x − j + 1) x � j=1 �x−1 j−1 �λj j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (D) �������������������������������������������������������� � ∞ �k−1 i=1 τi te−jz(1 − e−z)x−jdt, (113) where z is given by (67).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Let w = e−z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Then, D see (81) = Bθ(j,x − j + 1)�k−1 i=1 τi rk + Γ(j)2θj 3 ˜F2(j,j,j − x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='j + 1,j + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='θ) r2 k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (114) Therefore E(t) = �x j=1 (x−1 j−1)λj j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' � rkBθ(j,x − j + 1)�k−1 i=1 τi + Γ(j)2θj 3 ˜F2(j,j,j − x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='j + 1,j + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='θ) � rk �x j=1 (x−1 j−1)λj j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Bθ(j,x − j + 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (115) Recall that for the single-phase process, θ = θ′ = 1, so E(t) = ψ(x + 1) r1 − �x j=1 λj j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' ψ(j) r1 � eλ − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (116) Variance The variance is given by E(t2) − E(t)2, where E(t2) = rk �x j=1 (x−1 j−1)λj j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Bθ(j,x − j + 1) x � j=1 �x−1 j−1 �λj j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (D) ������������������������������������������������������������ � ∞ �k−1 i=1 τi t2e−jz(1 − e−z)x−jdt, (117) where z is given by (67).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' 40 Let w = e−z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Then, D see (85) = � ∂2 ∂n2 + 2 ∂2 ∂n∂x + ∂2 ∂x2 � Bθ(n,x − n + 1) rk3 + 2lnθ′Γ(n)2θn 3 ˜F2(n,n,n − x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='n + 1,n + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='θ) r3 k + Bθ(j,x − j + 1) � �������� ��k−1 i=1 τi �2 − ��k−1 i=1 riτi rk �2 rk � �������� (118) where θ′ is given by (82).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Plugging (118) into (117), we obtain E(t2) = 1 r2 k �x j=1 (x−1 j−1)λj j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Bθ(j,x − j + 1) x � j=1 �x−1 j−1 �λj j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' � ����� � ∂2 ∂n2 + 2 ∂2 ∂n∂x + ∂2 ∂x2 � Bθ(n,x − n + 1) + 2lnθ′Γ(n)2θn 3 ˜F2(n,n,n − x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='n + 1,n + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='θ) + r2 k Bθ(j,x − j + 1) � �������� � ������ k−1 � i=1 τi � ������ 2 − � ������ k−1 � i=1 riτi rk � ������ 2� �������� � �����.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (119) For the single-phase process, the variance is given by Var(t) = (eλ − 1)�x j=1 λj j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' � ψ1(j) + ψ(j)2 � − � ����� �x j=1 λj j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' ψ(j) � ����� 2 � r1(eλ − 1) �2 − ψ1(x + 1) r2 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (120) cdf The cdf of the Ct value is given by F(t|λ,⃗r,⃗τ,x) = rk �x j=1 (x−1 j−1)λj j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' � t�k−1 i=1 τi e−jz′(1 − e−z′)x−jds �x j=1 (x−1 j−1)λj j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Bθ(j,x − j + 1) , (121) where z′ = rks + �k−1 i=1(ri − rk)τi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' 41 Let w = e−z′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Then, F(t|λ,⃗r,⃗τ,x) = �x j=1 (x−1 j−1)λj j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' � θ e−z wj−1(1 − w)x−jdw �x j=1 (x−1 j−1)λj j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Bθ(j,x − j + 1) = �x j=1 (x−1 j−1)λj j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' �� θ 0 wj−1(1 − w)x−jdw − � e−z 0 wj−1(1 − w)x−jdw � �x j=1 (x−1 j−1)λj j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Bθ(j,x − j + 1) = �x j=1 (x−1 j−1)λj j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' � Bθ(j,x − j + 1) − Be−z(j,x − j + 1) � �x j=1 (x−1 j−1)λj j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Bθ(j,x − j + 1) = 1 − �x j=1 (x−1 j−1)λj j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Be−z(j,x − j + 1) �x j=1 (x−1 j−1)λj j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Bθ(j,x − j + 1) , (122) where θ is given by (70).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' For the single-phase process, using (110), we simplify the cdf to obtain F(t|λ,r1,x) = 1 − x�x j=1 (x−1 j−1)λj j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Be−r1t(j,x − j + 1) eλ − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (123) Probability density of λ We conclude by deriving the probability density of λ, P(λ|r1,t,x), for the single-phase process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' This density can be used to estimate λ from measured Ct values, and for calculating both the LoD and the LoQ of a PCR process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' The steps described below can also be used to derive P(λ|⃗r,t,⃗τ), for a PCR process with an arbitrary number of phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' By Bayes’ Theorem, we have P(λ|r1,t,x) ∝ �x j=1 (x−1 j−1) j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' � λw 1−w �j eλ − 1 , (124) where w = e−r1t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' 42 The normalizing constant is given by C = � ∞ 0 �x j=1 (x−1 j−1) j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' � λw 1−w �j eλ − 1 dλ = x � j=1 �x−1 j−1 � j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' � w 1 − w �j � ∞ 0 λj eλ − 1dλ = x � j=1 �x−1 j−1 � j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' � w 1 − w �j Γ(j + 1)ζ(j + 1) = x � j=1 �x − 1 j − 1 �� w 1 − w �j ζ(j + 1), (125) where ζ(j) is the Riemann zeta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Therefore, the probability density of λ is given by P(λ|r1,t,x) = �x j=1 (x−1 j−1) j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' � λw 1−w �j (eλ − 1)�x j=1 �x−1 j−1 �� w 1−w �j ζ(j + 1) = λw 1F1(1 − x,2, −λw 1−w ) (eλ − 1)(1 − w)�x j=1 �x−1 j−1 �� w 1−w �j ζ(j + 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (126) The probability that λ takes values between a and b is given by P(a ≤ λ ≤ b | r1,t,x) = �x j=1 (x−1 j−1) j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' � w 1−w �j � b a sj es−1ds �x j=1 �x−1 j−1 �� w 1−w �j ζ(j + 1) = �x j=1 �x−1 j−1 �� w 1−w �j [ζb(j + 1) − ζa(j + 1)] �x j=1 �x−1 j−1 �� w 1−w �j ζ(j + 1) , (127) where ζλ(·) is the incomplete Riemann zeta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' It follows that the cumulative density function of λ is given by F(λ | r1,t,x) = �x j=1 �x−1 j−1 �� w 1−w �j ζλ(j + 1) �x j=1 �x−1 j−1 �� w 1−w �j ζ(j + 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (128) As discussed earlier in relation to P(n|⃗r,t,⃗τ,x), Equation (126) can be interpreted as follows: Suppose that a Ct value t is produced by a PCR process with expected number of input molecules λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' If we replace λ by an estimate ˆλ, then (126) gives the 43 likelihood of ˆλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' For practical purposes (eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' to determine the LoQ), it is useful to calculate the probability that ˆλ will be obtained as the estimate of λ from any data t that can be produced by a PCR process with expected number of input molecules λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' This probability is given by P � ˆλ|λ,r1,x � = � ∞ �k−1 i=1 τi P � ˆλ,t|λ,r1,x � dt = � ∞ �k−1 i=1 τi P � ˆλ|λ,r1,t,x � P (t|λ,r1,x)dt = � ∞ �k−1 i=1 τi P � ˆλ|r1,t,x � P (t|λ,r1,x)dt = r1x � eλ − 1 �� e ˆλ − 1 � � ∞ �k−1 i=1 τi (1 − w)x ��x j=1 (x−1 j−1) j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' � λw 1−w �j ���x j=1 (x−1 j−1) j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' � ˆλw 1−w �j � �x j=1 �x−1 j−1 �� w 1−w �j ζ(j + 1) dt = x � eλ − 1 �� e ˆλ − 1 � � θ 0 (1 − w)x ��x j=1 (x−1 j−1) j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' � λw 1−w �j ���x j=1 (x−1 j−1) j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' � ˆλw 1−w �j � w�x j=1 �x−1 j−1 �� w 1−w �j ζ(j + 1) dw, (129) where θ is given by (70) and we have assumed that ˆλ is conditionally independent of λ given t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' It follows that the t-independent probability that ˆλ will take values between a and b is given by P(a ≤ ˆλ ≤ b | λ,r1,x) = � b a P � ˆλ|λ,r1,x � d ˆλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' (130) 44 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='2 Supplementary Figures Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='1: Limit of detection of the single-phase process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' The LoD was determined while accounting for either sampling noise alone (solid green line), amplification noise alone (solid red line), or both sampling noise and amplification noise (solid blue line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' It was then plotted versus amplification efficiency, which is expressed on a base-2 scale as a percentage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' The LoD based on sampling noise alone equals 3, whereas the LoD is highest when accounting for both types of noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' In the latter case, it ranges from 157, when the efficiency is only 80%, to 6, when the efficiency is 100%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' The plot shows a strong dependence of the LoD on efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' 45 150 - 100 Noise type LoD Sampling noise only Amplification noise only Sampling + amplification noise 50 0 - 80 85 90 95 100 Amplification efficiency (%)Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='2: Ratio of expected versus estimated fraction of positive partitions in digital PCR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' For different expected numbers of input molecules λ, the ratio of the fraction of digital PCR partitions expected to test positive was calculated using Equation (24) and divided by the standard estimate based on the Poisson distribution (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' 1−e−λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' The result was plotted versus the amplification efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' While the efficiency used in calculations is always expressed on a base-e scale, for ease of comprehension it was converted into a base-2 scale and displayed as a percentage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' 46 1.' metadata={'source': 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quantifying dna concentrations using fluorometry: A comparison of fluo- rophores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Molecular Vision, 8:416–421, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' [25] Harold Jeffreys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' An invariant form for the prior probability in estimation problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Proceedings of the Royal Society of London.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Series A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' Mathematical and Physical Sciences, 186(1007):453–461, 1946.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} +page_content=' 49' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tA0T4oBgHgl3EQfNf_3/content/2301.02149v1.pdf'} diff --git a/59AyT4oBgHgl3EQf2fl-/content/2301.00752v1.pdf b/59AyT4oBgHgl3EQf2fl-/content/2301.00752v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..b0aa5b8344b3f41d4ecb96f7f1be166832d25753 --- /dev/null +++ b/59AyT4oBgHgl3EQf2fl-/content/2301.00752v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:eff653fab82ec2d941a0ec167edb0ebd36779b7cb5434d9fe491708014d071e2 +size 2741386 diff --git a/59AyT4oBgHgl3EQf2fl-/vector_store/index.faiss b/59AyT4oBgHgl3EQf2fl-/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..2bce3a0fa50fa8fdd3bae604168b3fc825fbac4a --- /dev/null +++ b/59AyT4oBgHgl3EQf2fl-/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:9f1434180ae96c633941b8c3c86c6e83a761efff4c0a2ef990336846c4c8eec5 +size 6291501 diff --git a/5dAyT4oBgHgl3EQfcfda/content/tmp_files/2301.00283v1.pdf.txt b/5dAyT4oBgHgl3EQfcfda/content/tmp_files/2301.00283v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..96d412b4dbaf995a760b345fd6b8f87fda021e13 --- /dev/null +++ b/5dAyT4oBgHgl3EQfcfda/content/tmp_files/2301.00283v1.pdf.txt @@ -0,0 +1,755 @@ +arXiv:2301.00283v1 [quant-ph] 31 Dec 2022 +Scaling limit of the time averaged distribution for +continuous time quantum walk and Szegedy’s walk on the path +Yusuke Ide +Department of Mathematics, College of Humanities and Sciences, Nihon University +3-25-40 Sakura-josui, Setagaya-ku, Tokyo 156-8550, Japan +e-mail: ide.yusuke@nihon-u.ac.jp +Abstract +In this paper, we consider Szegedy’s walk, a type of discrete time quantum walk, and corresponding continuous time +quantum walk related to the birth and death chain. We show that the scaling limit of time averaged distribution for +the continuous time quantum walk induces that of Szegedy’s walk if there exists the spectral gap on so-called the +corresponding Jacobi matrix . +1 +Introduction +Quantum walks, a quantum counterpart of random walks have been extensively developed in various fields +during the last two decades. Since quantum walks are very simple models therefore they play fundamental +and important roles in both theoretical fields and applications. There are good review articles for these +developments such as Kempe [6], Kendon [7], Venegas-Andraca [14,15], Konno [8], Manouchehri and Wang +[9], and Portugal [11]. +We investigate the time averaged distribution of a variant of discrete time quantum walk (DTQW) +so-called Szegedy’s walk [13]. On the path graph, the spectral properties of Szegedy’s walk are directly +connected to the theory of (finite type) orthogonal polynomials. There are studies of the distribution of +Szegedy’s walk on the path graph for example [1–3,5,10,12]. +In this paper, we focus on scaling limit of the time averaged distributions of both Szegedy’s walk and +corresponding continuous time quantum walk on the path graph related to the random walk with reflecting +walls. In order to our main theorem (Theorem 4.1), if there exists the spectral gap, i.e., the limit superior in +the size of the path graph tends to infinity of the second largest eigenvalue of the Jacobi matrix is less than +one (the largest eigenvalue), then the scaling limit of Szegedy’s walk is the same as that of corresponding +continuous time quantum walk. We should note that existence of the spectral gap of the Jacobi matrix is +equivalent to that of the transition matrix of corresponding random walk. A typical example of this case +is space homogeneous random walk with pR +j = p case (the second largest eigenvalue is 2 +� +p(1 − p) cos π/n) +treated in [5] except for the symmetric random walk with pR +j = 1/2. Unfortunately we have not been covered +with non-spectral gap cases including symmetric random walk and the Ehrenfest model (the second largest +eigenvalue is 1 − 2/n) treated in [3]. To reveal non-spectral gap case is one of interesting future problems. +The rest of this paper is organized as follows. In Sec. 2, we define our setting of discrete time random +walk, continuous time quantum walk and discrete time quantum walk on the path graph. Sec. 3 is devoted to +show relationships between the time averaged distribution of Szegedy’s walk and continuous time quantum +walk. In the last section, we state our main theorem (Theorem 4.1) and prove it. +2 +Definition of the models +In this paper, we consider the path graph Pn+1 = (V (Pn+1), E(Pn+1)) with the vertex set V (Pn+1) = +{0, 1, . . ., n} and the (undirected) edge set E(Pn+1) = {(j, j + 1) : j = 0, 1, . . . , n − 1}. On the path graph +Keywords: +birth and death chain, Szegedy’s walk, continuous time quantum walk, scaling limit, time averaged distribution +1 + +Pn+1, we define a discrete time random walk (DTRW) with reflecting walls as follows: +Let pL +j be the transition probability of the random walker at the vertex j ∈ V (Pn+1) to the left (j − 1 ∈ +V (Pn+1)). Also let pR +j = 1−pL +j be the transition probability of the random walker at the vertex j ∈ V (Pn+1) +to the right (j + 1 ∈ V (Pn+1)). For the sake of simplicity, we assume 0 < pL +j , pR +j < 1 except for j = 0, n. We +put the reflecting walls at the vertex 0 ∈ V (Pn+1) and the vertex n ∈ V (Pn+1), i.e., we set pR +0 = pL +n = 1. +We also call this type of DTRW as the birth and death chain. +Let a positive constant Cπ be +Cπ := 1 + +n +� +j=1 +pR +0 · pR +1 · · · pR +j−1 +pL +1 · pL +2 · · · pL +j +then we can define the stationary distribution {π(0), π(1), . . . , π(n)} as +π(j) = + + + +1 +Cπ +if j = 0, +1 +Cπ · +pR +0 ·pR +1 ···pR +j−1 +pL +1 ·pL +2 ···pL +j +if j = 1, 2, . . ., n. +Note that π(j) > 0 for all j ∈ V (Pn+1) and the stationary distribution is satisfied with so-called the detailed +balance condition, +π(j) · pR +j = pL +j+1 · π(j + 1), +for j = 0, 1, . . .n − 1. +In order to define a continuous time quantum walk (CTQW) corresponding to the DTRW, we intro- +duce the normalized Laplacian matrix L. +Let P be the transition matrix of the DTRW. Also we de- +fine diagonal matrices D1/2 +π +:= diag +�� +π(0), +� +π(1), . . . , +� +π(n) +� +and D−1/2 +π += +� +D1/2 +π +�−1 +. +Note that +D−1/2 +π += diag +� +1/ +� +π(0), 1/ +� +π(1), . . . , 1/ +� +π(n) +� +by the definition. The normalized Laplacian matrix L +is given by +L := D1/2 +π +(In+1 − P) D−1/2 +π += In+1 − D1/2 +π +PD−1/2 +π +, +where In+1 be the (n + 1) × (n + 1) identity matrix. We should remark that the matrix +J := D1/2 +π +PD−1/2 +π +, +is referred as the Jacobi matrix. So we can rewrite L as L = In+1 − J. +By using the detailed balance condition, we obtain +Jj,k = Jk,j = +�� +pR +j pL +j+1, +if k = j + 1, +0, +otherwise. +Thus L = In+1 − J is an Hermitian matrix (real symmetric matrix). The CTQW which is discussed in this +paper is driven by the time evolution operator (unitary matrix) +UCT QW (t) := exp (itL) := +∞ +� +k=0 +(it)k +k! Lk, +where i is the imaginary unit. Let XC +t +(t ≥ 0) be the random variable representing the position of the +CTQWer at time t. The distribution of XC +t is determined by +P +� +XC +t = k|XC +0 = j +� +:= |⟨k|UCT QW (t)|j⟩|2 = +���(UCT QW (t))k,j +��� +2 +, +where |j⟩ is the (n + 1)-dimensional unit vector (column vector) which j-th component equals 1 and the +other components are 0 and ⟨v| is the transpose of |v⟩, i.e., ⟨v| = T |v⟩. +2 + +Hereafter we only consider XC +0 = 0 , i.e., the CTQWer starts from the left most vertex 0 ∈ V (Pn+1), +cases. The time averaged distribution ¯pC of the CTQW is defined by +¯pC(j) := lim +T →∞ +1 +T +� T +0 +P +� +XC +t = j|XC +0 = 0 +� +dt, +for each vertex j ∈ V (Pn+1). We define a random variable ¯XC +n as P +� ¯XC +n = j +� += ¯pC(j). +In this paper, we also deal with a type of discrete time quantum walk (DTQW) corresponding to the +DTRW so-called Szegedy’s walk. The time evolution operator for the DTQW is defined by U = SC with +the coin operator C and the shift operator (flip-flop type shift) S. The coin operator C is defined by +C = |0⟩⟨0| ⊗ I2 + +n−1 +� +j=1 +|j⟩⟨j| ⊗ Cj + |n⟩⟨n| ⊗ I2, +where I2 is the 2 × 2 identity matrix and ⊗ is the tensor product. The local coin operator Cj is defined by +Cj = 2|φj⟩⟨φj| − I2, +|φj⟩ = +� +pL +j |L⟩ + +� +pR +j |R⟩, +where |L⟩ = T [1 0] and |R⟩ = T [0 1]. The shift operator S is given by +S (|j⟩ ⊗ |L⟩) = |j − 1⟩ ⊗ |R⟩, +S (|j⟩ ⊗ |R⟩) = |j + 1⟩ ⊗ |L⟩. +Let XD +t (t = 0, 1, . . .) be the random variable representing the position of the DTQWer at time t. In this +paper, we only consider XD +0 = 0 cases. The distribution of XD +t +is defined by +P +� +XD +t = j|XD +0 = 0 +� +: = ∥(⟨j| ⊗ I2) UDT QW (t) (|0⟩ ⊗ |R⟩)∥2 += |(⟨j| ⊗ ⟨L|) UDT QW (t) (|0⟩ ⊗ |R⟩)|2 + |(⟨j| ⊗ ⟨R|) UDT QW (t) (|0⟩ ⊗ |R⟩)|2 . +We also consider the time averaged distribution ¯pD of the DTQW defined by +¯pD(j) := lim +T →∞ +1 +T +T −1 +� +t=0 +P +� +XD +t = j|XD +0 = 0 +� +, +for each vertex j ∈ V (Pn+1). We define a random variable ¯XD +n as P +� ¯XD +n = j +� += ¯pD(j). +3 +Relations between ¯XC +n and ¯XD +n +Since the Jacobi matrix J is a real symmetric matrix with simple [4] and symmetric [3] eigenvalues, we +obtain eigenvalues 1 = λ0 > λ1 > · · · > λn−1 > λn = −1 and corresponding eigenvectors {|vℓ⟩}n +ℓ=0 as an +orthonormal basis of n-dimensional complex vector space Cn. Thus we have the spectral decomposition +J = +n +� +ℓ=0 +λℓ|vℓ⟩⟨vℓ|. +Noting that L = In+1 − J, the spectral decomposition of UCT QW (t) is given by +UCT QW (t) = +n +� +ℓ=0 +exp [it (1 − λℓ)] |vℓ⟩⟨vℓ| = eit +n +� +ℓ=0 +e−itλℓ|vℓ⟩⟨vℓ|. +Because of simple eigenvalues of the Jacobi matrix J, the time averaged distribution ¯pC is expressed by +¯pC(j) = +n +� +ℓ=0 +|⟨j|vℓ⟩|2 |⟨vℓ|0⟩|2 = +n +� +ℓ=0 +|vℓ(j)|2 |vℓ(0)|2 , +3 + +where vℓ(j) is the jth component of |vℓ⟩. +On the other hand, the spectral decomposition of UDT QW (t) is given (see e.g. [3,5,12,13]) by +UDT QW (t) = µ0|u0⟩⟨u0| + +n−1 +� +ℓ=1 +� +1 +2(1 − λ2 +ℓ) +� +± +µ±ℓ|u±ℓ⟩⟨u±ℓ| +� ++ µn|un⟩⟨un|, +where + + + + + +µ0 = λ0 = 1, +|u0⟩ = |v0⟩, +µ±ℓ = exp +� +±i cos−1 λℓ +� +, +|u±ℓ⟩ = |vℓ⟩ − µ±ℓ S|vℓ⟩, +µn = λn = −1, +|un−1⟩ = |vn−1⟩, +with +|vℓ⟩ = vℓ(0)|0⟩ ⊗ |R⟩ + +n−1 +� +j=1 +vℓ(j)|j⟩ ⊗ |φj⟩ + vℓ(n)|n⟩ ⊗ |L⟩. +All the eigenvalues of UDT QW (t) are also simple, the time averaged distribution ¯pD is expressed by +¯pD(j) = +� +|(⟨j| ⊗ ⟨L|) |u0⟩|2 + |(⟨j| ⊗ ⟨R|) |u0⟩|2� +|⟨u0| (|0⟩ ⊗ |R⟩)|2 ++ +n−1 +� +ℓ=1 +� +1 +2(1 − λ2 +ℓ) +� +± +� +|(⟨j| ⊗ ⟨L|) |u±ℓ⟩|2 + |(⟨j| ⊗ ⟨R|) |u±ℓ⟩|2� +|⟨u±ℓ| (|0⟩ ⊗ |R⟩)|2 +� ++ +� +|(⟨j| ⊗ ⟨L|) |un⟩|2 + |(⟨j| ⊗ ⟨R|) |un⟩|2� +|⟨un| (|0⟩ ⊗ |R⟩)|2 . +More concrete expression of ¯pD in terms of eigenvalues and eigenvectors of the Jacobi matrix J is given as +follows (rearrangement of Eq.(10) in [3]): +¯pD(j) = 1 +2 |v0(j)|2 |v0(0)|2 + 1 +2 |vn(j)|2 |vn(0)|2 ++ 1 +2 +n +� +ℓ=0 +|vℓ(j)|2 |vℓ(0)|2 ++ 1 +2 +n−1 +� +ℓ=1 +1 +1 − λ2 +ℓ +� +pR +j−1 |vℓ(j − 1)|2 − λ2 +ℓ |vℓ(j)|2 + pL +j+1 |vℓ(j + 1)|2� +|vℓ(0)|2 , +with conventions pR +−1 = vℓ(−1) = pL +n+1 = vℓ(n + 1) = 0. +Now we consider the distribution functions ¯F C +n (x) := P +� ¯XC +n ≤ x +� += � +j≤x ¯pC(j) of ¯XC +n and ¯F D +n (x) := +P +� ¯XD +n ≤ x +� += � +j≤x ¯pD(j) of ¯XD +n . For each integer 0 ≤ k ≤ n − 1, we have +¯F C +n (k) = +k +� +j=0 +¯pC(j) = +k +� +j=0 +� n +� +ℓ=0 +|vℓ(j)|2 |vℓ(0)|2 +� +. +4 + +We also obtain the following expression by using pL +j + pR +j = 1, pR +0 = 1 and pL +1 |vℓ(1)|2 = λ2 +ℓ |vℓ(0)|2: +¯F D +n (k) = +k +� +j=0 +¯pD(j) += 1 +2 +k +� +j=0 +|v0(j)|2 |v0(0)|2 + 1 +2 +k +� +j=0 +|vn(j)|2 |vn(0)|2 ++ 1 +2 +k +� +j=0 +� n +� +ℓ=0 +|vℓ(j)|2 |vℓ(0)|2 +� ++ 1 +2 +k +� +j=1 +�n−1 +� +ℓ=1 +|vℓ(j)|2 |vℓ(0)|2 +� ++ 1 +2 +n−1 +� +ℓ=1 +1 +1 − λ2 +ℓ +� +pR +0 |vℓ(0)|2 − pL +1 |vℓ(1)|2 − pR +k |vℓ(k)|2 + pL +k+1 |vℓ(k + 1)|2� +|vℓ(0)|2 += +k +� +j=0 +� n +� +ℓ=0 +|vℓ(j)|2 |vℓ(0)|2 +� ++ 1 +2 +n−1 +� +ℓ=1 +1 +1 − λ2 +ℓ +� +−pR +k |vℓ(k)|2 + pL +k+1 |vℓ(k + 1)|2� +|vℓ(0)|2 += ¯F C +n (k) + 1 +2 +n−1 +� +ℓ=1 +1 +1 − λ2 +ℓ +� +−pR +k |vℓ(k)|2 + pL +k+1 |vℓ(k + 1)|2� +|vℓ(0)|2 . +4 +Scaling limit +In this section, we state our main result and prove it. +Theorem 4.1 Assume that there exists the spectral gap, i.e., lim supn→∞ λ1 < 1 = λ0. If +¯ +XC +n +n +converges +weakly to the random variable ¯X as n → ∞ then +¯ +XD +n +n +also converges weakly to the same random variable ¯X. +Proof of Theorem 4.1 +Let ¯F be the distribution function of the random variable ¯X. We assume that +lim +n→∞ P +� ¯XC +n +n +≤ x +� += ¯F(x) +(4.1) +for all points x at which ¯F is continuous. Hereafter we assume ¯F is continuous at x (0 ≤ x ≤ 1). Remark +that from the definition, Eq. (4.1) means that +lim +n→∞ +¯F C +n (nx) = lim +n→∞ +¯F C +n (⌊nx⌋) = lim +n→∞ +⌊nx⌋ +� +j=0 +� n +� +ℓ=0 +|vℓ(j)|2 |vℓ(0)|2 +� += ¯F(x), +(4.2) +where ⌊a⌋ denotes the biggest integer which is not greater than a. +From Eq. (4.2) and the relation +P +� ¯XD +n +n +≤ x +� += ¯F D +n (nx) = ¯F D +n (⌊nx⌋) += ¯F C +n (⌊nx⌋) + 1 +2 +n−1 +� +ℓ=1 +1 +1 − λ2 +ℓ +� +− pR +⌊nx⌋ |vℓ(⌊nx⌋)|2 + pL +⌊nx⌋+1 |vℓ(⌊nx⌋ + 1)|2 +� +|vℓ(0)|2 , +if we can prove +lim +n→∞ +n−1 +� +ℓ=1 +1 +1 − λ2 +ℓ +|vℓ(⌊nx⌋)|2 |vℓ(0)|2 = lim +n→∞ +n−1 +� +ℓ=1 +1 +1 − λ2 +ℓ +|vℓ(⌊nx⌋ + 1)|2 |vℓ(0)|2 = 0, +(4.3) +5 + +then we can conclude +lim +n→∞ P +� ¯XD +n +n +≤ x +� += ¯F(x), +for all points at which ¯F is continuous. +From Eq.(4.2), we obtain +0 ≤ +⌊nx⌋ +� +j=0 +�n−1 +� +ℓ=1 +|vℓ(j)|2 |vℓ(0)|2 +� +≤ ¯F C +n (⌊nx⌋) +n→∞ +−−−−→ ¯F(x). +Also we have +0 ≤ +⌊nx⌋+1 +� +j=0 +�n−1 +� +ℓ=1 +|vℓ(j)|2 |vℓ(0)|2 +� +≤ ¯F C +n +�� +n +� +x + 1 +n +��� +n→∞ +−−−−→ ¯F(x), +from continuity of ¯F at x. These mean that +lim +n→∞ +n−1 +� +ℓ=1 +|vℓ(⌊nx⌋)|2 |vℓ(0)|2 = lim +n→∞ +n−1 +� +ℓ=1 +|vℓ(⌊nx⌋ + 1)|2 |vℓ(0)|2 = 0. +(4.4) +Therefore combining with Eq. (4.4), we obtain Eq. (4.3) as follows: +lim sup +n→∞ +n−1 +� +ℓ=1 +1 +1 − λ2 +ℓ +|vℓ(⌊nx⌋)|2 |vℓ(0)|2 ≤ lim sup +n→∞ +1 +1 − λ2 +1 +n−1 +� +ℓ=1 +|vℓ(⌊nx⌋)|2 |vℓ(0)|2 +≤ +1 +1 − lim supn→∞ λ2 +1 +× lim +n→∞ +n−1 +� +ℓ=1 +|vℓ(⌊nx⌋)|2 |vℓ(0)|2 += 0, +lim sup +n→∞ +n−1 +� +ℓ=1 +1 +1 − λ2 +ℓ +|vℓ(⌊nx⌋ + 1)|2 |vℓ(0)|2 ≤ lim sup +n→∞ +1 +1 − λ2 +1 +n−1 +� +ℓ=1 +|vℓ(⌊nx⌋ + 1)|2 |vℓ(0)|2 +≤ +1 +1 − lim supn→∞ λ2 +1 +× lim +n→∞ +n−1 +� +ℓ=1 +|vℓ(⌊nx⌋ + 1)|2 |vℓ(0)|2 += 0. +This completes the proof. +✷ +References +[1] Anahara, Y., Konno, N., Morioka, H., Segawa, E.: Comfortable place for quantum walker on finite path. +Quantum Inf. Process. 21, 242 (2022). +[2] Higuchi, K., Komatsu, T., Konno, N., Morioka, H., Segawa, E.: A discontinuity of the energy of quantum walk +in impurities. Symmetry 13, 1134 (2022). +[3] Ho, C.-L., Ide, Y., Konno, N., Segawa, E., Takumi, K.: A spectral analysis of discrete-time quantum walks +related to the birth and death chains. J. Stat. Phys. 171, 207–219 (2018). +[4] Hora, A., Obata, N.: Quantum Probability and Spectral Analysis of Graphs. Springer (2007). +[5] Ide, Y., Konno, N., Segawa, E.: Time averaged distribution of a discrete-time quantum walk on the path. +Quantum Inf. Process. 11 (5), 1207–1218 (2012). +[6] Kempe, J.: Quantum random walks - an introductory overview. Contemporary Physics 44, 307–327 (2003). +6 + +[7] Kendon, V.: Decoherence in quantum walks - a review. Math. Struct. in Comp. Sci. 17, 1169–1220 (2007). +[8] Konno, N.: Quantum Walks. In: Quantum Potential Theory, Franz, U., and Sch¨urmann, M., Eds., Lecture +Notes in Mathematics: Vol. 1954, pp. 309–452, Springer-Verlag, Heidelberg (2008). +[9] Manouchehri, K., Wang, J.: Physical Implementation of Quantum Walks, Springer (2013). +[10] Marquezino, F. L., Portugal, R., Abal, G., Donangelo, R.: Mixing times in quantum walks on the hypercube. +Phys. Rev. A 77, 042312 (2008). +[11] Portugal, R.: Quantum Walks and Search Algorithms, Springer (2013). +[12] Segawa, E.: Localization of quantum walks induced by recurrence properties of random walks. J. Comput. +Nanosci. 10, 1583–1590 (2013). +[13] Szegedy, M.: Quantum speed-up of Markov chain based algorithms. Proc. of the 45th Annual IEEE Symposium +on Foundations of Computer Science (FOCS’04), 32–41 (2004). +[14] Venegas-Andraca, S. E.: Quantum Walks for Computer Scientists, Morgan and Claypool (2008). +[15] Venegas-Andraca, S. E.: Quantum walks: a comprehensive review, Quantum Inf. Process. 11, 1015–1106 +(2012). +7 + diff --git a/5dAyT4oBgHgl3EQfcfda/content/tmp_files/load_file.txt b/5dAyT4oBgHgl3EQfcfda/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4e37ad39cb75f1ff6db89502919459b1426350a1 --- /dev/null +++ b/5dAyT4oBgHgl3EQfcfda/content/tmp_files/load_file.txt @@ -0,0 +1,317 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf,len=316 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='00283v1 [quant-ph] 31 Dec 2022 Scaling limit of the time averaged distribution for continuous time quantum walk and Szegedy’s walk on the path Yusuke Ide Department of Mathematics, College of Humanities and Sciences, Nihon University 3-25-40 Sakura-josui, Setagaya-ku, Tokyo 156-8550, Japan e-mail: ide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='yusuke@nihon-u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='jp Abstract In this paper, we consider Szegedy’s walk, a type of discrete time quantum walk, and corresponding continuous time quantum walk related to the birth and death chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' We show that the scaling limit of time averaged distribution for the continuous time quantum walk induces that of Szegedy’s walk if there exists the spectral gap on so-called the corresponding Jacobi matrix .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' 1 Introduction Quantum walks, a quantum counterpart of random walks have been extensively developed in various fields during the last two decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' Since quantum walks are very simple models therefore they play fundamental and important roles in both theoretical fields and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' There are good review articles for these developments such as Kempe [6], Kendon [7], Venegas-Andraca [14,15], Konno [8], Manouchehri and Wang [9], and Portugal [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' We investigate the time averaged distribution of a variant of discrete time quantum walk (DTQW) so-called Szegedy’s walk [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' On the path graph, the spectral properties of Szegedy’s walk are directly connected to the theory of (finite type) orthogonal polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' There are studies of the distribution of Szegedy’s walk on the path graph for example [1–3,5,10,12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' In this paper, we focus on scaling limit of the time averaged distributions of both Szegedy’s walk and corresponding continuous time quantum walk on the path graph related to the random walk with reflecting walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' In order to our main theorem (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='1), if there exists the spectral gap, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=', the limit superior in the size of the path graph tends to infinity of the second largest eigenvalue of the Jacobi matrix is less than one (the largest eigenvalue), then the scaling limit of Szegedy’s walk is the same as that of corresponding continuous time quantum walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' We should note that existence of the spectral gap of the Jacobi matrix is equivalent to that of the transition matrix of corresponding random walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' A typical example of this case is space homogeneous random walk with pR j = p case (the second largest eigenvalue is 2 � p(1 − p) cos π/n) treated in [5] except for the symmetric random walk with pR j = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' Unfortunately we have not been covered with non-spectral gap cases including symmetric random walk and the Ehrenfest model (the second largest eigenvalue is 1 − 2/n) treated in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' To reveal non-spectral gap case is one of interesting future problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' 2, we define our setting of discrete time random walk, continuous time quantum walk and discrete time quantum walk on the path graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' 3 is devoted to show relationships between the time averaged distribution of Szegedy’s walk and continuous time quantum walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' In the last section, we state our main theorem (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='1) and prove it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' 2 Definition of the models In this paper, we consider the path graph Pn+1 = (V (Pn+1), E(Pn+1)) with the vertex set V (Pn+1) = {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=', n} and the (undirected) edge set E(Pn+1) = {(j, j + 1) : j = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' , n − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' On the path graph Keywords: birth and death chain, Szegedy’s walk, continuous time quantum walk, scaling limit, time averaged distribution 1 Pn+1, we define a discrete time random walk (DTRW) with reflecting walls as follows: Let pL j be the transition probability of the random walker at the vertex j ∈ V (Pn+1) to the left (j − 1 ∈ V (Pn+1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' Also let pR j = 1−pL j be the transition probability of the random walker at the vertex j ∈ V (Pn+1) to the right (j + 1 ∈ V (Pn+1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' For the sake of simplicity, we assume 0 < pL j , pR j < 1 except for j = 0, n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' We put the reflecting walls at the vertex 0 ∈ V (Pn+1) and the vertex n ∈ V (Pn+1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=', we set pR 0 = pL n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' We also call this type of DTRW as the birth and death chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' Let a positive constant Cπ be Cπ := 1 + n � j=1 pR 0 · pR 1 · · · pR j−1 pL 1 · pL 2 · · · pL j then we can define the stationary distribution {π(0), π(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' , π(n)} as π(j) = \uf8f1 \uf8f2 \uf8f3 1 Cπ if j = 0, 1 Cπ · pR 0 ·pR 1 ···pR j−1 pL 1 ·pL 2 ···pL j if j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=', n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' Note that π(j) > 0 for all j ∈ V (Pn+1) and the stationary distribution is satisfied with so-called the detailed balance condition, π(j) · pR j = pL j+1 · π(j + 1), for j = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' In order to define a continuous time quantum walk (CTQW) corresponding to the DTRW, we intro- duce the normalized Laplacian matrix L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' Let P be the transition matrix of the DTRW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' Also we de- fine diagonal matrices D1/2 π := diag �� π(0), � π(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' , � π(n) � and D−1/2 π = � D1/2 π �−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' Note that D−1/2 π = diag � 1/ � π(0), 1/ � π(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' , 1/ � π(n) � by the definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' The normalized Laplacian matrix L is given by L := D1/2 π (In+1 − P) D−1/2 π = In+1 − D1/2 π PD−1/2 π , where In+1 be the (n + 1) × (n + 1) identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' We should remark that the matrix J := D1/2 π PD−1/2 π , is referred as the Jacobi matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' So we can rewrite L as L = In+1 − J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' By using the detailed balance condition, we obtain Jj,k = Jk,j = �� pR j pL j+1, if k = j + 1, 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' Thus L = In+1 − J is an Hermitian matrix (real symmetric matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' The CTQW which is discussed in this paper is driven by the time evolution operator (unitary matrix) UCT QW (t) := exp (itL) := ∞ � k=0 (it)k k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' Lk, where i is the imaginary unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' Let XC t (t ≥ 0) be the random variable representing the position of the CTQWer at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' The distribution of XC t is determined by P � XC t = k|XC 0 = j � := |⟨k|UCT QW (t)|j⟩|2 = ���(UCT QW (t))k,j ��� 2 , where |j⟩ is the (n + 1)-dimensional unit vector (column vector) which j-th component equals 1 and the other components are 0 and ⟨v| is the transpose of |v⟩, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=', ⟨v| = T |v⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' 2 Hereafter we only consider XC 0 = 0 , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=', the CTQWer starts from the left most vertex 0 ∈ V (Pn+1), cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' The time averaged distribution ¯pC of the CTQW is defined by ¯pC(j) := lim T →∞ 1 T � T 0 P � XC t = j|XC 0 = 0 � dt, for each vertex j ∈ V (Pn+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' We define a random variable ¯XC n as P � ¯XC n = j � = ¯pC(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' In this paper, we also deal with a type of discrete time quantum walk (DTQW) corresponding to the DTRW so-called Szegedy’s walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' The time evolution operator for the DTQW is defined by U = SC with the coin operator C and the shift operator (flip-flop type shift) S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' The coin operator C is defined by C = |0⟩⟨0| ⊗ I2 + n−1 � j=1 |j⟩⟨j| ⊗ Cj + |n⟩⟨n| ⊗ I2, where I2 is the 2 × 2 identity matrix and ⊗ is the tensor product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' The local coin operator Cj is defined by Cj = 2|φj⟩⟨φj| − I2, |φj⟩ = � pL j |L⟩ + � pR j |R⟩, where |L⟩ = T [1 0] and |R⟩ = T [0 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' The shift operator S is given by S (|j⟩ ⊗ |L⟩) = |j − 1⟩ ⊗ |R⟩, S (|j⟩ ⊗ |R⟩) = |j + 1⟩ ⊗ |L⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' Let XD t (t = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=') be the random variable representing the position of the DTQWer at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' In this paper, we only consider XD 0 = 0 cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' The distribution of XD t is defined by P � XD t = j|XD 0 = 0 � : = ∥(⟨j| ⊗ I2) UDT QW (t) (|0⟩ ⊗ |R⟩)∥2 = |(⟨j| ⊗ ⟨L|) UDT QW (t) (|0⟩ ⊗ |R⟩)|2 + |(⟨j| ⊗ ⟨R|) UDT QW (t) (|0⟩ ⊗ |R⟩)|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' We also consider the time averaged distribution ¯pD of the DTQW defined by ¯pD(j) := lim T →∞ 1 T T −1 � t=0 P � XD t = j|XD 0 = 0 � , for each vertex j ∈ V (Pn+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' We define a random variable ¯XD n as P � ¯XD n = j � = ¯pD(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' 3 Relations between ¯XC n and ¯XD n Since the Jacobi matrix J is a real symmetric matrix with simple [4] and symmetric [3] eigenvalues, we obtain eigenvalues 1 = λ0 > λ1 > · · · > λn−1 > λn = −1 and corresponding eigenvectors {|vℓ⟩}n ℓ=0 as an orthonormal basis of n-dimensional complex vector space Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' Thus we have the spectral decomposition J = n � ℓ=0 λℓ|vℓ⟩⟨vℓ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' Noting that L = In+1 − J, the spectral decomposition of UCT QW (t) is given by UCT QW (t) = n � ℓ=0 exp [it (1 − λℓ)] |vℓ⟩⟨vℓ| = eit n � ℓ=0 e−itλℓ|vℓ⟩⟨vℓ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' Because of simple eigenvalues of the Jacobi matrix J, the time averaged distribution ¯pC is expressed by ¯pC(j) = n � ℓ=0 |⟨j|vℓ⟩|2 |⟨vℓ|0⟩|2 = n � ℓ=0 |vℓ(j)|2 |vℓ(0)|2 , 3 where vℓ(j) is the jth component of |vℓ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' On the other hand, the spectral decomposition of UDT QW (t) is given (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' [3,5,12,13]) by UDT QW (t) = µ0|u0⟩⟨u0| + n−1 � ℓ=1 � 1 2(1 − λ2 ℓ) � ± µ±ℓ|u±ℓ⟩⟨u±ℓ| � + µn|un⟩⟨un|, where \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 µ0 = λ0 = 1, |u0⟩ = |v0⟩, µ±ℓ = exp � ±i cos−1 λℓ � , |u±ℓ⟩ = |vℓ⟩ − µ±ℓ S|vℓ⟩, µn = λn = −1, |un−1⟩ = |vn−1⟩, with |vℓ⟩ = vℓ(0)|0⟩ ⊗ |R⟩ + n−1 � j=1 vℓ(j)|j⟩ ⊗ |φj⟩ + vℓ(n)|n⟩ ⊗ |L⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' All the eigenvalues of UDT QW (t) are also simple, the time averaged distribution ¯pD is expressed by ¯pD(j) = � |(⟨j| ⊗ ⟨L|) |u0⟩|2 + |(⟨j| ⊗ ⟨R|) |u0⟩|2� |⟨u0| (|0⟩ ⊗ |R⟩)|2 + n−1 � ℓ=1 � 1 2(1 − λ2 ℓ) � ± � |(⟨j| ⊗ ⟨L|) |u±ℓ⟩|2 + |(⟨j| ⊗ ⟨R|) |u±ℓ⟩|2� |⟨u±ℓ| (|0⟩ ⊗ |R⟩)|2 � + � |(⟨j| ⊗ ⟨L|) |un⟩|2 + |(⟨j| ⊗ ⟨R|) |un⟩|2� |⟨un| (|0⟩ ⊗ |R⟩)|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' More concrete expression of ¯pD in terms of eigenvalues and eigenvectors of the Jacobi matrix J is given as follows (rearrangement of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' (10) in [3]): ¯pD(j) = 1 2 |v0(j)|2 |v0(0)|2 + 1 2 |vn(j)|2 |vn(0)|2 + 1 2 n � ℓ=0 |vℓ(j)|2 |vℓ(0)|2 + 1 2 n−1 � ℓ=1 1 1 − λ2 ℓ � pR j−1 |vℓ(j − 1)|2 − λ2 ℓ |vℓ(j)|2 + pL j+1 |vℓ(j + 1)|2� |vℓ(0)|2 , with conventions pR −1 = vℓ(−1) = pL n+1 = vℓ(n + 1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' Now we consider the distribution functions ¯F C n (x) := P � ¯XC n ≤ x � = � j≤x ¯pC(j) of ¯XC n and ¯F D n (x) := P � ¯XD n ≤ x � = � j≤x ¯pD(j) of ¯XD n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' For each integer 0 ≤ k ≤ n − 1, we have ¯F C n (k) = k � j=0 ¯pC(j) = k � j=0 � n � ℓ=0 |vℓ(j)|2 |vℓ(0)|2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' 4 We also obtain the following expression by using pL j + pR j = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' pR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='0 = 1 and pL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='1 |vℓ(1)|2 = λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='ℓ |vℓ(0)|2: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='¯F D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='n (k) = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='j=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='¯pD(j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='= 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='j=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='|v0(j)|2 |v0(0)|2 + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='j=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='|vn(j)|2 |vn(0)|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='j=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='� n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='ℓ=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='|vℓ(j)|2 |vℓ(0)|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='j=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='�n−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='ℓ=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='|vℓ(j)|2 |vℓ(0)|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='n−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='ℓ=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='1 − λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='ℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='pR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='0 |vℓ(0)|2 − pL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='1 |vℓ(1)|2 − pR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='k |vℓ(k)|2 + pL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='k+1 |vℓ(k + 1)|2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='|vℓ(0)|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='j=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='� n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='ℓ=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='|vℓ(j)|2 |vℓ(0)|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='n−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='ℓ=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='1 − λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='ℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='−pR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='k |vℓ(k)|2 + pL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='k+1 |vℓ(k + 1)|2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='|vℓ(0)|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='= ¯F C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='n (k) + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='n−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='ℓ=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='1 − λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='ℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='−pR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='k |vℓ(k)|2 + pL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='k+1 |vℓ(k + 1)|2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='|vℓ(0)|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' 4 Scaling limit In this section, we state our main result and prove it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='1 Assume that there exists the spectral gap, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=', lim supn→∞ λ1 < 1 = λ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' If ¯ XC n n converges weakly to the random variable ¯X as n → ∞ then ¯ XD n n also converges weakly to the same random variable ¯X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='1 Let ¯F be the distribution function of the random variable ¯X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' We assume that lim n→∞ P � ¯XC n n ≤ x � = ¯F(x) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='1) for all points x at which ¯F is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' Hereafter we assume ¯F is continuous at x (0 ≤ x ≤ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' Remark that from the definition, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='1) means that lim n→∞ ¯F C n (nx) = lim n→∞ ¯F C n (⌊nx⌋) = lim n→∞ ⌊nx⌋ � j=0 � n � ℓ=0 |vℓ(j)|2 |vℓ(0)|2 � = ¯F(x), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='2) where ⌊a⌋ denotes the biggest integer which is not greater than a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='2) and the relation P � ¯XD n n ≤ x � = ¯F D n (nx) = ¯F D n (⌊nx⌋) = ¯F C n (⌊nx⌋) + 1 2 n−1 � ℓ=1 1 1 − λ2 ℓ � − pR ⌊nx⌋ |vℓ(⌊nx⌋)|2 + pL ⌊nx⌋+1 |vℓ(⌊nx⌋ + 1)|2 � |vℓ(0)|2 , if we can prove lim n→∞ n−1 � ℓ=1 1 1 − λ2 ℓ |vℓ(⌊nx⌋)|2 |vℓ(0)|2 = lim n→∞ n−1 � ℓ=1 1 1 − λ2 ℓ |vℓ(⌊nx⌋ + 1)|2 |vℓ(0)|2 = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='3) 5 then we can conclude lim n→∞ P � ¯XD n n ≤ x � = ¯F(x), for all points at which ¯F is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='2), we obtain 0 ≤ ⌊nx⌋ � j=0 �n−1 � ℓ=1 |vℓ(j)|2 |vℓ(0)|2 � ≤ ¯F C n (⌊nx⌋) n→∞ −−−−→ ¯F(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' Also we have 0 ≤ ⌊nx⌋+1 � j=0 �n−1 � ℓ=1 |vℓ(j)|2 |vℓ(0)|2 � ≤ ¯F C n �� n � x + 1 n ��� n→∞ −−−−→ ¯F(x), from continuity of ¯F at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' These mean that lim n→∞ n−1 � ℓ=1 |vℓ(⌊nx⌋)|2 |vℓ(0)|2 = lim n→∞ n−1 � ℓ=1 |vℓ(⌊nx⌋ + 1)|2 |vℓ(0)|2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='4) Therefore combining with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='4), we obtain Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='3) as follows: lim sup n→∞ n−1 � ℓ=1 1 1 − λ2 ℓ |vℓ(⌊nx⌋)|2 |vℓ(0)|2 ≤ lim sup n→∞ 1 1 − λ2 1 n−1 � ℓ=1 |vℓ(⌊nx⌋)|2 |vℓ(0)|2 ≤ 1 1 − lim supn→∞ λ2 1 × lim n→∞ n−1 � ℓ=1 |vℓ(⌊nx⌋)|2 |vℓ(0)|2 = 0, lim sup n→∞ n−1 � ℓ=1 1 1 − λ2 ℓ |vℓ(⌊nx⌋ + 1)|2 |vℓ(0)|2 ≤ lim sup n→∞ 1 1 − λ2 1 n−1 � ℓ=1 |vℓ(⌊nx⌋ + 1)|2 |vℓ(0)|2 ≤ 1 1 − lim supn→∞ λ2 1 × lim n→∞ n−1 � ℓ=1 |vℓ(⌊nx⌋ + 1)|2 |vℓ(0)|2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' ✷ References [1] Anahara, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=', Konno, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=', Morioka, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=', Segawa, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=': Comfortable place for quantum walker on finite path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' Quantum Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' 21, 242 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' [2] Higuchi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=', Komatsu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=', Konno, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=', Morioka, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=', Segawa, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=': A discontinuity of the energy of quantum walk in impurities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' Symmetry 13, 1134 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' [3] Ho, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=', Ide, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=', Konno, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=', Segawa, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=', Takumi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=': A spectral analysis of discrete-time quantum walks related to the birth and death chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQfcfda/content/2301.00283v1.pdf'} +page_content=' J.' metadata={'source': 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diff --git a/6dE1T4oBgHgl3EQfmwTN/content/tmp_files/2301.03302v1.pdf.txt b/6dE1T4oBgHgl3EQfmwTN/content/tmp_files/2301.03302v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..5bcf392a4779c75a79176ba01659b8665c173878 --- /dev/null +++ b/6dE1T4oBgHgl3EQfmwTN/content/tmp_files/2301.03302v1.pdf.txt @@ -0,0 +1,2603 @@ +arXiv:2301.03302v1 [eess.SY] 9 Jan 2023 +A Rolling Horizon Game Considering Network Effect +in Cluster Forming for Dynamic Resilient +Multiagent Systems +Yurid Nugraha a, Ahmet Cetinkaya b, Tomohisa Hayakawa a, Hideaki Ishii c, +Quanyan Zhu d +aDepartment of Systems and Control Engineering, Tokyo Institute of Technology, Tokyo 152-8552, Japan +bDepartment of Functional Control Systems, Shibaura Institute of Technology, Tokyo, 135-8548, Japan +cDepartment of Computer Science, Tokyo Insitute of Technology, Yokohama 226-8502, Japan +dDepartment of Electrical and Computer Engineering, New York University, Brooklyn NY, 11201, USA +Abstract +A two-player game-theoretic problem on resilient graphs in a multiagent consensus setting is formulated. An attacker is +capable to disable some of the edges of the network with the objective to divide the agents into clusters by emitting jamming +signals while, in response, the defender recovers some of the edges by increasing the transmission power for the communication +signals. Specifically, we consider repeated games between the attacker and the defender where the optimal strategies for the +two players are derived in a rolling horizon fashion based on utility functions that take both the agents’ states and the sizes of +clusters (known as network effect) into account. The players’ actions at each discrete-time step are constrained by their energy +for transmissions of the signals, with a less strict constraint for the attacker. Necessary conditions and sufficient conditions +of agent consensus are derived, which are influenced by the energy constraints. The number of clusters of agents at infinite +time in the face of attacks and recoveries are also characterized. Simulation results are provided to demonstrate the effects of +players’ actions on the cluster forming and to illustrate the players’ performance for different horizon parameters. +Key words: Multiagent Systems, Cybersecurity, Game Theory, Consensus, Cluster Forming, Network Effect/Network +Externality +1 +Introduction +Applications of large-scale networked systems have +rapidly grown in various areas of critical infrastructures +including power grids and transportation systems. Such +systems can be considered as multiagent systems where +a number of agents capable of making local decisions +interact over a network and exchange information to +reach a common goal [2]. While wireless communica- +tion plays an important role for the functionality of the +network, it is also prone to cyber attacks initiated by +malicious adversaries [11,25]. +Email addresses: yurid@dsl.sc.e.titech.ac.jp (Yurid +Nugraha), ahmet@shibaura-it.ac.jp (Ahmet Cetinkaya), +hayakawa@sc.e.titech.ac.jp (Tomohisa Hayakawa), +ishii@c.titech.ac.jp (Hideaki Ishii), +quanyan.zhu@nyu.edu (Quanyan Zhu). +Jamming attacks in consensus problems of multiagent +systems have been studied in [3, 5, 28]. Noncooperative +games between attackers and other players protecting +the network are widely used to analyze security prob- +lems, including jamming attacks [12, 17] and injection +attacks [18,24,26]. +In a jamming attack formulation, it is natural to consider +that the jammer/the attacker has an energy constraint +such that, if it is not connected to energy sources, it is +impossible to attack all communication links of the net- +work at all times [4,5]. In the context of game-theoretical +approaches, this constraint becomes important to char- +acterize the strategic behaviors of the players [17]. +When the links in the network are attacked, the agents +may become disconnected from other agents, resulting +in several groups of connected agents, or clusters. The + +work [13] proposed the notion of network effect/network +externality, which refers to the utility of an agent in +a certain cluster depending on how many other agents +belong to that particular cluster. Such a concept has +been used to analyze grouping of agents on, e.g., social +networks and computer networks, as discussed in [10,16]. +Rolling horizon control has been used to handle sys- +tems with uncertainties. It is also studied in the context +of networked control [15,30], where there may be addi- +tional uncertainties related to communications among +agents in the networks. Rolling horizon approaches are +also discussed in noncooperative security game settings +in [34,35], where horizon lengths affect the resilience of +the system. Rolling horizon approaches have also been +used to handle the constraints in the system, e.g., in an +agent with obstacle avoidance constraints [14,27]. +In this paper, we consider a security problem in a two- +player game setting between an attacker, who is moti- +vated to disrupt the communication among agents by +attacking communication links, and a defender, who at- +tempts to recover some of the attacked links. We for- +mulate the problem based on [6, 20], which use graph +connectivity to characterize the game and the players’ +strategies. The game in this paper is played repeatedly +over discrete time in the context of multiagent consen- +sus. +As a results of these persistent attacks and recover- +ies, under consensus protocol cluster forming emerges +among the agents of the networks with different clus- +ters having different agents’ states. Cluster forming in +multiagent systems has been studied in, e.g., [1, 7, 29], +where the relations among certain agents may be hos- +tile. In this paper, we approach clustering from a dif- +ferent viewpoint based on a game-theoretic formulation. +Specifically, the players of the game consider network +effect/network externality [13] to form clusters among +agents. Their utilities are determined by how the net- +work is disconnected into groups of agents as well as how +the players’ actions affect the states of the agents at each +time. Under this setting, the number and the size of the +clusters are influenced by how strong the attacks are; +the stronger attacker is supposed to be able to separate +agents into more smaller clusters, and vice versa. +In the resilient network setting, it is common that there +exists a network manager who is aware of the incoming +attack, since the agents try to communicate with their +neighbor agents at all time and thus quickly know if some +of their neighbors do not send any signal. The network +manager then tries to prepare a defense plan to quickly +recover from such attacks and to repel the subsequent +attacks. +From the attacker’s viewpoint, it is also common that the +attacker knows which edges of the network are the most +vulnerable as well as how powerful the network manager +is, e.g., the manager’s remaining resources. Therefore, +we believe that this sequential model can be applied to +several real-world settings. +The main contribution of this paper is that we introduce +a repeated game played repeatedly over time to model +the decision making process between the attacker and +the defender in the context of network security. It is then +natural to explore how these games affect the networks +and state evolution of the agents. Consensus protocol is +considered due to its simple characterization, where all +agents should converge in the case of no attack. More +specifically, in comparison to [6, 20], our contribution +is threefold: (i) We introduce more options for the at- +tacker’s jamming signal strengths; (ii) the game consists +of multiple attack-recovery actions, resulting in more +complicated strategies; and (iii) we consider a rolling +horizon approach for the players so that their strategies +may be modified as they obtain new knowledge of the +status of the system. +More specifically, it is now possible for the attacker to +disable links with stronger intensity of attack signals +so that the defender is unable to recover those links +(the decision on which edges are to be attacked with +stronger attack signals is made at the same time as the +decision on which edges are to be attacked with nor- +mal attack signals); this feature is motivated by [32,33]. +In practice, this is possible when the attacker emits +stronger jamming signals that takes more resource that +results in much lower signal-to-interference-plus-noise +ratio (SINR) so that it is not possible for the defender to +recover the communication on those links with its lim- +ited recovery strength. On the other hand, we consider +games consisting of multiple parts, where the players +need to consider their future utilities and energy con- +straints when deciding their strategies at any point in +time. This setting enables the the players to think fur- +ther ahead and prioritize their long-term payoffs, com- +pared to in a single-step case. The players recalculate and +may override their strategies as time goes on, according +to the rolling horizon approach. A related formulation +without rolling horizon is discussed in [19], where the +players are not able to change their strategies decided at +earlier times. +The paper is organized as follows. In Section 2, we in- +troduce the framework for the attack-recovery sequence, +cluster forming among agents, and energy consumption +models of the players. The utility functions of the games +in rolling horizon approach of the repeated games is dis- +cussed in Section 3, whereas the game structure is char- +acterized in Section 4. In Section 5, we analyze some con- +ditions of consensus among agents, which are related to +the parameters of the underlying graph and the players’ +energy constraints. We continue by discussing the clus- +ter forming of agents when consensus is not achieved in +Section 6. The equilibrium characterization of the game +under certain conditions is discussed in Section 7. We +2 + +then provide numerical examples on consensus and clus- +ter forming in Section 8 and conclude the paper in Sec- +tion 9. The conference version of this paper appeared +in [21], where we consider a more restricted situation on +how often players update their strategies. +The notations used in this paper are fairly standard. We +denote by |·| the cardinality of a set. The floor function +and the ceiling function are denoted by ⌊·⌋ and ⌈·⌉, re- +spectively. The sets of positive and nonnegative integers +are denoted by N and N0, respectively. +2 +Attack/Recovery Characterization for Multi- +agent Systems Under Consensus Dynamics +We consider a multiagent system of n agents communi- +cating to each other in discrete time in the face of jam- +ming attacks. The agents are aiming to converge to a +consensus state by interacting with each other over the +communication network. The network topology for the +normal operation is given by an undirected and con- +nected graph G = (V, E). The graph consists of the set V +of vertices representing the agents and the set E ⊆ V ×V +of edges representing the communication links. The edge +connectivity [2] of the connected graph G is denoted by +λ. +Each agent i has the scalar state xi[k] following the +discrete-time update rule at time k ∈ N0 given by +xi[k + 1] = xi[k] + ui[k], +x[0] = x0, +(1) +where ui[k] denotes the control input applied to agent i. +We assume that ui[k] is constructed as the weighted sum +of the state differences between agent i and its neighbor +agents, commonly used in, e.g., [8], which is given by +ui[k] = +� +j∈Ni[k] +aij(xj[k] − xi[k]), +(2) +where Ni[k] denotes the set of agents that can communi- +cate with agent i at time k, and aij represents the weight +of edge (i, j) ∈ E such that Σn +j=1,j̸=iaij < 1, i ∈ V to +ensure that the agents achieve consensus without any +attack. +We assume that the jamming attacks on an edge affect +the communication between the two agents connected +by that attacked edge. As a result, the set Ni[k] may +change, and the resulting communication topology can +be disconnected at time k. Such jamming attacks are +represented by the removal of edges in G. On the other +hand, within the system there is a defender that may be +capable of maintaining the communication among the +agents, e.g., by asking agents to send stronger commu- +nication signals to overcome the jamming signals. This +action is represented as rebuilding some of the attacked +edges. +From this sequence of attacks and recoveries, we charac- +terize the attack-recovery process as a two-player game +between the attacker and the defender in terms of the +communication links in the network. In other words, the +graph characterizing the networked system is resilient if +the group of agents is able to recover from the damages +caused by the attacker. However, there may be cases +where the resiliency level of the graph is reduced if the +jamming signals are sufficiently strong such that the de- +fender cannot recover. Note that to achieve consensus, +the agents need not be connected for all time. +In this paper, we consider the case where the attacker has +two types of jamming signals in terms of their strength, +strong and normal. The defender is able to recover only +the edges that are attacked with normal strength. In the +following subsections, we first describe the sequence of +attacks and recoveries and characterize some constraints +on the players’ energy and computational ability that +we need to impose as well as how the objective of the +problem is formulated. +2.1 +Attack-Recovery Sequence +In our setting, at each discrete time k, the players (the +attacker and the defender) decide to attack/recover cer- +tain edges in two stages, with the attacker acting first +and then the defender. Specifically, at time k the at- +tacker attacks G by deleting the edges EA +k +⊆ E with +normal jamming signals and E +A +k ⊆ E with strong jam- +ming signals with EA +k ∩ E +A +k = ∅, whereas the defender +recovers ED +k ⊆ EA +k . As mentioned earlier, the defender +is not able to recover the edges attacked with strong +jamming signals, i.e., ED +k ∩ E +A +k += ∅. Due to the at- +tacks and then the recoveries, the network changes from +G to GA +k := (V, E \ (EA +k ∪ E +A +k )) and further to GD +k := +(V, (E \ (EA +k ∪ E +A +k )) ∪ ED +k ) at time k. The agents then +communicate to their neighbors Ni[k] based on this re- +sulting graph GD +k . +In this game, the players attempt to choose the best +strategies in terms of edges attacked/recovered (E +A +k , EA +k ) +and ED +k to maximize their own utility functions. Here, +the games are played every game period T time steps +and the lth game is defined over the horizon of h steps +from time (l − 1)T to (l − 1)T + h − 1, with l ∈ N and +1 ≤ T ≤ h. The players make decisions in a rolling hori- +zon fashion; the optimal strategies obtained at (l − 1)T +for the future time may be overridden when the play- +ers recalculate their strategies at time lT when the next +game starts. Fig. 1 illustrates the discussed sequence over +time with h = 8 and T = 4, where the filled circles in- +dicate the implemented strategies and the empty circles +3 + +PSfrag replacements +k +Edge +0 +1T +2T +l = 1 +l = 2 +l = 3 +horizon length h +2nd game +horizon length h +game period T +Fig. 1. Illustration of the games played over discrete time k with rolling +horizon approaches by the players. +PSfrag replacements +0 +1 +2 +3 +k +4 +Energy +κA +– Time +Fig. 2. Energy constraint of the attacker considered +in the formulation. The dashed line represents the +total supplied energy to spend. The filled circles +representing the actual energy consumed by the +player should be below the dashed line. +indicate the strategies of the game that are discarded. +In this setting, the horizon length h indicates the com- +putational ability, i.e., how long in the future the play- +ers can plan their strategies, whereas the game period +T ≤ h indicates the players’ adaptability, i.e., how long +the players apply the obtained strategies without updat- +ing (shorter T means that a player is more adaptable). +The rolling horizon game structure will be discussed in +Section 4 in more detail. +2.2 +Energy Constraints +The actions of the attacker and the defender are af- +fected by the constraints on their energy resources. It +is assumed that the total supplied energy for the play- +ers increases linearly in time; furthermore, the energy +consumed by the players is proportional to the number +of attacked/recovered edges. Here we suppose that the +players initially possess certain amount of energy κA and +κD for the attacker and the defender, respectively. More- +over, the players are assumed to be able to supply en- +ergy wirelessly to devices that obstruct/retain commu- +nication signals between the agents so that the energy +supply rates to these devices are limited by the constant +values of ρA and ρD every discrete time step. These de- +vices are supposed to have unlimited battery capacity +and thus can be supplied constantly by the players with +a linear rate ρA or ρD. +For the attacker, the strong attacks on E +A +k take β +A > +0 energy per edge per unit time whereas the normal +attacks on EA +k take βA > 0 cost per edge, with β +A > βA. +The total energy used by the attacker is constrained as +k +� +m=0 +(β +A|E +A +m|+βA|EA +m|) ≤ κA + ρAk +(3) +for any time k, where κA ≥ ρA > 0. This implies that +the total energy spent by the attacker cannot exceed the +available energy characterized as the sum of the initial +energy κA and the supplied energy ρAk by time k. This +energy constraint restricts the number of edges that the +attacker can attack. Note that the attacker’s available +energy increases by ρA at each k. The condition κA ≥ +ρA allows the attacker to have at least the same attack +ability at time k = 0. +Fig. 2 illustrates the energy constraint of the attacker, +where the dashed line with slope ρA represents the total +supplied energy and the filled circles indicate the total +energy spent. A critical case is when βA < ρA, since it is +possible for the attacker to attack at least one edge for all +times. This will have implications on the consensus and +cluster forming of the agents, as we will discuss later. +The energy constraint for the defender is similar to (3): +k +� +m=0 +βD|ED +m|≤ κD + ρDk, +(4) +with κD ≥ ρD > 0 and βD > 0. Note that there is a +single term on the left-hand side because there is only +one type of recovery signals for the agents. +3 +Utility Functions with Cluster Forming and +Agent-group Index Considerations +In our game setting, the attacker tries to make the +graph disconnected to separate the agents into clusters. +Here, we introduce a few notions related to group- +ing/clustering of agents. In a given subgraph G′ = (V, E′) +of G, the agents may be divided into n(G′) number +of groups, with the groups V′ +1, V′ +2, . . . , V′ +n(G′) being a +partition of V with ∪n(G′) +p=1 V′ +p = V and V′ +p ∩ V′ +q = ∅, if +p ̸= q. There is no edge connecting different groups, i.e., +ei′,j′ /∈ E′, ∀i′ ∈ V′ +p, j′ ∈ V′ +q. We also call each subset of +agents taking the same state at infinite time as a clus- +4 + +ter, i.e., limk→∞(xi[k] − xj[k]) = 0 implies that agents +i and j belong to the same cluster. +In the considered game, the attacker and the defender +are concerned about the number of agents in each +group. Specifically, we follow the notion of network ef- +fect/network externality [13], where the utility of an +agent in a certain group depends on how many other +agents belong to that particular group. In the context +of this game, the attacker wants to isolate agents so +that fewer agents are in each group, while the defender +wants as many agents as possible in the same group. We +then represent the level of grouping in the graph G′ by +the function c(·), which we call the agent-group index, +given by +c(G′) := +n(G′) +� +p=1 +|V′ +p|2−|V|2 +(≤ 0). +(5) +The value of c(G′) is 0 if G′ is connected, since there is +only one group (i.e., n(G′) = 1). A larger value (closer to +0) of c(G′) implies that there are fewer groups in graph +G′, and/or each group has more agents. The agent-group +indices of some graphs are shown in Fig. 3. Here, it is +interesting that c(GD) is smaller than c(GC), even though +GC has more groups. It is because the largest cluster is +constituted by more agents in GC than the case of GD. +Thus, for an attacker who tries to reduce the number of +agents in one cluster, GD is preferable to GC. +In our problem setting, the players also consider the ef- +fects of their actions on the agent states when attack- +ing/recovering. For example, the attacker may want to +separate agents having state values with more differences +in different groups. We specify the agents’ state differ- +ence zk as +zk(E +A +k , EA +k , ED +k ) := xT[k + 1]Lcx[k + 1], +(6) +with Lc, for simplicity, being the Laplacian matrix of the +complete graph with n agents. That is, (6) represents the +sum of squares of the state differences of all the agent +pairs. This implies that all state differences between any +pair of agents are worth the same and thus the players +do not prioritize any connection between agents. +The attacked and recovered edges (E +A +k , EA +k , ED +k ) will af- +fect x[k + 1] in accordance with (1) and (2), and in turn +the value of zk. Note that the value of zk is nonincreas- +ing over time [2] even if some agents are left discon- +nected from other agents under attacks. This sum-of- +square characterization of the agents’ state difference is +commonly used and essentially the same to our previous +work [19] for the continuous-time setting; here, we ex- +tend the formulation to comply with the discrete-time +1 +2 +3 +4 +5 +6 +7 +1 +2 +3 +4 +5 +6 +7 +1 +2 +3 +4 +5 +6 +7 +1 +2 +3 +4 +5 +6 +7 +PSfrag replacements +(a) GA +(b) GB +(c) GC +(d) GD +Fig. 3. Graphs and their agent-group indices: (a) c(GA) = 0, +(b) c(GB) = −12, (c) c(GC) = −22, and (d) c(GD) = −24. +Note that c(GC) is larger than c(GD), even with more number +of groups. +setting by considering the states at one time step ahead +k + 1. +Now, we combine the two measures in (5) and (6) to +construct the utility functions for the game in a zero- +sum manner. Specifically, for the lth game starting at +time k = (l−1)T , the attacker and the defender’s utility +functions take account of the agent-group index c(·) and +the difference zk of agents’ states over h horizon length +from time (l − 1)T to (l − 1)T + h − 1. With weights +a, b ≥ 0, the utilities for the lth game U A +l for the attacker +and U D +l +for the defender are, respectively, defined by +U A +l := +(l−1)T +h−1 +� +k=(l−1)T +(azk − bc(GD +k )), +(7) +U D +l := −U A +l . +(8) +In our setting both players attempt to maximize their +utilities at the start of each game l. The values of a +and b represent the preference of the players towards +either a long-term agent clustering or a short-term agent- +grouping. A higher value of a implies that the players +prefer to focus on the agent states and the subsequent +cluster forming, whereas a higher value of b implies that +they focus on the agent-grouping more. We suppose that +both players know the underlying topology G as well as +the states of all agents xi[k]. +4 +Rolling Horizon Game Structure +We are interested in finding the subgame perfect equi- +librium [9] of this game outlined in Section 3. To this +end, the game is divided into some subgames/decision- +making points. The subgame perfect equilibrium must +be an equilibrium in every subgame. The optimal strat- +egy of each player is obtained by using a backward in- +duction approach, i.e., by finding the equilibrium from +the smallest subgames. The tie-break condition happens +when the players’ strategies result in the same utility. In +this case, we suppose that the players choose to save their +energy by attacking/recovering less edges unless they +have enough energy to attack/recover all edges in ev- +ery subsequent steps, in which case they attack/recover +more edges. +Due to the nature of the rolling horizon approach, the +5 + +strategies obtained from the lth game, i.e., attacked and +recovered edges, are applied only from time (l − 1)T to +lT − 1. Specifically, in the lth game for time (l − 1)T +to (l − 1)T + h − 1, the strategies of both players +are denoted by ((E +A +l,1, EA +l,1, ED +l,1), . . . , (E +A +l,h, EA +l,h, ED +l,h)), +with (E +A +l,α, EA +l,α, ED +l,α) indicating the strategies at the +αth step of the lth game with α ∈ {1, . . ., h}. Note +that here we show the strategies with two subscripts +representing the game and the step indices along +the time axis. From the above set of strategies, only +((E +A +l,1, EA +l,1, ED +l,1), . . . , (E +A +l,T , EA +l,T , ED +l,T )) +is +applied. +Re- +call that h is taken to be greater than or equal to +T . Therefore, for the lth game from time (l − 1)T +to lT − 1, the strategy applied will be written as +((E +A +(l−1)T , EA +(l−1)T , ED +(l−1)T ), . . . , (E +A +lT −1, EA +lT −1, ED +lT −1)) := +((E +A +l,1, EA +l,1, ED +l,1), . . . , (E +A +l,T , EA +l,T , ED +l,T )). +We look at how the optimal edges can be found by an +example with h = 2 and T = 1 or 2. In this case, for the +lth game over time (l−1)T and (l−1)T +1, the optimal +strategies of the players are given by +ED∗ +l,2 (E +A +l,2, EA +l,2) ∈ arg max +ED +l,2 +U D +l,2, +(9) +(E +A∗ +l,2 (ED +l,1), EA∗ +l,2 (ED +l,1)) ∈ arg +max +(E +A +l,2,EA +l,2) +U A +l,2, +(10) +ED∗ +l,1 (E +A +l,1, EA +l,1) ∈ arg max +ED +l,1 +U D +l , +(11) +(E +A∗ +l,1 , EA∗ +l,1 ) ∈ arg +max +(E +A +l,1,EA +l,1) +U A +l , +(12) +where U A +l,α and U D +l,α are defined as parts of U A +l +and +U D +l , respectively, calculated from the αth step to the +last (hth) step of the lth game, i.e., U A +l,α = −U D +l,α := +�(l−1)T +h−1 +(l−1)T +α−1(azk − bc(GD +k )). In this case with h = 2, +the functions U A +l,2 and U D +l,2 are based on the values of +azk and bGD +k at k = (l − 1)T + 1 only. Note that to +find (E +A∗ +l,1 , EA∗ +l,1 ), one needs to obtain ED∗ +l,1 (E +A +l,1, EA +l,1) be- +forehand. Likewise, to find ED∗ +l,1 , one needs to obtain +(E +A∗ +l,2 (ED +l,1), EA∗ +l,2 (ED +l,1)). Similarly, to find (E +A∗ +l,2 , EA∗ +l,2 ), the +edges ED∗ +l,2 (E +A +l,2, EA +l,2) must be obtained beforehand. Note +that deriving the optimal strategies above is subject to +the energy constraints (3) and (4). +For h > 2, the players’ optimal strategies consist of 2h +parts similar to those in (9)–(12), with one time step +consisting of two parts of strategies corresponding to the +number of players. They are solved by the players at +every time k = (l − 1)T of the lth game, l ∈ N. With +T = h, the players do not have chance to override their +strategies, which removes the rolling horizon aspect of +the game. +We will find the optimal strategies of the players by +computing all possible combinations, since the choices of +edges are finite. From the optimization problems speci- +fied above, the players examine at most 3|E|2|E|h number +of combinations of attacked and recovered edges for util- +ity evaluations, since they have to foresee the opponent’s +response as well. Note that the attacker has three pos- +sible actions on an edge: no attack, attack with normal +signals, and attack with strong signals, whereas the de- +fender has only two actions: recover or not recover. Here +we can see that the number of computation increases ex- +ponentially with respect to the number of edges in the +underlying graph. To address scalability issues, we may +find edges that are easier to attack first, i.e., edges that +result in the formation of new groups if attacked, and +limit the strategy choices over those edges only. +Our previous works [19, 20] considered related games +in continuous time, where the timings for launching at- +tack/defense actions are also part of the decision vari- +ables. This aspect complicated the formulation, making +it difficult to study games over a time horizon. In this pa- +per, we simplify the timing issue and instead introduce +the rolling horizon feature. This enables the players to +consider the cluster forming in a longer time range, which +is especially important when consensus among agents is +obstructed by adversaries. +With this rolling horizon setting, it is important for a +player to know what the opponent’s previous action at +the previous step of the game is in order to know its +position at the game tree, i.e., which subgame is the +player’s playing. For example, if the defender does not +know which edges are previously attacked, then it cannot +properly calculate the value of the utility function (8). +5 +Consensus Analysis +In this section, we examine the effect of the game struc- +ture and players’ energy constraints on consensus. +We will begin the analysis by looking at the case of +certain energy conditions of the players. Specifically, if +a player has enough energy to attack/recover all edges +from a certain step of the game, then it will use all of +their energy to attack/recover as many edges as they +can in the subsequent steps. We will confirm this point +formally in the following. For simplicity, we denote the +total energy that the defender consumed before the lth +game as ˜βD +l := �(l−1)T −1 +k=0 +βD|ED +k | and the total energy +that the defender may consume from the 1st to the αth +step of the lth game as ˆβD +α := �α +m=1 βD|ED +l,m|, where +we omit the index l from the left-hand side, with a +slight abuse of notation. Similarly, for the attacker we +denote ˜βA +l +:= �(l−1)T −1 +m=0 +(βA|EA +m|+β +A|E +A +m|) and ˆβA +α := +�α +m=1(βA|EA +l,m|+β +A|E +A +l,m|). +6 + +We discuss in Lemma 1 (resp., Lemma 2) the optimal +strategy of the defender (resp., attacker) at the αth step +of the game given certain energy conditions mentioned +in Section 2. This characterization of optimal strategy +of the defender (resp., attacker) will be useful to obtain +the necessary (resp., sufficient) conditions for consensus +not to happen. +5.1 +Necessary Conditions for not Reaching Consensus +This subsection discusses necessary conditions for the +agents to be separated into different clusters for infinitely +long duration without achieving overall consensus. We +first discuss the defender’s optimal strategy on some +games with specific conditions in Lemmas 1 and 2. In +Lemma 1, we state the defender’s optimal strategy at +any step of the lth game given a certain energy condition. +Lemma 1 If the defender’s total energy ˜βD +l + ˆβD +ˆα−1 con- +sumed before the ˆαth step of the lth game satisfies +˜βD +l + ˆβD +ˆα−1 +≤ κD + ρD((l − 1)T + ˆα − 1) − (h − ˆα + 1)|E|βD, +(13) +then ED∗ +l,α = EA∗ +l,α for all α ≥ ˆα, i.e., the defender will +recover all normally attacked edges from the ˆαth step. +PROOF. We first look at the last (hth) step of the lth +game. Since the game consists of a horizon of h steps, the +last step of the game corresponds to the last decision- +making point, in which the players’ strategies cannot in- +fluence the decision already made in the previous steps of +the same game. Hence, in the last step of the lth game the +players do not save their energy by attacking/recovering +less edges. +From the defender’s energy constraint (4), it is clear that +at any time k, the set of edges that the defender recov- +ers is bounded as |ED +k |≤ +κD+ρDk−�k−1 +m=0 βD|ED +m| +βD +. Thus, at +the hth step, recovered edges satisfy |ED +l,h|≤ |ED′ +l,h| with +|ED′ +l,h|:= min{⌊ +κD+ρD((l−1)T +h−1)−( ˜βD +l + ˆβD +h−1) +βD +⌋, |EA∗ +l,h |}. +Depending on which edges are normally attacked, +the defender may not recover the maximum num- +ber |ED′ +l,h| of edges. If the defender’s optimal strat- +egy given normally attacked edges EA +l,h is not to re- +cover |ED′ +l,h| number of edges, i.e., recover less, then +the defender will be able to obtain more utility +U D +l,h(E +A +l,h, EA +l,h, ED +l,h) > U D +l,h(E +A +l,h, EA +l,h, ED′ +l,h). However, un- +der (13) with α = h the defender has sufficiently high en- +ergy, and thus the utility becomes U D +l,h(E +A +l,h, EA +l,h, ED +l,h) > +U D +l,h(E +A +l,h, EA +l,h, EA +l,h) = U D +l,h(E +A +l,h, ∅, ∅). It then follows +that as long as the defender has enough energy, it will +recover all optimal edges attacked normally at the hth +step, i.e., ED∗ +l,h = EA∗ +l,h . +Next, we investigate the effect of this property on the +earlier steps of the lth game. Since the defender’s strat- +egy at the hth step is not affected by its strategy at the +previous (i.e., (h−1)th) step when κD +ρD((l −1)T +h +−1) − (˜βD +l + ˆβD +h−1) ≥ βD|E|, here the defender does not +need to recover fewer edges at the (h − 1)th step to save +energy; this is because it already has enough energy to +recover EA∗ +l,h at the hth step. +Now, we derive that if κD + ρD((l − 1)T + h − 2)− +(˜βD +l + ˆβD +h−2) ≥ 2βD|E| at the (h − 1)th step, then the +defender will also recover ED∗ +l,h−1 = EA∗ +l,h−1. To recover all +attacked edges at steps α ≥ ˆα, it is then sufficient that +the defender’s energy satisfies (13) so that κD + ρD((l − +1)T + α − 1) ≥ ˜βD +l + ˆβD +α−1 + βD|E|, i.e., the worst-case +scenario of the energy constraint (4) when the defender +recovers all edges, is always satisfied when α ≥ ˆα. +□ +From the proof above, note that if the defender’s strat- +egy is not to recover all normally attacked edges given +even if (13) is satisfied, i.e., EA +l,α = ˆEA ̸= ED +l,α, then +the attacker will not attack ˆEA set of edges in the first +place. This is because by attacking ˆEA (and consid- +ering ED +l,α ̸= ˆEA) the attacker’s utility for step α ≥ +ˆα becomes U A +l,α(·, ˆEA, ED +l,α ̸= ˆEA) < U A +l,α(·, ∅, ∅), since +U D +l,α(·, ˆEA, ED +l,α ̸= ˆEA) > U D +l,α(·, ∅, ∅) = U D +l,α(·, ˆEA, ˆEA) +and U D +l = −U A +l . +We also remark that in order to derive the same optimal +strategy for the defender the quantity (h − α + 1)|E| +in the right-hand side of inequality (13) can be relaxed +to the maximum number of edges that the attacker can +attack from step ˆα to step h given its energy condition. +However, this number of edges may change every game, +making the inequality complicated to express. +Lemma 2 gives an interval over which, at least once, +either not attacking with normal signals or recovering +nonzero edges is optimal. +Lemma 2 There is at least one occurrence of either +ED +k ̸= ∅ or EA +k = ∅ every ⌈ h|E|βD−ρD +ρDT ++ 1⌉ time steps. +PROOF. It follows from Lemma 1 that in a game with +index l′ where (13) is satisfied for α = 1, the defender +always recovers edges that are attacked normally in the +1st step, i.e., ED +l′,1 ̸= ∅ if EA +l′,1 ̸= ∅. We then investigate +in which game inequality (13) is satisfied for α = 1. +Since the defender gains ρD every time k, if ED +k = ∅ for +7 + +any k ∈ {0, . . ., (l′ − 1)T − 1}, then (13) at the first +step of the l′th game can be written as κD+ρD(l′−1)T +βD +≤ +h|E|. With κD = ρD as a worst-case scenario, the left- +hand side becomes +ρD(1+(l′−1)T ) +βD +, and we then obtain +l′ ≥ ⌈ h|E|βD−ρD +ρDT ++ 1⌉. +Note that the above fact holds when the defender +does not recover any edge for any k ∈ {(j − 1)(l′ − +1)T, . . . , j(l′ − 1)T − 1}, j +∈ +N. If the defender +recovers one or more attacked edges at any k +∈ +{0, . . ., (l′ − 1)T − 1}, then the above result may +not hold, i.e., the defender may not be able to re- +cover all EA +l′ . However, it follows that during time +k ∈ {(j − 1)(l′ − 1)T, . . . , j(l′ − 1)T − 1}, either 1) the +defender recovers nonzero edges (ED +k +̸= ∅), or 2) the +attacker attacks no edges with normal signals (EA +k = ∅) +at least once. +□ +Lemmas 1 and 2 above imply that the defender is guar- +anteed to make recoveries from normal attacks every cer- +tain interval. Hence, the attacker needs to attack some +edges strongly to prevent the recovery in order to sepa- +rate agents into different clusters, as we discuss next. +The following two results provide necessary conditions +for consensus not to take place. We consider a more gen- +eral condition in Proposition 3, whereas in Theorem 4 +we consider a more specific situation for the utility func- +tions that leads to a tighter condition. Recall that λ rep- +resents the connectivity of G. +Proposition 3 A necessary condition for consensus not +to happen is ⌊ρA/βA⌋ ≥ λ. +PROOF. In deriving this necessary condition, we sup- +pose that there is no recovery by the defender at any time +k. Without any recovery from the defender (ED +k = ∅), +the attacker must attack at least λ number of edges with +normal signals (which take less energy) at any time k to +make GD +k disconnected at all times. Otherwise, there will +be time steps where the graph GD +k is connected, which +implies that consensus will still be reached. +If the attacker attacks λ edges with normal jamming +signals at all times, the energy constraint (3) becomes +(βAλ − ρA)k ≤ κA. Thus, the condition ρA/βA ≥ λ has +to be satisfied to ensure that the attacker can make GD +k +disconnected for all k. Note that, if the attacker does not +have enough energy to disconnect GD +k given no recovery, +then it definitely cannot disconnect GD +k in the face of +recovery by the defender. +□ +We now limit the class of utility functions in (7), (8) to +the case of b = 0 in the weights. This means that the +players do not take account of the agent-group index in +the graph, but only the states in consensus. In this case, +the attacker may need more energy to prevent consensus +as shown in the next theorem. +Theorem 4 Suppose that b = 0. A necessary condition +for consensus not to happen is ρA/β +A ≥ λ. +PROOF. We prove by contrapositive; especially, we +prove that consensus always happens if ρA/β +A < λ. +We first suppose that the attacker attempts to attack +λ edges strongly at all times to disconnect the graph +GD +k . From (3), the energy constraint of the attacker at +time k becomes (β +Aλ − ρA)k ≤ κA. This inequality is +not satisfied for sufficiently large k if ρA/β +A < λ, since +β +Aλ − ρA becomes positive and κA is finite. Therefore, +the attacker cannot attack λ edges strongly at all times +if ρA/β +A < λ, and is forced to disconnect the graph by +attacking with normal jamming signals instead. +Next, by Lemma 2 above, we show that there exists an +interval of time where the defender always recovers if +there are edges attacked normally, i.e., ED +l′ ̸= ∅ is optimal +given that EA +l′ ̸= ∅. +From the definitions in (7), (8), given that b = 0, we can +see that the defender obtains a higher utility if the agents +are closer. This means that given a nonzero number +of edges to recover (at time jl′T described above), the +defender recovers the edges connecting further agents. +Specifically, for some i ∈ N, for interval [jl′T, (j +i)l′T ], +there is a time step where U D +l (ED +k += E1) ≥ U D +l (E2), +with edges E1 connecting agents with further states than +agents connected by E2. This fact implies that when re- +covering, the defender always chooses the further dis- +connected agents. Since by communicating with the con- +sensus protocol as in (1) the agents’ states are getting +closer, the defender will choose different edges to re- +cover if the states of agents connected by recovered edges +ED +k become close enough. Consequently, if ρA/β +A < λ, +then there exists i ∈ N where the union of graphs, +i.e., the graph having the union of the edges of each +graph (V, �((E \(E +A +k ∪EA +k ))∪ED +k )) over the time interval +[j(l′ − 1)T, (j + i)(l′ − 1)T ], becomes a connected graph, +where l′ = ⌈ h|E|βD−ρD +ρDT ++ 1⌉ as in Lemma 2 above. These +intervals [j(l′−1)T, (j+i)(l′−1)T ] occur infinitely many +times, since the defender’s energy bound keeps increas- +ing over time. +It is shown in [31] that with protocol (1), the agents +achieve consensus in the time-varying graph as long as +the union of the graphs over bounded time intervals +is a connected graph. This implies that consensus is +8 + +achieved if (V, �((E \(E +A +k ∪EA +k ))∪ED +k )) is connected over +[l′ +i, l′ +i + 1, . . . , l′ +i+j]. Thus, if ρA/β +A < λ then consensus +is achieved. +□ +The result in Theorem 4 only holds for b = 0, since with +b > 0 the defender may choose to recover the edges con- +necting agents that already have similar states to maxi- +mize c(GD +k ) (instead of those connecting further agents). +In such a case, the network may remain disconnected and +thus the agents may converge to different states. As we +see from these results, the weight values affect the nec- +essary conditions to prevent consensus, whereas the ef- +fect of the weights on the sufficient condition (discussed +later) is less straightforward. The effect of the values of +a and b on consensus is illustrated in Section 8. +5.2 +Sufficient Condition to Prevent Consensus +The next result provides a sufficient condition for pre- +venting consensus. It shows that the attacker can pre- +vent consensus if it has sufficiently large recharge rate ρA +given the network topology G. We first state Lemma 5 +about the attacker’s optimal strategy under some energy +conditions, similar to the discussion on the defender’s +case above. +Lemma 5 The attacker’s optimal strategy is E +A∗ +l,α = E if +• the attacker’s recharge rate satisfies ρA/β +A ≥ |E|, +or +• the attacker’s total energy ˜βA +l + ˆβA +α−1 that it con- +sumes before αth step of the lth game satisfies +˜βA +l + ˆβA +α−1 +≤ κA + ρA((l − 1)T + α − 1) − (h − α + 1)β +A|E|. +(14) +PROOF. We first observe that in the hth step of the lth +game the attacker does not save their energy by attack- +ing fewer edges. Since zl,h(E, ∅, ∅) > zl,h(E +A +l,h, EA +l,h, ED +l,h) +and c((V, ∅)) ≥ c((V, (E \ (E +A +l,h ∪ EA +l,h) ∪ ED +l,h))) are al- +ways satisfied for any edges E +A +l,h, EA +l,h, ED +l,h, the function +U A +h always has the highest value if the attacker strongly +attacks all edges E. It then follows that the attacker with +enough energy, i.e., κA + ρA((l − 1)T + h − 1) − (˜βA +l + +ˆβA +h−1) ≥ β +A|E| is satisfied, will choose to attack all edges +with strong signals. +Similar to the proof in Lemma 1, inequalities zl,α(E, ∅, +∅) > zl,α(E +A +l,α, EA +l,α, ED +l,α) and c((V, ∅)) ≥ c((V, (E\(E +A +l,α∪ +EA +l,α) ∪ ED +l,α))) are always satisfied for any step α. Hence, +the attacker will choose to attack all edges with strong +signals in any step α given enough energy. This can be +achieved if the attacker has high enough stored energy, +i.e., (14) is satisfied, or if the attacker has high enough +recharge rate, i.e., ρA ≥ β +A|E|. These conditions enable +the attacker to attack all edges strongly while still sat- +isfying the energy constraint (3) above for all steps. +□ +Proposition 6 A sufficient condition for all agents not +to achieve consensus at infinite time is that the attacker’s +parameters satisfy ρA/β +A ≥ |E|. +PROOF. By Lemma 5, the attacker always strongly at- +tacks all edges with strong signals in a game at any step +α given either sufficient recharge rate or sufficient stored +energy at the beginning of the game. Consequently, if +the attacker’s recharge rate satisfies ρA/β +A ≥ |E|, the +attacker will attack E with stronger jamming signals at +all steps of all games, separating every agent at all times. +As a result, there are n clusters formed, and hence, ob- +viously, consensus is not reached. +□ +Remark 7 Note that the necessary conditions and the +sufficient condition above consider zk = xTLcx in (6) +which is a nonincreasing function. It is possible to con- +sider other Laplacian matrices, e.g., Laplacian of the un- +derlying graph G, however the function zk may not be non- +increasing anymore. For example, we consider a simple +path graph 1-2-3 with initial states x0 = [10, 0, −5]T and +Laplacian of graph G considered in state difference func- +tion zk. With weights of the utility functions (7) and (8) +a = 1 and b = 0 and under consensus protocol (1) and (2) +with weights a12 = 0.1 and a23 = 0.8, the players’ utili- +ties in the first game with h = 1 are U A +1 = −U D +1 = 148 +without any attacks, and U A +1 = U A +0 = −U D +1 = 125 if both +edges are attacked. This implies that not attacking any +edge may actually be optimal for the attacker even with +large enough energy. As a consequence, with Laplacian +of graph G considered in state difference function zk, the +analysis becomes more complicated and some of the the- +oretical results do not hold anymore, e.g., the sufficient +condition in Proposition 6. +5.3 +Example on a Gap Between Necessary Condition +and Sufficient Condition +In this subsection we provide an example that illustrates +the gap between the necessary condition for preventing +consensus in Theorem 4 and the sufficient condition in +Proposition 6. Here we suppose that the defender has a +very high recharge rate (i.e., ρD is much larger than βD) +such that it can recover any normally-attacked edges at +any k (note that the condition in Theorem 4 only consists +of the attacker’s parameters). This will force the attacker +9 + +1 +2 +3 +4 +Fig. 4. Graph G used in the case study. +Table 1 +Agent state difference for various values of ρA/β +A +ρA/β +A +z20 +Consensus +1 +0.113 +Yes +1.1 +0.115 +Yes +1.2 +1.3405 +No +1.4 +235.345 +No +1.8 +706.8 +No +2 +1354 +No +to attack with strong jamming signals to disconnect any +agent. +We consider a graph G as in Fig. 4, with x[0] = +[−5, 0, −20, 10], h = 2, and κA = ρA. The weight of the +utility functions are set to be a = 1 and b = 0. We test +various values of 1 ≤ ρA/β +A ≤ 2, implying that the +attacker can attack one edge with strong signals at all +time without running out of energy. Thus, the attacker +needs to attack e12 (min-cut edge of G) at all times in +order to prevent consensus, since it is the only edge +which, if attacked, will make the graph disconnected. +Note that this ratio 1 ≤ ρA/β +A ≤ 2 satisfies the neces- +sary condition for preventing consensus in Theorem 4, +but not the sufficient condition in Proposition 6. +Specifically in this example we test whether consensus +is prevented or not for various value of ρA/β +A based on +agent states at time k = 20. It is interesting to note +from Table 1 that even with a relatively small value of +ρA/β +A < |E|, consensus can still be prevented by the +attacker. +From this example, we observe that there is a gap be- +tween the necessary condition and the sufficient condi- +tion. Note that this gap may be larger for a more con- +nected G as well as for network consisting of more agents, +where typically |E|>> λ. Later in Section 8, we pro- +vide more detailed examples which illustrate the effect +of these parameters’ values on consensus. +As the last result of the section, we state that for a special +case with the complete graph under b = 0 and h = +1, i.e., a single-step game without rolling horizon, the +condition in Theorem 4 is also sufficient, i.e., there is no +gap between the necessary condition and the sufficient +condition. +Proposition 8 Suppose that b = 0 and h = 1. In the +complete graph G, a sufficient condition for consensus +not to happen is ρA/β +A ≥ n − 1. +PROOF. With h = 1, the attacker will spend all of its +energy at the only step of the game. With ρA/β +A ≥ n−1, +the attacker is always able to disconnect the complete +graph G. +In the complete graph G, every agent is connected to +all other agents regardless of their states, implying that +there is no agent that can be prioritized to be isolated by +the attacker (different from the example above). Then, +with b = 0, the attacker is ensured to separate the fur- +thest agent. This implies that, at each game (and at each +k), the attacker will always attack the same edges, re- +sulting in disconnected GD +k at each time. +□ +We note that in different class of graphs (including in +other symmetric graphs such as cycle graphs or star +graphs), it is more challenging to derive a tighter suffi- +cient condition. This is because agents have direct access +only to some other agents which makes cluster forming +based on the agent states more difficult. +6 +Clustering Analysis +In this section, we derive some results on the number of +formed clusters of agents at infinite time. From Propo- +sition 6, the result implies the simple case where if the +attacker has enough energy such that ρA/β +A ≥ |E|, then +the attacker can attack all the edges of the underlying +topology G so that the number of clusters is n (i.e., all +the agents are separated). +The next result discusses a relation between the at- +tacker’s cost and energy recharge rate with the maxi- +mum number of clusters that the attacker may create +through jamming. In the subsequent results of this sec- +tion, we suppose that b = 0. +We first define a vector which characterizes the maxi- +mum number of clusters of G, given the parameters ρA +and β +A. Specifically, we define a vector Θ ∈ R|E| with el- +ements Θj := max|EA|=j n(V, E \ EA), with n(V, E \ EA) +being the number of agent groups of (V, E \ EA). +Proposition 9 An upper bound on the number of +formed clusters at infinite time is Θ⌊ρA/β +A⌋. +PROOF. The vector Θ consists of the maximum num- +ber of formed groups n(V, E \ EA) given the number of +10 + +attacked edges as the element index. Since some edges +need to be attacked consistently in order to divide the +agents into different clusters, the number of formed clus- +ters at infinite time is never more than the maximum +number of groups at any time k given the same number +of strongly attacked edges. +Recall that ⌊ρA/β +A⌋ is the maximum achievable num- +ber of edges that can be strongly attacked at all times. +Given the known graph topology G, we then can imply +that Θ⌊ρA/β +A⌋ gives the maximum number of clusters at +infinite time. +□ +We continue by addressing a special case where all the +agents in the network are connected with each other. +Corollary 10 In the complete graph G, the attacker can- +not divide the agents into more than +1 + +(n−1) +� +j=1 +min +� +1, +� +2ρA +jβ +A(2n − j − 1) +�� +(15) +number of clusters. +PROOF. In the complete graph, every agent is con- +nected to all other n − 1 agents. From Proposition 9, we +can derive the vector Θ of the complete graph G as +Θ =[1, . . . , 1, 2, . . ., 2, 3, . . . , n − 1, n]T, +where the value of the (n−1)th entry is 2, the value of the +((n−1)+(n−2))th entry is 3, and so on. This is because +in the complete graph G the attacker needs to attack +(n−1) number of edges to disconnect the graph, further +(n − 2) number of edges to make three groups of agents, +further (n − 3) number of edges to make four groups of +agents, and so on, until (n − 1) + (n − 2) + · · · + 1 = +n(n − 1)/2 agents to make n groups. The value of the +⌊ρA/β +A⌋th entry of this Θ matrix for the complete graph +can be written as in (15). This value determines the +upper bound of the number of clusters. +□ +In Proposition 9, we use the information of the graph +structure to obtain the vector Θ. We remark that if the graph +structure G is not known, then the number of clusters at +infinite time is in general upper bounded by ⌊ρA/β +A⌋ + 1. +This is because the attacker can attack continuously at all +time at most ⌊ρA/β +A⌋ number of edges, and in the most +vulnerable graph with λ = 1, i.e., tree graphs, any attacked +edge will result in a new group. +To illustrate the relationship between Θ and ρA/β +A, we look +Table 2 +Possible cases of attack and recovery actions +Case +c(GA +l,α) +c(GD +l,α) +1 +c(GA +l,α) = c(G) +c(GD +l,α) = c(GA +l,α) +2 +c(GA +l,α) < c(G) +c(GD +l,α) = c(GA +l,α) +3 +c(GA +l,α) < c(G) +c(GD +l,α) > c(GA +l,α) +7 +Equilibrium Characterization +In this game the strategy choices are all finite in form of +edges attacked and recovered. Here, we characterize the +equilibrium/optimal strategies of the players in certain +situations for the case where the players’ horizon length +is 1 so that they myopically update their strategies every +time step. +In this section, we state some results when a = 0, i.e., +when the players do not consider the agents’ states but +agent-group index in determining their strategies so that +the defender (resp., attacker) has higher (resp., lower) +utility when more agents belong to the same group. Sim- +ilar to the analysis in [20], here we explore some possible +optimal strategy candidates for the players in a game. +However, since a game consists of several steps in this +formulation, the subgame perfect equilibrium is more in- +volved to characterize, compared to the case of a game +consisting of one step as in [20]. +In the αth step of each game, there are three possibilities +in function c(·) as shown in Table 2 (Cases 1, 2, and 3). +From this table, we characterize the optimal strategies +of both players in each case: +• Case 1: When c(G) = c(GD +l,α), the attacker’s utility +in one time step is c(G), which implies that the at- +tacker should not attack any edge either with nor- +mal signals or strong signals, with the utilities of +both players equal to zero. The players’ strategies +in this case are called Combined Strategy 1. +• Case 2: When c(GD +l,α) = c(GA +l,α), the defender does +not recover any attacked edge, whereas the attacker +should attack some edges either with strong or nor- +mal signals. The players’ strategies in this case are +classified as Combined Strategy 2. +• Case 3: Here both players will attack/recover +nonzero number of edges. In particular, the at- +tacker will attack with normal signals and poten- +tially with strong signals. The players’ strategies +here are called Combined Strategy 3. +at the graph in Fig. 4 from the last section. Here, Θ = +[2, 2, 3, 4]T, whereas the values of ⌊ρA/β +A⌋ + 1 are 2, 3, 4, +5 for ρA/β +A = 1, 2, 3, and 4, respectively. Note that for +any value of ρA/β +A, inequality Θ⌊ρA/βA⌋ ≤ ⌊ρA/β +A⌋ + 1 is +always satisfied, indicating that knowing the graph structure +helps to better estimate the upper bound of the number of +clusters. +11 + +We will then discuss the equilibrium for this game in +Proposition 11 below. For simplicity, we only consider +the case when h = 1. The case of h > 1 can be examined +based on the characterization here for h = 1. +Proposition 11 The optimal strategies for the players +with h = 1 satisfy the following: +(1) Combined Strategy 1 if ˜βA +l +βA > κA +ρA(l −1)T , +(2) Otherwise, +(a) Combined Strategy 2 if +(i) ˜βD +l + βD > κD + ρD(l − 1)T , or +(ii) ˜βD +l ++ βD +≤ +κD + ρD(l − 1)T +and +U A +l (⌊(κA + ρA(l − 1)T − ˜βA +l )/β +A⌋, ∅, ∅) = +maxE +A +k ,EA +k ,ED +k U A +l (E +A +k , EA +k , ED +k ), +(b) Combined Strategy 3 if ˜βD +l ++ βD ≤ κD + +ρD(l − 1)T and U A +l (⌊(κA + ρA(l − 1)T − +˜βA +l )/β +A⌋, ∅, ∅) ̸= maxE +A +k ,EA +k ,ED +k U A +l (E +A +k , EA +k , ED +k ). +PROOF. With a = 0, we observe that the defender al- +ways recovers from the optimal attack at the last step +given sufficient energy, which implies that it always re- +covers for h = 1 if ˜βD +l + βD ≤ κD + ρD((l − 1)T ) is sat- +isfied. Similar to the defender, the attacker obtains the +least utility, i.e., zero, by not attacking for the case of +h = 1. Therefore, the attacker will attack at least one +edge as long as it has enough energy to do so. We prove +each point of the proposition statement as below. +(1): We now suppose that ˜βA +l + βA > κA + ρA((l − 1)T ) +(point (1) in the statement) is satisfied, i.e., the attacker +does not have enough energy to even attack one edge +normally. In this case, Combined Strategy 1 becomes +optimal since there is no other choice, i.e., the attacker +cannot attack even one edge with normal signals. In the +rest of the proof, we assume that ˜βA +l +βA ≤ κA+ρA((l− +1)T ) is satisfied. +(2a(i)): We now continue by providing the conditions +for Combined Strategy 2. Similarly to the attacker +above, we observe that the defender cannot recover any +edge if ˜βD +l + βD > κD + ρD((l − 1)T ), implying that +c(GA +l,α) < c(G) and c(GD +l,α) = c(GA +l,α) (corresponds to +point (2a(i))). +(2a(ii)): We then suppose that ˜βD +l ++ βD ≤ κD + +ρD((l − 1)T ) is satisfied. It then follows that given +enough energy for the defender, the attacker needs to +attack nonzero number of edges with strong signals to +satisfy c(GA +l,α) < c(G) and c(GD +l,α) = c(GA +l,α). In order for +Combined Strategy 2 to be optimal, the attacker then +needs to attack edges strongly without attacking with +normal signals at all, i.e., EA +k = ∅. Thus, β +A needs to be +sufficiently low to make strong attack feasible. Specif- +ically, U A +l (E +A′ +k , ∅, ∅) += +maxE +A +k ,EA +k ,ED +k U A +l (E +A +k , EA +k , ED +k ), +with |E +A′ +k |= ⌊(κA + ρA((l − 1)T ) − ˜βA +l )/β +A⌋ indicating +the maximum number of edges the attacker attacks +strongly. This corresponds to point (2a(ii)). +(2b): Consequently, if ˜βD +l + βD ≤ κD + ρD((l − 1)T ) +and U A +l (E +A′ +k , ∅, ∅) ̸= maxE +A +k ,EA +k ,ED +k U A +l (E +A +k , EA +k , ED +k ) are +true, then the attacker normally attacks nonzero num- +ber of edges and the defender recovers nonzero number +of edges, which imply that Combined Strategy 3 is op- +timal (point 2b). +□ +Remark 12 The characterization of optimal strategies +in Proposition 11 also holds for a more general class of +agent-group indices other than c(G′) defined in (5), as +long as the utility function structure (7) and (8) does not +change. Specifically, it holds for those indices that belong +to the class given by +C := {˜c : 2V × 2E → R : ˜c((V, E ∪ E′)) ≥ ˜c((V, E)), +E, E′ ⊆ E}. +(16) +The condition ˜c((V, E ∪ E′)) ≥ ˜c((V, E)) implies that not +attacking results in the maximum value of ˜c(GA +l,α) of the +attacker. Similarly, for the defender, this condition im- +plies that not recovering given the attacks results in the +minimum value of ˜c(GD +l,α). This condition is necessary +for ensuring the equilibrium as in Proposition 11, since it +guarantees that attacking/recovering nonzero number of +edges (corresponding to Combined Strategy 3) is always +optimal for the players as long as they have the energy to +do so. +In general, since the cases discussed above are for one +step only, for longer h > 1 the optimal strategies will +take form of a set of combined strategies. For exam- +ple, if h = 3, the sequence of optimal strategies may be +{Combined Strategy 1, Combined Strategy 2, Combined +Strategy 2}. On the other hand, for a > 0, the condition +in Proposition 11 becomes more complicated to charac- +terize since attacking more edges does not necessarily +result in the highest possible utility. +8 +Simulation Results +8.1 +Consensus and Clustering across Parameters +Here we show how the consensus varies across different +weights of the utility functions and the initial states. +8.1.1 +Varying Weights a and b +We consider the 4-agents line/path graph 1–2–3–4 with +initial states x0 = [1, 0.75, 0.75, −1]T. The parameters +12 + +0 +5 +10 +15 +20 +25 +30 +Time +-1 +-0.5 +0 +0.5 +1 +State +agent 1 +agent 2 +agent 3 +agent 4 +Fig. 5. Agent states with a = 0.1 and b = 0.9 +0 +5 +10 +15 +20 +25 +30 +Time +-1 +-0.5 +0 +0.5 +1 +State +agent 1 +agent 2 +agent 3 +agent 4 +Fig. 6. Agent states with a = 0.9 and b = 0.1 +0 +5 +10 +15 +20 +25 +30 +Time +Edges +Fig. 7. Attacked and recovered edges with a = 0.1 and b = 0.9 +0 +5 +10 +15 +20 +25 +30 +Time +Edges +Fig. 8. Attacked and recovered edges with a = 0.9 and b = 0.1 +are βA = βD = 1, h = β +A = 2, κA = ρA = 2.6, ρD = +0.3, and κD = 0.8, which satisfy the necessary condition +for preventing consensus in Proposition 3, but not the +sufficient condition in Proposition 6. With b = 1 − a, +Figs. 5 and 6 show the agent states with small a (at a = +0.1) and large a (at a = 0.9), respectively. Figs. 7 and 8 +illustrate the status of the edges in GD +k over discrete time +k. There, no line in the corresponding edge implies that +the edge is strongly attacked; likewise, dashed red lines: +normally attacked, dashed black lines: recovered, and +solid black lines: not attacked. +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +0 +2 +4 +6 +8 +10 +12 +170 +180 +190 +200 +210 +220 +230 +Fig. 9. Comparison of zk and − � c(GD +k ) (k = 20) versus a +1 +2 +3 +4 +6 +5 +7 +8 +9 +10 +Fig. 10. Graph used for simulation in Section 8.1.2 +We observe that for small a, the attacker more often +divides the agents into more groups, indicated by more +dashed red lines in Fig. 7. As a result, the attacker fails +to prevent consensus among the agents (Fig. 5), despite +the condition in Proposition 3 being satisfied. On the +other hand, with large a, the attacker is more focused +to make the difference among agents’ states larger while +separating the agents into fewer groups compared to the +case with small a. These features can be seen in Fig. 8, +where there are no black lines in the edge e34, and thus +no consensus among the agents in Fig. 6. +We next present a comparison in the optimal state dif- +ference zk(E +A∗ +k , EA∗ +k , ED∗ +k ) and agent-group index c(GD +k ) +across different a and b = 1 − a in Fig. 9. We observe +that with larger a, the attacker successfully prevents con- +sensus among agents (shown with larger value of zk) at +time k = 20. On the other hand, with smaller a (corre- +sponding to larger b), the attacker obtains higher c(GD +k ) +at the cost of low zk, implying that the attacker fails to +prevent consensus. It is interesting that the values of zk +and � c(GD +k ) remain almost constant for some different +a, implying that there is a critical value of weights a and +b that determine the consensus and the number of clus- +ters at infinite time; in this case, the critical value of a +is located in 0.4 < a < 0.5. +8.1.2 +Varying Initial States x0 +We also observe how the initial states x0 affect the agent- +group index of the agents. We consider the graph shown +in Fig. 10, which consists of 10 agents. All parameters +other than the initial states are set to be the same and +satisfy the conditions in Proposition 3. Specifically, we +set βA = βD = 1, β +A = 2, κA = ρA = 2.1, κD = ρD = +0.7, and a = 1 − b = 0.9. The state trajectories of the +agents with varying x0 are shown in Figs. 11–13. Here +we consider three cases of initial states x0: +13 + +(1) x0 = [1, 0.9, 0.8, 0.4, 0.44, 0.35, 0.48, 0.2, 0.19, 0.28]T, +(2) x0 = [1, 0.9, 0.8, 0.4, 0.44, 0.35, 0.48, −0.5, −0.1, −0.2]T, +(3) x0 = [0.6, 0.5, 0.8, 0.4, 0.44, 0.35, 0.48, 0.58, 0.8, 0.75]T. +Note that in Case (1), agents 1–3 have closer initial states +and are far from the other agents. Similarly, in Case (2), +agents 8–10 have initial states that are different from +the other agents. However, in Case (3), agent states are +distributed approximately evenly in the range [0.35, 0.8] +so that it is hard for the attacker to divide them into +clusters. +From Fig. 11, we can see that in Case (1), agents 1–3, +which have weak connection to other agents (only con- +nected by one edge), are grouped together and converge +to the same state. This occurs by attacking the edge con- +necting agents 3 and 5. On the other hand, in Fig. 12 +for Case (2), agents 8–10 are separated from the others +because the edge connecting agents 5 and 8 is attacked +continuously. Clearly, in Cases (1) and (2) it is easier for +the attacker to separate agents since their initial states +form clusters matching the network topology. +In Case (3), however, the initial state values do not ex- +hibit such properties and as a result, the states converge +towards the same value as shown in Fig. 13. In this sim- +ulation, the attacker is not able to effectively attack cer- +tain edges at all times; as a consequence, the agents are +not divided into clusters and thus consensus happens. +The attacker may be able to prevent consensus with +higher weight a, as discussed in Section 8.1.1 above. +For obtaining Figs. 11–13, we solve combinatorial opti- +mization problems to find optimal strategies of the play- +ers. We remark that the computational complexity of +this problem depends on the number of edges E of G. We +have reduced the complexity by disregarding some com- +binations of edges that are clearly not optimal; for ex- +ample, attacking only the edge connecting agents 4 and +7 does not disconnect the graph, and thus cannot be the +best move for the attacker. +8.1.3 +Varying Energy and Cost Parameters +We continue by discussing the effect of the attacker’s +recharge rate ρA and unit costs of attacks βA and β +A +on the consensus and cluster forming. Recall that in the +theoretical results in Sections 5 and 6, the ratios of ρA +to β +A and ρA to βA are used to derive the necessary con- +ditions and sufficient conditions for preventing consen- +sus as well as the upper bound of the number of clusters +formed at infinite time. +Assuming that b = 0, the number of clusters is dictated +by ρA/β +A as discussed in Proposition 9. We show the +number of clusters over different topologies of the un- +derlying graph G in Fig. 14. We consider networks with +0 +5 +10 +15 +20 +25 +30 +Time +0 +0.2 +0.4 +0.6 +0.8 +1 +State +Fig. 11. Agent states in Case 1 +0 +5 +10 +15 +20 +25 +30 +Time +-0.5 +0 +0.5 +1 +State +Fig. 12. Agent states in Case 2 +0 +5 +10 +15 +20 +25 +30 +Time +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +State +Fig. 13. Agent states in Case 3 +n = 5, with the edges positioned to yield the most con- +nected topology, i.e., maximum λ, given the same num- +ber of edges |E|. Note that, with n = 5, there are at +most n(n−1)/2 = 10 number of edges in the underlying +graph G (which happens for the complete graph G). We +observe that with ρA/β +A ≥ |E|, the agents are divided +into 5 clusters (all agents are separated) as shown in the +upper left area of the figure indicated by “5” as derived +in Proposition 6 whereas in the lower right area indi- +cated by “1” the agents converge to the same cluster. It +is clear that in a more connected graph, the agents are +more likely to converge to a fewer number of clusters. +8.2 +Players’ Performance Under Varying Horizon +Length and Game Period +In this subsection, we evaluate the players’ performance +under varying horizon length h and game period T . To +evaluate the performance of the players, we introduce +the applied utilities ˆU A +k := azk(E +A∗ +k , EA∗ +k , ED∗ +k )−bc(GD∗ +k ) +and +ˆU D +k +:= +−azk(E +A∗ +k , EA∗ +k , ED∗ +k ) + bc(GD∗ +k ), with +GD∗ +k += (V, ((E \ (E +A∗ +k +∪ EA∗ +k )) ∪ ED∗ +k ). These are el- +14 + +PSfrag replacements +|E| +ρA +β +A +Fig. 14. Number of clusters at k = 50 with b = 0. The un- +derlying graphs used are those with 5 agents with maximum +10 edges. +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +Time +0 +5 +10 +15 +20 +25 +30 +Fig. 15. � +k ˆU A +k in the path graph (solid lines) and the com- +plete graph (dashed lines) for varying value of h. The ap- +plied utility for h = 2 and h = 3 in the path graph is almost +identical. +ements of utility functions U A +l +and U D +l +correspond- +ing to the αth step, α += +k mod T + 1, of the +game with index l = ⌊k/T ⌋ + 1, where the obtained +strategies (E +A∗ +(l−1)T +α−1, EA∗ +(l−1)T +α−1, ED∗ +(l−1)T +α−1) += +(E +A∗ +l,α, EA∗ +l,α, ED∗ +l,α) are applied. Since U A +l += −U D +l , having +higher applied utility for the attacker implies lower ap- +plied utility for the defender. Note that the values of h +and T are uniform among the players. +In this subsection, we consider the weight aij = ˆa, ˆa < +1/n in (2) which implies that different agents have dif- +ferent convergence speeds depending on the number of +their neighbors. Furthermore, we consider various ini- +tial states x0 for the agents in order to more accurately +evaluate the attacker’s performance and the pattern of +applied utilities ˆU A +k . We use up to 1000 randomly gener- +ated initial states in this simulation for each agent rang- +ing from −1 to 1. Throughout this subsection, we use +parameters n = 3, ρA = 1.1, κA = 7, β +A = 2βA = 1. +8.2.1 +Players’ Performance Under Varying Horizon +Length +We now consider the case of varying value of horizon +length h when the network is a path graph and a com- +plete graph. Note that the value of h is still uniform +among the attacker and the defender. The evolutions of +the attacker’s applied utility ˆU A +k with varying h (with +T = 1 for every h) are shown in Fig. 15. +Table 3 +Difference in the optimal actions and the resulting utilities +in the path graph G between h = 2 and h = 3 +Initial states +|E +A∗ +0 | +�19 +k=0 ˆU A +k +h = 2 +h = 3 +h = 2 +h = 3 +[0.824, −0.798, −0.413]T +2 +2 +37.74 +[−0.983, 0.649, 0.535]T +2 +2 +39.89 +[−0.787, −0.786, −0.265]T +2 +1 +28.41 +30.00 +[0.624, 0.629, −0.821]T +2 +1 +37.92 +43.45 +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +Time +0 +5 +10 +15 +20 +25 +30 +Fig. 16. � +k ˆU A +k in the path graph (solid lines) and the com- +plete graph (dashed lines) for varying T . The applied utility +for T = 1 and T = 2 in the path graph is almost identical. +Since the path graph is the least connected graph, the +attacker will be able to make multiple groups of agents +relatively easily compared to more connected graphs. As +a result, the attacker may not need to have a very long +horizon length h to improve its utility since it does not +need to save energy as much compared to the case of +the complete graph. This is shown with the overlapping +red and yellow solid lines in the Fig. 15, implying that +the horizon length h = 3 is already as good as the case +of h = 2. On the other hand, the blue solid line is far +below the red and the yellow ones, implying that having +h being too short can result in a worse utility for the +attacker over time. +The differences of the attacker’s strategies for some no- +table cases in the path graph G between h = 2 and +h = 3 are shown in Table 3. Here, we see the difference +in the optimal actions between the attacker with h = 2 +and h = 3 in the path graph G even though the plots +of applied utilities in Fig. 15 are very similar. We ob- +serve that when the initial states of some agents are suf- +ficiently close, the attacker with h = 2 keeps attacking +both edges at k = 0, whereas the attacker with h = 3 +chooses to save its energy by attacking fewer edges. At +k = 19 the attacker with h = 3 obtains higher applied +utility, indicating that it is able to better use its energy +than the attacker with h = 2 by attacking later. +On the other hand, since the complete graph is the most +connected graph, here the attacker will need more energy +to disconnect the graph and obtain some utility. Conse- +quently, even with longer h, the difference of � ˆU A +k is +smaller compared to the path graph case. The difference +15 + +5 +5 +5 +5 +5 +4 +5 +4 +3 +4 +3 +310 +5 +5 +5 +5 +5 +5 +5 +9 +5 +5 +5 +5 +5 +5 +5 +8 +5 +5 +5 +5 +5 +5 +5 +7 +5 +5 +5 +5 +5 +5 +53 +3 +2 +3 +2 +2 +2 +2 +2 +2 +2 +1 +1 +1 +1 +1 +1 +1 +8 +6 +109 +5 +5 +5 +5 +5 +5 +4 +a +5 +5 +5 +5 +5 +5 +4 +3 +4 +5 +5 +5 +5 +4 +3 +3 +3 +5 +5 +5 +4 +3 +3 +2 +2 +5 +5 +4 +3 +2 +2 +2 +1 +5 +4 +3 +2 +1 +1 +1 +1 +2 +3 +4 +5 +9 +7 +ed6between the red and the yellow dashed lines is clearer +however, suggesting that the attacker still benefits by +having h = 3 (compared to the very little difference in +the path graph case). The attacker’s different behavior +for the path graph and the complete graph G suggests +that in a less connected graph, the effectiveness of longer +h may saturate from a lower value compared to the one +in a more connected graph G, given the attacker’s energy +parameters. +In general, we observe that having a longer h may re- +sult in a better applied utility for the attacker over time +due to its role as a leader of the game, i.e., the attacker +moves first and is able to choose its strategy that min- +imizes the defender’s best response. Additionally, there +is also a clear pattern on when � ˆU A +k increases; this im- +plies that the variation of initial states may not affect +the attacker’s optimal strategy, except in some cases as +explained above. +We also remark that the effect of different values of h is +also influenced by the underlying graph G. Specifically, +in a less connected graph G, having a very short horizon +may even be more harmful compared to the case with a +more connected G. For example, in Fig. 16, the difference +of � ˆU A +k in the path graph between h = 1 and h = 2 is +much more apparent than in the complete graph. The +possible reason is that in the path graph, it is easier for +the attacker to disconnect all agents and make n groups +at some time steps. Thus, with large enough h, the at- +tacker can save enough energy to make n groups more +often. On the other hand, we also observe that increasing +horizon length from h = 2 to h = 3 has minimal effect +on the attacker’s utility for the path graph, indicating +that increasing horizon length past a certain value may +not be beneficial anymore. As we see later, the similar +phenomenon also happens for varying values of T . +8.2.2 +Players’ Performance Under Varying Game Pe- +riod +We then continue by simulating the case of varying value +of game period T (value of h is set to be h = 3 for both +players so that the assumption T ≤ h is always satisfied). +The average value of � ˆU A +k over time is shown in Fig. 16, +where in general, the attacker with shorter game period +T has higher applied utility especially at later time for +both the path graph and the complete graph G. +The attacker with shorter T will be more adaptive to +the changes of the agents’ and players’ conditions. In the +context of this game, the attacker with shorter T may +delay the attack further to maximize its utility later. +This in turn increases the attacker’s utility at later time, +similar to the case of longer h discussed above. Note that +the yellow dashed and solid lines are the same as the +yellow lines in Fig. 15, and we observe that the green +and the purple lines do not differ as much as the red and +Table 4 +Average total number of edges attacked in the path graph G +h +T +�k +m=0|E A∗ +m | (Normal) +�k +m=0|E +A∗ +m | (Strong) +k = 9 +k = 19 +k = 9 +k = 19 +1 +1 +7 +16 +5 +6 +2 +0 +0 +8 +13.959 +3 +0 +0 +7.993 +13.971 +2 +0 +0 +8 +13.970 +3 +2.970 +4.970 +7.003 +11.015 +the blue lines in Fig. 15, indicating that for the attacker, +having a large value of T may not be as disadvantageous +as having short h. +Table 4 shows the average number of edges attacked by +normal and strong jamming signals given different values +of h and T . It is interesting to note that for h > T , +the attacker never attacks any edge with normal signals, +indicating that it prefers to save its energy to use it later +for more powerful attacks. Consequently, the number of +edges attacked strongly with h > T becomes more than +those in the case of h = T , which results in the larger +applied utilities as described above. We can also observe +that in the case of h = 3 and T = 1, the attacker is +able to strongly attack more edges than the other cases +in Table 4 in average at k = 19, even though at k = 9 +it attacks slightly fewer edges than the case of closer +values of h and T . This suggests that the attacker tends +to save its energy more in the case of larger value of h +and smaller T . +9 +Conclusion +We have formulated a two-player game in a cluster form- +ing of resilient multiagent systems played over time. The +players consider the impact of their actions on future +communication topology and agent states, and adjust +their strategies according to a rolling horizon approach. +Necessary conditions and sufficient conditions for form- +ing clusters among agents have been derived. We have +discussed the effect of the weights of the utility func- +tions and different initial states on cluster forming, and +evaluated the effects of varying horizon length and game +period on the players’ performance. +Possible future extensions include the case where the +players’ utility functions are not zero-sum, the case +where the players do not have perfect knowledge, and +the setting where each agent is capable to decide its own +strategies in a decentralized way. We have also consid- +ered in [22] the case where the players’ horizon lengths +and game periods are not uniform. This case can be fur- +ther generalized to decentralized settings where agents +decide their own strategies in an asynchronous way. +16 + +Furthermore, it is also interesting to consider a case +where the players may not have a complete knowledge of +the other players. 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Control, 59(3):804–808, 2014. +17 + diff --git a/6dE1T4oBgHgl3EQfmwTN/content/tmp_files/load_file.txt b/6dE1T4oBgHgl3EQfmwTN/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..bf46c9e5f39a0dc45b02d66a13ba9ee295dbfb84 --- /dev/null +++ b/6dE1T4oBgHgl3EQfmwTN/content/tmp_files/load_file.txt @@ -0,0 +1,1191 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf,len=1190 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='03302v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='SY] 9 Jan 2023 A Rolling Horizon Game Considering Network Effect in Cluster Forming for Dynamic Resilient Multiagent Systems Yurid Nugraha a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Ahmet Cetinkaya b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Tomohisa Hayakawa a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Hideaki Ishii c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Quanyan Zhu d aDepartment of Systems and Control Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Tokyo Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Tokyo 152-8552,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Japan bDepartment of Functional Control Systems,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Shibaura Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Tokyo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' 135-8548,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Japan cDepartment of Computer Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Tokyo Insitute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Yokohama 226-8502,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Japan dDepartment of Electrical and Computer Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' New York University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Brooklyn NY,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' 11201,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' USA Abstract A two-player game-theoretic problem on resilient graphs in a multiagent consensus setting is formulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' An attacker is capable to disable some of the edges of the network with the objective to divide the agents into clusters by emitting jamming signals while, in response, the defender recovers some of the edges by increasing the transmission power for the communication signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Specifically, we consider repeated games between the attacker and the defender where the optimal strategies for the two players are derived in a rolling horizon fashion based on utility functions that take both the agents’ states and the sizes of clusters (known as network effect) into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' The players’ actions at each discrete-time step are constrained by their energy for transmissions of the signals, with a less strict constraint for the attacker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Necessary conditions and sufficient conditions of agent consensus are derived, which are influenced by the energy constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' The number of clusters of agents at infinite time in the face of attacks and recoveries are also characterized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Simulation results are provided to demonstrate the effects of players’ actions on the cluster forming and to illustrate the players’ performance for different horizon parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Key words: Multiagent Systems, Cybersecurity, Game Theory, Consensus, Cluster Forming, Network Effect/Network Externality 1 Introduction Applications of large-scale networked systems have rapidly grown in various areas of critical infrastructures including power grids and transportation systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Such systems can be considered as multiagent systems where a number of agents capable of making local decisions interact over a network and exchange information to reach a common goal [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' While wireless communica- tion plays an important role for the functionality of the network, it is also prone to cyber attacks initiated by malicious adversaries [11,25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Email addresses: yurid@dsl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='titech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='jp (Yurid Nugraha), ahmet@shibaura-it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='jp (Ahmet Cetinkaya), hayakawa@sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='titech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='jp (Tomohisa Hayakawa), ishii@c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='titech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='jp (Hideaki Ishii), quanyan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='zhu@nyu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='edu (Quanyan Zhu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Jamming attacks in consensus problems of multiagent systems have been studied in [3, 5, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Noncooperative games between attackers and other players protecting the network are widely used to analyze security prob- lems, including jamming attacks [12, 17] and injection attacks [18,24,26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' In a jamming attack formulation, it is natural to consider that the jammer/the attacker has an energy constraint such that, if it is not connected to energy sources, it is impossible to attack all communication links of the net- work at all times [4,5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' In the context of game-theoretical approaches, this constraint becomes important to char- acterize the strategic behaviors of the players [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' When the links in the network are attacked, the agents may become disconnected from other agents, resulting in several groups of connected agents, or clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' The work [13] proposed the notion of network effect/network externality, which refers to the utility of an agent in a certain cluster depending on how many other agents belong to that particular cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Such a concept has been used to analyze grouping of agents on, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=', social networks and computer networks, as discussed in [10,16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Rolling horizon control has been used to handle sys- tems with uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' It is also studied in the context of networked control [15,30], where there may be addi- tional uncertainties related to communications among agents in the networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Rolling horizon approaches are also discussed in noncooperative security game settings in [34,35], where horizon lengths affect the resilience of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Rolling horizon approaches have also been used to handle the constraints in the system, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=', in an agent with obstacle avoidance constraints [14,27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' In this paper, we consider a security problem in a two- player game setting between an attacker, who is moti- vated to disrupt the communication among agents by attacking communication links, and a defender, who at- tempts to recover some of the attacked links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' We for- mulate the problem based on [6, 20], which use graph connectivity to characterize the game and the players’ strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' The game in this paper is played repeatedly over discrete time in the context of multiagent consen- sus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' As a results of these persistent attacks and recover- ies, under consensus protocol cluster forming emerges among the agents of the networks with different clus- ters having different agents’ states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Cluster forming in multiagent systems has been studied in, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=', [1, 7, 29], where the relations among certain agents may be hos- tile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' In this paper, we approach clustering from a dif- ferent viewpoint based on a game-theoretic formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Specifically, the players of the game consider network effect/network externality [13] to form clusters among agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Their utilities are determined by how the net- work is disconnected into groups of agents as well as how the players’ actions affect the states of the agents at each time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Under this setting, the number and the size of the clusters are influenced by how strong the attacks are;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' the stronger attacker is supposed to be able to separate agents into more smaller clusters, and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' In the resilient network setting, it is common that there exists a network manager who is aware of the incoming attack, since the agents try to communicate with their neighbor agents at all time and thus quickly know if some of their neighbors do not send any signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' The network manager then tries to prepare a defense plan to quickly recover from such attacks and to repel the subsequent attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' From the attacker’s viewpoint, it is also common that the attacker knows which edges of the network are the most vulnerable as well as how powerful the network manager is, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=', the manager’s remaining resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Therefore, we believe that this sequential model can be applied to several real-world settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' The main contribution of this paper is that we introduce a repeated game played repeatedly over time to model the decision making process between the attacker and the defender in the context of network security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' It is then natural to explore how these games affect the networks and state evolution of the agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Consensus protocol is considered due to its simple characterization, where all agents should converge in the case of no attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' More specifically, in comparison to [6, 20], our contribution is threefold: (i) We introduce more options for the at- tacker’s jamming signal strengths;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' (ii) the game consists of multiple attack-recovery actions, resulting in more complicated strategies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' and (iii) we consider a rolling horizon approach for the players so that their strategies may be modified as they obtain new knowledge of the status of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' More specifically, it is now possible for the attacker to disable links with stronger intensity of attack signals so that the defender is unable to recover those links (the decision on which edges are to be attacked with stronger attack signals is made at the same time as the decision on which edges are to be attacked with nor- mal attack signals);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' this feature is motivated by [32,33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' In practice, this is possible when the attacker emits stronger jamming signals that takes more resource that results in much lower signal-to-interference-plus-noise ratio (SINR) so that it is not possible for the defender to recover the communication on those links with its lim- ited recovery strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' On the other hand, we consider games consisting of multiple parts, where the players need to consider their future utilities and energy con- straints when deciding their strategies at any point in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' This setting enables the the players to think fur- ther ahead and prioritize their long-term payoffs, com- pared to in a single-step case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' The players recalculate and may override their strategies as time goes on, according to the rolling horizon approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' A related formulation without rolling horizon is discussed in [19], where the players are not able to change their strategies decided at earlier times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' In Section 2, we in- troduce the framework for the attack-recovery sequence, cluster forming among agents, and energy consumption models of the players.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' The utility functions of the games in rolling horizon approach of the repeated games is dis- cussed in Section 3, whereas the game structure is char- acterized in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' In Section 5, we analyze some con- ditions of consensus among agents, which are related to the parameters of the underlying graph and the players’ energy constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' We continue by discussing the clus- ter forming of agents when consensus is not achieved in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' The equilibrium characterization of the game under certain conditions is discussed in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' We 2 then provide numerical examples on consensus and clus- ter forming in Section 8 and conclude the paper in Sec- tion 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' The conference version of this paper appeared in [21], where we consider a more restricted situation on how often players update their strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' The notations used in this paper are fairly standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' We denote by |·| the cardinality of a set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' The floor function and the ceiling function are denoted by ⌊·⌋ and ⌈·⌉, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' The sets of positive and nonnegative integers are denoted by N and N0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' 2 Attack/Recovery Characterization for Multi- agent Systems Under Consensus Dynamics We consider a multiagent system of n agents communi- cating to each other in discrete time in the face of jam- ming attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' The agents are aiming to converge to a consensus state by interacting with each other over the communication network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' The network topology for the normal operation is given by an undirected and con- nected graph G = (V, E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' The graph consists of the set V of vertices representing the agents and the set E ⊆ V ×V of edges representing the communication links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' The edge connectivity [2] of the connected graph G is denoted by λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Each agent i has the scalar state xi[k] following the discrete-time update rule at time k ∈ N0 given by xi[k + 1] = xi[k] + ui[k], x[0] = x0, (1) where ui[k] denotes the control input applied to agent i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' We assume that ui[k] is constructed as the weighted sum of the state differences between agent i and its neighbor agents, commonly used in, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=', [8], which is given by ui[k] = � j∈Ni[k] aij(xj[k] − xi[k]), (2) where Ni[k] denotes the set of agents that can communi- cate with agent i at time k, and aij represents the weight of edge (i, j) ∈ E such that Σn j=1,j̸=iaij < 1, i ∈ V to ensure that the agents achieve consensus without any attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' We assume that the jamming attacks on an edge affect the communication between the two agents connected by that attacked edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' As a result, the set Ni[k] may change, and the resulting communication topology can be disconnected at time k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Such jamming attacks are represented by the removal of edges in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' On the other hand, within the system there is a defender that may be capable of maintaining the communication among the agents, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=', by asking agents to send stronger commu- nication signals to overcome the jamming signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' This action is represented as rebuilding some of the attacked edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' From this sequence of attacks and recoveries, we charac- terize the attack-recovery process as a two-player game between the attacker and the defender in terms of the communication links in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' In other words, the graph characterizing the networked system is resilient if the group of agents is able to recover from the damages caused by the attacker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' However, there may be cases where the resiliency level of the graph is reduced if the jamming signals are sufficiently strong such that the de- fender cannot recover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Note that to achieve consensus, the agents need not be connected for all time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' In this paper, we consider the case where the attacker has two types of jamming signals in terms of their strength, strong and normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' The defender is able to recover only the edges that are attacked with normal strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' In the following subsections, we first describe the sequence of attacks and recoveries and characterize some constraints on the players’ energy and computational ability that we need to impose as well as how the objective of the problem is formulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='1 Attack-Recovery Sequence In our setting, at each discrete time k, the players (the attacker and the defender) decide to attack/recover cer- tain edges in two stages, with the attacker acting first and then the defender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Specifically, at time k the at- tacker attacks G by deleting the edges EA k ⊆ E with normal jamming signals and E A k ⊆ E with strong jam- ming signals with EA k ∩ E A k = ∅, whereas the defender recovers ED k ⊆ EA k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' As mentioned earlier, the defender is not able to recover the edges attacked with strong jamming signals, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=', ED k ∩ E A k = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Due to the at- tacks and then the recoveries, the network changes from G to GA k := (V, E \\ (EA k ∪ E A k )) and further to GD k := (V, (E \\ (EA k ∪ E A k )) ∪ ED k ) at time k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' The agents then communicate to their neighbors Ni[k] based on this re- sulting graph GD k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' In this game, the players attempt to choose the best strategies in terms of edges attacked/recovered (E A k , EA k ) and ED k to maximize their own utility functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Here, the games are played every game period T time steps and the lth game is defined over the horizon of h steps from time (l − 1)T to (l − 1)T + h − 1, with l ∈ N and 1 ≤ T ≤ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' The players make decisions in a rolling hori- zon fashion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' the optimal strategies obtained at (l − 1)T for the future time may be overridden when the play- ers recalculate their strategies at time lT when the next game starts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' 1 illustrates the discussed sequence over time with h = 8 and T = 4, where the filled circles in- dicate the implemented strategies and the empty circles 3 PSfrag replacements k Edge 0 1T 2T l = 1 l = 2 l = 3 horizon length h 2nd game horizon length h game period T Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Illustration of the games played over discrete time k with rolling horizon approaches by the players.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' PSfrag replacements 0 1 2 3 k 4 Energy κA – Time Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Energy constraint of the attacker considered in the formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' The dashed line represents the total supplied energy to spend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' The filled circles representing the actual energy consumed by the player should be below the dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' indicate the strategies of the game that are discarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' In this setting, the horizon length h indicates the com- putational ability, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=', how long in the future the play- ers can plan their strategies, whereas the game period T ≤ h indicates the players’ adaptability, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=', how long the players apply the obtained strategies without updat- ing (shorter T means that a player is more adaptable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' The rolling horizon game structure will be discussed in Section 4 in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='2 Energy Constraints The actions of the attacker and the defender are af- fected by the constraints on their energy resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' It is assumed that the total supplied energy for the play- ers increases linearly in time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' furthermore, the energy consumed by the players is proportional to the number of attacked/recovered edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Here we suppose that the players initially possess certain amount of energy κA and κD for the attacker and the defender, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' More- over, the players are assumed to be able to supply en- ergy wirelessly to devices that obstruct/retain commu- nication signals between the agents so that the energy supply rates to these devices are limited by the constant values of ρA and ρD every discrete time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' These de- vices are supposed to have unlimited battery capacity and thus can be supplied constantly by the players with a linear rate ρA or ρD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' For the attacker, the strong attacks on E A k take β A > 0 energy per edge per unit time whereas the normal attacks on EA k take βA > 0 cost per edge, with β A > βA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' The total energy used by the attacker is constrained as k � m=0 (β A|E A m|+βA|EA m|) ≤ κA + ρAk (3) for any time k, where κA ≥ ρA > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' This implies that the total energy spent by the attacker cannot exceed the available energy characterized as the sum of the initial energy κA and the supplied energy ρAk by time k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' This energy constraint restricts the number of edges that the attacker can attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Note that the attacker’s available energy increases by ρA at each k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' The condition κA ≥ ρA allows the attacker to have at least the same attack ability at time k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' 2 illustrates the energy constraint of the attacker, where the dashed line with slope ρA represents the total supplied energy and the filled circles indicate the total energy spent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' A critical case is when βA < ρA, since it is possible for the attacker to attack at least one edge for all times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' This will have implications on the consensus and cluster forming of the agents, as we will discuss later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' The energy constraint for the defender is similar to (3): k � m=0 βD|ED m|≤ κD + ρDk, (4) with κD ≥ ρD > 0 and βD > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Note that there is a single term on the left-hand side because there is only one type of recovery signals for the agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' 3 Utility Functions with Cluster Forming and Agent-group Index Considerations In our game setting, the attacker tries to make the graph disconnected to separate the agents into clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Here, we introduce a few notions related to group- ing/clustering of agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' In a given subgraph G′ = (V, E′) of G, the agents may be divided into n(G′) number of groups, with the groups V′ 1, V′ 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' , V′ n(G′) being a partition of V with ∪n(G′) p=1 V′ p = V and V′ p ∩ V′ q = ∅, if p ̸= q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' There is no edge connecting different groups, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=', ei′,j′ /∈ E′, ∀i′ ∈ V′ p, j′ ∈ V′ q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' We also call each subset of agents taking the same state at infinite time as a clus- 4 ter, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=', limk→∞(xi[k] − xj[k]) = 0 implies that agents i and j belong to the same cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' In the considered game, the attacker and the defender are concerned about the number of agents in each group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Specifically, we follow the notion of network ef- fect/network externality [13], where the utility of an agent in a certain group depends on how many other agents belong to that particular group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' In the context of this game, the attacker wants to isolate agents so that fewer agents are in each group, while the defender wants as many agents as possible in the same group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' We then represent the level of grouping in the graph G′ by the function c(·), which we call the agent-group index, given by c(G′) := n(G′) � p=1 |V′ p|2−|V|2 (≤ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' (5) The value of c(G′) is 0 if G′ is connected, since there is only one group (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=', n(G′) = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' A larger value (closer to 0) of c(G′) implies that there are fewer groups in graph G′, and/or each group has more agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' The agent-group indices of some graphs are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Here, it is interesting that c(GD) is smaller than c(GC), even though GC has more groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' It is because the largest cluster is constituted by more agents in GC than the case of GD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Thus, for an attacker who tries to reduce the number of agents in one cluster, GD is preferable to GC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' In our problem setting, the players also consider the ef- fects of their actions on the agent states when attack- ing/recovering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' For example, the attacker may want to separate agents having state values with more differences in different groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' We specify the agents’ state differ- ence zk as zk(E A k , EA k , ED k ) := xT[k + 1]Lcx[k + 1], (6) with Lc, for simplicity, being the Laplacian matrix of the complete graph with n agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' That is, (6) represents the sum of squares of the state differences of all the agent pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' This implies that all state differences between any pair of agents are worth the same and thus the players do not prioritize any connection between agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' The attacked and recovered edges (E A k , EA k , ED k ) will af- fect x[k + 1] in accordance with (1) and (2), and in turn the value of zk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Note that the value of zk is nonincreas- ing over time [2] even if some agents are left discon- nected from other agents under attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' This sum-of- square characterization of the agents’ state difference is commonly used and essentially the same to our previous work [19] for the continuous-time setting;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' here, we ex- tend the formulation to comply with the discrete-time 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 PSfrag replacements (a) GA (b) GB (c) GC (d) GD Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Graphs and their agent-group indices: (a) c(GA) = 0, (b) c(GB) = −12, (c) c(GC) = −22, and (d) c(GD) = −24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Note that c(GC) is larger than c(GD), even with more number of groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' setting by considering the states at one time step ahead k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Now, we combine the two measures in (5) and (6) to construct the utility functions for the game in a zero- sum manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Specifically, for the lth game starting at time k = (l−1)T , the attacker and the defender’s utility functions take account of the agent-group index c(·) and the difference zk of agents’ states over h horizon length from time (l − 1)T to (l − 1)T + h − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' With weights a, b ≥ 0, the utilities for the lth game U A l for the attacker and U D l for the defender are, respectively, defined by U A l := (l−1)T +h−1 � k=(l−1)T (azk − bc(GD k )), (7) U D l := −U A l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' (8) In our setting both players attempt to maximize their utilities at the start of each game l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' The values of a and b represent the preference of the players towards either a long-term agent clustering or a short-term agent- grouping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' A higher value of a implies that the players prefer to focus on the agent states and the subsequent cluster forming, whereas a higher value of b implies that they focus on the agent-grouping more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' We suppose that both players know the underlying topology G as well as the states of all agents xi[k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' 4 Rolling Horizon Game Structure We are interested in finding the subgame perfect equi- librium [9] of this game outlined in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' To this end, the game is divided into some subgames/decision- making points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' The subgame perfect equilibrium must be an equilibrium in every subgame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' The optimal strat- egy of each player is obtained by using a backward in- duction approach, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=', by finding the equilibrium from the smallest subgames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' The tie-break condition happens when the players’ strategies result in the same utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' In this case, we suppose that the players choose to save their energy by attacking/recovering less edges unless they have enough energy to attack/recover all edges in ev- ery subsequent steps, in which case they attack/recover more edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Due to the nature of the rolling horizon approach, the 5 strategies obtained from the lth game, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=', attacked and recovered edges, are applied only from time (l − 1)T to lT − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Specifically, in the lth game for time (l − 1)T to (l − 1)T + h − 1, the strategies of both players are denoted by ((E A l,1, EA l,1, ED l,1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' , (E A l,h, EA l,h, ED l,h)), with (E A l,α, EA l,α, ED l,α) indicating the strategies at the αth step of the lth game with α ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=', h}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Note that here we show the strategies with two subscripts representing the game and the step indices along the time axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' From the above set of strategies, only ((E A l,1, EA l,1, ED l,1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' , (E A l,T , EA l,T , ED l,T )) is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Re- call that h is taken to be greater than or equal to T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Therefore, for the lth game from time (l − 1)T to lT − 1, the strategy applied will be written as ((E A (l−1)T , EA (l−1)T , ED (l−1)T ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' , (E A lT −1, EA lT −1, ED lT −1)) := ((E A l,1, EA l,1, ED l,1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' , (E A l,T , EA l,T , ED l,T )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' We look at how the optimal edges can be found by an example with h = 2 and T = 1 or 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' In this case,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' for the lth game over time (l−1)T and (l−1)T +1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' the optimal strategies of the players are given by ED∗ l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='2 (E A l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' EA l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='2) ∈ arg max ED l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='2 U D l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' (9) (E A∗ l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='2 (ED l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' EA∗ l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='2 (ED l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='1)) ∈ arg max (E A l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='EA l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='2) U A l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' (10) ED∗ l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='1 (E A l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' EA l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='1) ∈ arg max ED l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='1 U D l ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' (11) (E A∗ l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' EA∗ l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='1 ) ∈ arg max (E A l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='EA l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='1) U A l ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' (12) where U A l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='α and U D l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='α are defined as parts of U A l and U D l ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' calculated from the αth step to the last (hth) step of the lth game,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=', U A l,α = −U D l,α := �(l−1)T +h−1 (l−1)T +α−1(azk − bc(GD k )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' In this case with h = 2, the functions U A l,2 and U D l,2 are based on the values of azk and bGD k at k = (l − 1)T + 1 only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Note that to find (E A∗ l,1 , EA∗ l,1 ), one needs to obtain ED∗ l,1 (E A l,1, EA l,1) be- forehand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Likewise, to find ED∗ l,1 , one needs to obtain (E A∗ l,2 (ED l,1), EA∗ l,2 (ED l,1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Similarly, to find (E A∗ l,2 , EA∗ l,2 ), the edges ED∗ l,2 (E A l,2, EA l,2) must be obtained beforehand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Note that deriving the optimal strategies above is subject to the energy constraints (3) and (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' For h > 2, the players’ optimal strategies consist of 2h parts similar to those in (9)–(12), with one time step consisting of two parts of strategies corresponding to the number of players.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' They are solved by the players at every time k = (l − 1)T of the lth game, l ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' With T = h, the players do not have chance to override their strategies, which removes the rolling horizon aspect of the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' We will find the optimal strategies of the players by computing all possible combinations, since the choices of edges are finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' From the optimization problems speci- fied above, the players examine at most 3|E|2|E|h number of combinations of attacked and recovered edges for util- ity evaluations, since they have to foresee the opponent’s response as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Note that the attacker has three pos- sible actions on an edge: no attack, attack with normal signals, and attack with strong signals, whereas the de- fender has only two actions: recover or not recover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Here we can see that the number of computation increases ex- ponentially with respect to the number of edges in the underlying graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' To address scalability issues, we may find edges that are easier to attack first, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=', edges that result in the formation of new groups if attacked, and limit the strategy choices over those edges only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Our previous works [19, 20] considered related games in continuous time, where the timings for launching at- tack/defense actions are also part of the decision vari- ables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' This aspect complicated the formulation, making it difficult to study games over a time horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' In this pa- per, we simplify the timing issue and instead introduce the rolling horizon feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' This enables the players to consider the cluster forming in a longer time range, which is especially important when consensus among agents is obstructed by adversaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' With this rolling horizon setting, it is important for a player to know what the opponent’s previous action at the previous step of the game is in order to know its position at the game tree, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=', which subgame is the player’s playing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' For example, if the defender does not know which edges are previously attacked, then it cannot properly calculate the value of the utility function (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' 5 Consensus Analysis In this section, we examine the effect of the game struc- ture and players’ energy constraints on consensus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' We will begin the analysis by looking at the case of certain energy conditions of the players.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Specifically, if a player has enough energy to attack/recover all edges from a certain step of the game, then it will use all of their energy to attack/recover as many edges as they can in the subsequent steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' We will confirm this point formally in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' For simplicity, we denote the total energy that the defender consumed before the lth game as ˜βD l := �(l−1)T −1 k=0 βD|ED k | and the total energy that the defender may consume from the 1st to the αth step of the lth game as ˆβD α := �α m=1 βD|ED l,m|, where we omit the index l from the left-hand side, with a slight abuse of notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Similarly, for the attacker we denote ˜βA l := �(l−1)T −1 m=0 (βA|EA m|+β A|E A m|) and ˆβA α := �α m=1(βA|EA l,m|+β A|E A l,m|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' 6 We discuss in Lemma 1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=', Lemma 2) the optimal strategy of the defender (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=', attacker) at the αth step of the game given certain energy conditions mentioned in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' This characterization of optimal strategy of the defender (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=', attacker) will be useful to obtain the necessary (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=', sufficient) conditions for consensus not to happen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='1 Necessary Conditions for not Reaching Consensus This subsection discusses necessary conditions for the agents to be separated into different clusters for infinitely long duration without achieving overall consensus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' We first discuss the defender’s optimal strategy on some games with specific conditions in Lemmas 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' In Lemma 1, we state the defender’s optimal strategy at any step of the lth game given a certain energy condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Lemma 1 If the defender’s total energy ˜βD l + ˆβD ˆα−1 con- sumed before the ˆαth step of the lth game satisfies ˜βD l + ˆβD ˆα−1 ≤ κD + ρD((l − 1)T + ˆα − 1) − (h − ˆα + 1)|E|βD, (13) then ED∗ l,α = EA∗ l,α for all α ≥ ˆα, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=', the defender will recover all normally attacked edges from the ˆαth step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' PROOF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' We first look at the last (hth) step of the lth game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Since the game consists of a horizon of h steps, the last step of the game corresponds to the last decision- making point, in which the players’ strategies cannot in- fluence the decision already made in the previous steps of the same game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Hence, in the last step of the lth game the players do not save their energy by attacking/recovering less edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' From the defender’s energy constraint (4), it is clear that at any time k, the set of edges that the defender recov- ers is bounded as |ED k |≤ κD+ρDk−�k−1 m=0 βD|ED m| βD .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Thus, at the hth step, recovered edges satisfy |ED l,h|≤ |ED′ l,h| with |ED′ l,h|:= min{⌊ κD+ρD((l−1)T +h−1)−( ˜βD l + ˆβD h−1) βD ⌋, |EA∗ l,h |}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Depending on which edges are normally attacked, the defender may not recover the maximum num- ber |ED′ l,h| of edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' If the defender’s optimal strat- egy given normally attacked edges EA l,h is not to re- cover |ED′ l,h| number of edges, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=', recover less, then the defender will be able to obtain more utility U D l,h(E A l,h, EA l,h, ED l,h) > U D l,h(E A l,h, EA l,h, ED′ l,h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' However, un- der (13) with α = h the defender has sufficiently high en- ergy, and thus the utility becomes U D l,h(E A l,h, EA l,h, ED l,h) > U D l,h(E A l,h, EA l,h, EA l,h) = U D l,h(E A l,h, ∅, ∅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' It then follows that as long as the defender has enough energy, it will recover all optimal edges attacked normally at the hth step, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=', ED∗ l,h = EA∗ l,h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Next, we investigate the effect of this property on the earlier steps of the lth game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Since the defender’s strat- egy at the hth step is not affected by its strategy at the previous (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=', (h−1)th) step when κD +ρD((l −1)T +h −1) − (˜βD l + ˆβD h−1) ≥ βD|E|, here the defender does not need to recover fewer edges at the (h − 1)th step to save energy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' this is because it already has enough energy to recover EA∗ l,h at the hth step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Now, we derive that if κD + ρD((l − 1)T + h − 2)− (˜βD l + ˆβD h−2) ≥ 2βD|E| at the (h − 1)th step, then the defender will also recover ED∗ l,h−1 = EA∗ l,h−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' To recover all attacked edges at steps α ≥ ˆα, it is then sufficient that the defender’s energy satisfies (13) so that κD + ρD((l − 1)T + α − 1) ≥ ˜βD l + ˆβD α−1 + βD|E|, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=', the worst-case scenario of the energy constraint (4) when the defender recovers all edges, is always satisfied when α ≥ ˆα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' □ From the proof above, note that if the defender’s strat- egy is not to recover all normally attacked edges given even if (13) is satisfied, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=', EA l,α = ˆEA ̸= ED l,α, then the attacker will not attack ˆEA set of edges in the first place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' This is because by attacking ˆEA (and consid- ering ED l,α ̸= ˆEA) the attacker’s utility for step α ≥ ˆα becomes U A l,α(·, ˆEA, ED l,α ̸= ˆEA) < U A l,α(·, ∅, ∅), since U D l,α(·, ˆEA, ED l,α ̸= ˆEA) > U D l,α(·, ∅, ∅) = U D l,α(·, ˆEA, ˆEA) and U D l = −U A l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' We also remark that in order to derive the same optimal strategy for the defender the quantity (h − α + 1)|E| in the right-hand side of inequality (13) can be relaxed to the maximum number of edges that the attacker can attack from step ˆα to step h given its energy condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' However, this number of edges may change every game, making the inequality complicated to express.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Lemma 2 gives an interval over which, at least once, either not attacking with normal signals or recovering nonzero edges is optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Lemma 2 There is at least one occurrence of either ED k ̸= ∅ or EA k = ∅ every ⌈ h|E|βD−ρD ρDT + 1⌉ time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' PROOF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' It follows from Lemma 1 that in a game with index l′ where (13) is satisfied for α = 1, the defender always recovers edges that are attacked normally in the 1st step, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=', ED l′,1 ̸= ∅ if EA l′,1 ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' We then investigate in which game inequality (13) is satisfied for α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Since the defender gains ρD every time k, if ED k = ∅ for 7 any k ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=', (l′ − 1)T − 1}, then (13) at the first step of the l′th game can be written as κD+ρD(l′−1)T βD ≤ h|E|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' With κD = ρD as a worst-case scenario, the left- hand side becomes ρD(1+(l′−1)T ) βD , and we then obtain l′ ≥ ⌈ h|E|βD−ρD ρDT + 1⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Note that the above fact holds when the defender does not recover any edge for any k ∈ {(j − 1)(l′ − 1)T, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' , j(l′ − 1)T − 1}, j ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' If the defender recovers one or more attacked edges at any k ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=', (l′ − 1)T − 1}, then the above result may not hold, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=', the defender may not be able to re- cover all EA l′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' However, it follows that during time k ∈ {(j − 1)(l′ − 1)T, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' , j(l′ − 1)T − 1}, either 1) the defender recovers nonzero edges (ED k ̸= ∅), or 2) the attacker attacks no edges with normal signals (EA k = ∅) at least once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' □ Lemmas 1 and 2 above imply that the defender is guar- anteed to make recoveries from normal attacks every cer- tain interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Hence, the attacker needs to attack some edges strongly to prevent the recovery in order to sepa- rate agents into different clusters, as we discuss next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' The following two results provide necessary conditions for consensus not to take place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' We consider a more gen- eral condition in Proposition 3, whereas in Theorem 4 we consider a more specific situation for the utility func- tions that leads to a tighter condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Recall that λ rep- resents the connectivity of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Proposition 3 A necessary condition for consensus not to happen is ⌊ρA/βA⌋ ≥ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' PROOF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' In deriving this necessary condition, we sup- pose that there is no recovery by the defender at any time k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Without any recovery from the defender (ED k = ∅), the attacker must attack at least λ number of edges with normal signals (which take less energy) at any time k to make GD k disconnected at all times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Otherwise, there will be time steps where the graph GD k is connected, which implies that consensus will still be reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' If the attacker attacks λ edges with normal jamming signals at all times, the energy constraint (3) becomes (βAλ − ρA)k ≤ κA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Thus, the condition ρA/βA ≥ λ has to be satisfied to ensure that the attacker can make GD k disconnected for all k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Note that, if the attacker does not have enough energy to disconnect GD k given no recovery, then it definitely cannot disconnect GD k in the face of recovery by the defender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' □ We now limit the class of utility functions in (7), (8) to the case of b = 0 in the weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' This means that the players do not take account of the agent-group index in the graph, but only the states in consensus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' In this case, the attacker may need more energy to prevent consensus as shown in the next theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Theorem 4 Suppose that b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' A necessary condition for consensus not to happen is ρA/β A ≥ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' PROOF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' We prove by contrapositive;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' especially, we prove that consensus always happens if ρA/β A < λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' We first suppose that the attacker attempts to attack λ edges strongly at all times to disconnect the graph GD k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' From (3), the energy constraint of the attacker at time k becomes (β Aλ − ρA)k ≤ κA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' This inequality is not satisfied for sufficiently large k if ρA/β A < λ, since β Aλ − ρA becomes positive and κA is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Therefore, the attacker cannot attack λ edges strongly at all times if ρA/β A < λ, and is forced to disconnect the graph by attacking with normal jamming signals instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Next, by Lemma 2 above, we show that there exists an interval of time where the defender always recovers if there are edges attacked normally, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=', ED l′ ̸= ∅ is optimal given that EA l′ ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' From the definitions in (7), (8), given that b = 0, we can see that the defender obtains a higher utility if the agents are closer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' This means that given a nonzero number of edges to recover (at time jl′T described above), the defender recovers the edges connecting further agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Specifically, for some i ∈ N, for interval [jl′T, (j +i)l′T ], there is a time step where U D l (ED k = E1) ≥ U D l (E2), with edges E1 connecting agents with further states than agents connected by E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' This fact implies that when re- covering, the defender always chooses the further dis- connected agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Since by communicating with the con- sensus protocol as in (1) the agents’ states are getting closer, the defender will choose different edges to re- cover if the states of agents connected by recovered edges ED k become close enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Consequently, if ρA/β A < λ, then there exists i ∈ N where the union of graphs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=', the graph having the union of the edges of each graph (V, �((E \\(E A k ∪EA k ))∪ED k )) over the time interval [j(l′ − 1)T, (j + i)(l′ − 1)T ], becomes a connected graph, where l′ = ⌈ h|E|βD−ρD ρDT + 1⌉ as in Lemma 2 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' These intervals [j(l′−1)T, (j+i)(l′−1)T ] occur infinitely many times, since the defender’s energy bound keeps increas- ing over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' It is shown in [31] that with protocol (1), the agents achieve consensus in the time-varying graph as long as the union of the graphs over bounded time intervals is a connected graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' This implies that consensus is 8 achieved if (V, �((E \\(E A k ∪EA k ))∪ED k )) is connected over [l′ i, l′ i + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' , l′ i+j].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Thus, if ρA/β A < λ then consensus is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' □ The result in Theorem 4 only holds for b = 0, since with b > 0 the defender may choose to recover the edges con- necting agents that already have similar states to maxi- mize c(GD k ) (instead of those connecting further agents).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' In such a case, the network may remain disconnected and thus the agents may converge to different states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' As we see from these results, the weight values affect the nec- essary conditions to prevent consensus, whereas the ef- fect of the weights on the sufficient condition (discussed later) is less straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' The effect of the values of a and b on consensus is illustrated in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='2 Sufficient Condition to Prevent Consensus The next result provides a sufficient condition for pre- venting consensus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' It shows that the attacker can pre- vent consensus if it has sufficiently large recharge rate ρA given the network topology G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' We first state Lemma 5 about the attacker’s optimal strategy under some energy conditions, similar to the discussion on the defender’s case above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Lemma 5 The attacker’s optimal strategy is E A∗ l,α = E if the attacker’s recharge rate satisfies ρA/β A ≥ |E|, or the attacker’s total energy ˜βA l + ˆβA α−1 that it con- sumes before αth step of the lth game satisfies ˜βA l + ˆβA α−1 ≤ κA + ρA((l − 1)T + α − 1) − (h − α + 1)β A|E|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' (14) PROOF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' We first observe that in the hth step of the lth game the attacker does not save their energy by attack- ing fewer edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Since zl,h(E, ∅, ∅) > zl,h(E A l,h, EA l,h, ED l,h) and c((V, ∅)) ≥ c((V, (E \\ (E A l,h ∪ EA l,h) ∪ ED l,h))) are al- ways satisfied for any edges E A l,h, EA l,h, ED l,h, the function U A h always has the highest value if the attacker strongly attacks all edges E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' It then follows that the attacker with enough energy, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=', κA + ρA((l − 1)T + h − 1) − (˜βA l + ˆβA h−1) ≥ β A|E| is satisfied, will choose to attack all edges with strong signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Similar to the proof in Lemma 1, inequalities zl,α(E, ∅, ∅) > zl,α(E A l,α, EA l,α, ED l,α) and c((V, ∅)) ≥ c((V, (E\\(E A l,α∪ EA l,α) ∪ ED l,α))) are always satisfied for any step α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Hence, the attacker will choose to attack all edges with strong signals in any step α given enough energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' This can be achieved if the attacker has high enough stored energy, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=', (14) is satisfied, or if the attacker has high enough recharge rate, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=', ρA ≥ β A|E|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' These conditions enable the attacker to attack all edges strongly while still sat- isfying the energy constraint (3) above for all steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' □ Proposition 6 A sufficient condition for all agents not to achieve consensus at infinite time is that the attacker’s parameters satisfy ρA/β A ≥ |E|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' PROOF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' By Lemma 5, the attacker always strongly at- tacks all edges with strong signals in a game at any step α given either sufficient recharge rate or sufficient stored energy at the beginning of the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Consequently, if the attacker’s recharge rate satisfies ρA/β A ≥ |E|, the attacker will attack E with stronger jamming signals at all steps of all games, separating every agent at all times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' As a result, there are n clusters formed, and hence, ob- viously, consensus is not reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' □ Remark 7 Note that the necessary conditions and the sufficient condition above consider zk = xTLcx in (6) which is a nonincreasing function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' It is possible to con- sider other Laplacian matrices, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=', Laplacian of the un- derlying graph G, however the function zk may not be non- increasing anymore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' For example, we consider a simple path graph 1-2-3 with initial states x0 = [10, 0, −5]T and Laplacian of graph G considered in state difference func- tion zk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' With weights of the utility functions (7) and (8) a = 1 and b = 0 and under consensus protocol (1) and (2) with weights a12 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='1 and a23 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='8, the players’ utili- ties in the first game with h = 1 are U A 1 = −U D 1 = 148 without any attacks, and U A 1 = U A 0 = −U D 1 = 125 if both edges are attacked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' This implies that not attacking any edge may actually be optimal for the attacker even with large enough energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' As a consequence, with Laplacian of graph G considered in state difference function zk, the analysis becomes more complicated and some of the the- oretical results do not hold anymore, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=', the sufficient condition in Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='3 Example on a Gap Between Necessary Condition and Sufficient Condition In this subsection we provide an example that illustrates the gap between the necessary condition for preventing consensus in Theorem 4 and the sufficient condition in Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Here we suppose that the defender has a very high recharge rate (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=', ρD is much larger than βD) such that it can recover any normally-attacked edges at any k (note that the condition in Theorem 4 only consists of the attacker’s parameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' This will force the attacker 9 1 2 3 4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Graph G used in the case study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Table 1 Agent state difference for various values of ρA/β A ρA/β A z20 Consensus 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='113 Yes 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='115 Yes 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='3405 No 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='4 235.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='345 No 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='8 706.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='8 No 2 1354 No to attack with strong jamming signals to disconnect any agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' We consider a graph G as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' 4, with x[0] = [−5, 0, −20, 10], h = 2, and κA = ρA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' The weight of the utility functions are set to be a = 1 and b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' We test various values of 1 ≤ ρA/β A ≤ 2, implying that the attacker can attack one edge with strong signals at all time without running out of energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Thus, the attacker needs to attack e12 (min-cut edge of G) at all times in order to prevent consensus, since it is the only edge which, if attacked, will make the graph disconnected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Note that this ratio 1 ≤ ρA/β A ≤ 2 satisfies the neces- sary condition for preventing consensus in Theorem 4, but not the sufficient condition in Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Specifically in this example we test whether consensus is prevented or not for various value of ρA/β A based on agent states at time k = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' It is interesting to note from Table 1 that even with a relatively small value of ρA/β A < |E|, consensus can still be prevented by the attacker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' From this example, we observe that there is a gap be- tween the necessary condition and the sufficient condi- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Note that this gap may be larger for a more con- nected G as well as for network consisting of more agents, where typically |E|>> λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Later in Section 8, we pro- vide more detailed examples which illustrate the effect of these parameters’ values on consensus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' As the last result of the section, we state that for a special case with the complete graph under b = 0 and h = 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=', a single-step game without rolling horizon, the condition in Theorem 4 is also sufficient, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=', there is no gap between the necessary condition and the sufficient condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Proposition 8 Suppose that b = 0 and h = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' In the complete graph G, a sufficient condition for consensus not to happen is ρA/β A ≥ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' PROOF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' With h = 1, the attacker will spend all of its energy at the only step of the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' With ρA/β A ≥ n−1, the attacker is always able to disconnect the complete graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' In the complete graph G, every agent is connected to all other agents regardless of their states, implying that there is no agent that can be prioritized to be isolated by the attacker (different from the example above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Then, with b = 0, the attacker is ensured to separate the fur- thest agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' This implies that, at each game (and at each k), the attacker will always attack the same edges, re- sulting in disconnected GD k at each time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' □ We note that in different class of graphs (including in other symmetric graphs such as cycle graphs or star graphs), it is more challenging to derive a tighter suffi- cient condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' This is because agents have direct access only to some other agents which makes cluster forming based on the agent states more difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' 6 Clustering Analysis In this section, we derive some results on the number of formed clusters of agents at infinite time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' From Propo- sition 6, the result implies the simple case where if the attacker has enough energy such that ρA/β A ≥ |E|, then the attacker can attack all the edges of the underlying topology G so that the number of clusters is n (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=', all the agents are separated).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' The next result discusses a relation between the at- tacker’s cost and energy recharge rate with the maxi- mum number of clusters that the attacker may create through jamming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' In the subsequent results of this sec- tion, we suppose that b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' We first define a vector which characterizes the maxi- mum number of clusters of G, given the parameters ρA and β A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Specifically, we define a vector Θ ∈ R|E| with el- ements Θj := max|EA|=j n(V, E \\ EA), with n(V, E \\ EA) being the number of agent groups of (V, E \\ EA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Proposition 9 An upper bound on the number of formed clusters at infinite time is Θ⌊ρA/β A⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' PROOF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' The vector Θ consists of the maximum num- ber of formed groups n(V, E \\ EA) given the number of 10 attacked edges as the element index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Since some edges need to be attacked consistently in order to divide the agents into different clusters, the number of formed clus- ters at infinite time is never more than the maximum number of groups at any time k given the same number of strongly attacked edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Recall that ⌊ρA/β A⌋ is the maximum achievable num- ber of edges that can be strongly attacked at all times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Given the known graph topology G, we then can imply that Θ⌊ρA/β A⌋ gives the maximum number of clusters at infinite time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' □ We continue by addressing a special case where all the agents in the network are connected with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Corollary 10 In the complete graph G, the attacker can- not divide the agents into more than 1 + (n−1) � j=1 min � 1, � 2ρA jβ A(2n − j − 1) �� (15) number of clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' PROOF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' In the complete graph, every agent is con- nected to all other n − 1 agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' From Proposition 9, we can derive the vector Θ of the complete graph G as Θ =[1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' , 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=', 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' , n − 1, n]T, where the value of the (n−1)th entry is 2, the value of the ((n−1)+(n−2))th entry is 3, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' This is because in the complete graph G the attacker needs to attack (n−1) number of edges to disconnect the graph, further (n − 2) number of edges to make three groups of agents, further (n − 3) number of edges to make four groups of agents, and so on, until (n − 1) + (n − 2) + · · · + 1 = n(n − 1)/2 agents to make n groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' The value of the ⌊ρA/β A⌋th entry of this Θ matrix for the complete graph can be written as in (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' This value determines the upper bound of the number of clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' □ In Proposition 9, we use the information of the graph structure to obtain the vector Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' We remark that if the graph structure G is not known, then the number of clusters at infinite time is in general upper bounded by ⌊ρA/β A⌋ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' This is because the attacker can attack continuously at all time at most ⌊ρA/β A⌋ number of edges, and in the most vulnerable graph with λ = 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=', tree graphs, any attacked edge will result in a new group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' To illustrate the relationship between Θ and ρA/β A, we look Table 2 Possible cases of attack and recovery actions Case c(GA l,α) c(GD l,α) 1 c(GA l,α) = c(G) c(GD l,α) = c(GA l,α) 2 c(GA l,α) < c(G) c(GD l,α) = c(GA l,α) 3 c(GA l,α) < c(G) c(GD l,α) > c(GA l,α) 7 Equilibrium Characterization In this game the strategy choices are all finite in form of edges attacked and recovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Here, we characterize the equilibrium/optimal strategies of the players in certain situations for the case where the players’ horizon length is 1 so that they myopically update their strategies every time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' In this section, we state some results when a = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=', when the players do not consider the agents’ states but agent-group index in determining their strategies so that the defender (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=', attacker) has higher (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=', lower) utility when more agents belong to the same group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Sim- ilar to the analysis in [20], here we explore some possible optimal strategy candidates for the players in a game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' However, since a game consists of several steps in this formulation, the subgame perfect equilibrium is more in- volved to characterize, compared to the case of a game consisting of one step as in [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' In the αth step of each game, there are three possibilities in function c(·) as shown in Table 2 (Cases 1, 2, and 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' From this table, we characterize the optimal strategies of both players in each case: Case 1: When c(G) = c(GD l,α), the attacker’s utility in one time step is c(G), which implies that the at- tacker should not attack any edge either with nor- mal signals or strong signals, with the utilities of both players equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' The players’ strategies in this case are called Combined Strategy 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Case 2: When c(GD l,α) = c(GA l,α), the defender does not recover any attacked edge, whereas the attacker should attack some edges either with strong or nor- mal signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' The players’ strategies in this case are classified as Combined Strategy 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Case 3: Here both players will attack/recover nonzero number of edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' In particular, the at- tacker will attack with normal signals and poten- tially with strong signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' The players’ strategies here are called Combined Strategy 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' at the graph in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' 4 from the last section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Here, Θ = [2, 2, 3, 4]T, whereas the values of ⌊ρA/β A⌋ + 1 are 2, 3, 4, 5 for ρA/β A = 1, 2, 3, and 4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Note that for any value of ρA/β A, inequality Θ⌊ρA/βA⌋ ≤ ⌊ρA/β A⌋ + 1 is always satisfied, indicating that knowing the graph structure helps to better estimate the upper bound of the number of clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' 11 We will then discuss the equilibrium for this game in Proposition 11 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' For simplicity, we only consider the case when h = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' The case of h > 1 can be examined based on the characterization here for h = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Proposition 11 The optimal strategies for the players with h = 1 satisfy the following: (1) Combined Strategy 1 if ˜βA l +βA > κA +ρA(l −1)T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' (2) Otherwise,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' (a) Combined Strategy 2 if (i) ˜βD l + βD > κD + ρD(l − 1)T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' or (ii) ˜βD l + βD ≤ κD + ρD(l − 1)T and U A l (⌊(κA + ρA(l − 1)T − ˜βA l )/β A⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' ∅,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' ∅) = maxE A k ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='EA k ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='ED k U A l (E A k ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' EA k ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' ED k ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' (b) Combined Strategy 3 if ˜βD l + βD ≤ κD + ρD(l − 1)T and U A l (⌊(κA + ρA(l − 1)T − ˜βA l )/β A⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' ∅,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' ∅) ̸= maxE A k ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='EA k ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='ED k U A l (E A k ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' EA k ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' ED k ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' PROOF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' With a = 0, we observe that the defender al- ways recovers from the optimal attack at the last step given sufficient energy, which implies that it always re- covers for h = 1 if ˜βD l + βD ≤ κD + ρD((l − 1)T ) is sat- isfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Similar to the defender, the attacker obtains the least utility, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=', zero, by not attacking for the case of h = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Therefore, the attacker will attack at least one edge as long as it has enough energy to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' We prove each point of the proposition statement as below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' (1): We now suppose that ˜βA l + βA > κA + ρA((l − 1)T ) (point (1) in the statement) is satisfied, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=', the attacker does not have enough energy to even attack one edge normally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' In this case, Combined Strategy 1 becomes optimal since there is no other choice, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=', the attacker cannot attack even one edge with normal signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' In the rest of the proof, we assume that ˜βA l +βA ≤ κA+ρA((l− 1)T ) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' (2a(i)): We now continue by providing the conditions for Combined Strategy 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Similarly to the attacker above, we observe that the defender cannot recover any edge if ˜βD l + βD > κD + ρD((l − 1)T ), implying that c(GA l,α) < c(G) and c(GD l,α) = c(GA l,α) (corresponds to point (2a(i))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' (2a(ii)): We then suppose that ˜βD l + βD ≤ κD + ρD((l − 1)T ) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' It then follows that given enough energy for the defender, the attacker needs to attack nonzero number of edges with strong signals to satisfy c(GA l,α) < c(G) and c(GD l,α) = c(GA l,α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' In order for Combined Strategy 2 to be optimal, the attacker then needs to attack edges strongly without attacking with normal signals at all, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=', EA k = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Thus, β A needs to be sufficiently low to make strong attack feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Specif- ically, U A l (E A′ k , ∅, ∅) = maxE A k ,EA k ,ED k U A l (E A k , EA k , ED k ), with |E A′ k |= ⌊(κA + ρA((l − 1)T ) − ˜βA l )/β A⌋ indicating the maximum number of edges the attacker attacks strongly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' This corresponds to point (2a(ii)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' (2b): Consequently, if ˜βD l + βD ≤ κD + ρD((l − 1)T ) and U A l (E A′ k , ∅, ∅) ̸= maxE A k ,EA k ,ED k U A l (E A k , EA k , ED k ) are true, then the attacker normally attacks nonzero num- ber of edges and the defender recovers nonzero number of edges, which imply that Combined Strategy 3 is op- timal (point 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' □ Remark 12 The characterization of optimal strategies in Proposition 11 also holds for a more general class of agent-group indices other than c(G′) defined in (5), as long as the utility function structure (7) and (8) does not change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Specifically, it holds for those indices that belong to the class given by C := {˜c : 2V × 2E → R : ˜c((V, E ∪ E′)) ≥ ˜c((V, E)), E, E′ ⊆ E}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' (16) The condition ˜c((V, E ∪ E′)) ≥ ˜c((V, E)) implies that not attacking results in the maximum value of ˜c(GA l,α) of the attacker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Similarly, for the defender, this condition im- plies that not recovering given the attacks results in the minimum value of ˜c(GD l,α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' This condition is necessary for ensuring the equilibrium as in Proposition 11, since it guarantees that attacking/recovering nonzero number of edges (corresponding to Combined Strategy 3) is always optimal for the players as long as they have the energy to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' In general, since the cases discussed above are for one step only, for longer h > 1 the optimal strategies will take form of a set of combined strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' For exam- ple, if h = 3, the sequence of optimal strategies may be {Combined Strategy 1, Combined Strategy 2, Combined Strategy 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' On the other hand, for a > 0, the condition in Proposition 11 becomes more complicated to charac- terize since attacking more edges does not necessarily result in the highest possible utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' 8 Simulation Results 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='1 Consensus and Clustering across Parameters Here we show how the consensus varies across different weights of the utility functions and the initial states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='1 Varying Weights a and b We consider the 4-agents line/path graph 1–2–3–4 with initial states x0 = [1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='75, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='75, −1]T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' The parameters 12 0 5 10 15 20 25 30 Time 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='5 1 State agent 1 agent 2 agent 3 agent 4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Agent states with a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='1 and b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='9 0 5 10 15 20 25 30 Time 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='5 1 State agent 1 agent 2 agent 3 agent 4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Agent states with a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='9 and b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='1 0 5 10 15 20 25 30 Time Edges Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Attacked and recovered edges with a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='1 and b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='9 0 5 10 15 20 25 30 Time Edges Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Attacked and recovered edges with a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='9 and b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='1 are βA = βD = 1, h = β A = 2, κA = ρA = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='6, ρD = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='3, and κD = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='8, which satisfy the necessary condition for preventing consensus in Proposition 3, but not the sufficient condition in Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' With b = 1 − a, Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' 5 and 6 show the agent states with small a (at a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='1) and large a (at a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='9), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' 7 and 8 illustrate the status of the edges in GD k over discrete time k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' There, no line in the corresponding edge implies that the edge is strongly attacked;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' likewise, dashed red lines: normally attacked, dashed black lines: recovered, and solid black lines: not attacked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='9 0 2 4 6 8 10 12 170 180 190 200 210 220 230 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Comparison of zk and − � c(GD k ) (k = 20) versus a 1 2 3 4 6 5 7 8 9 10 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Graph used for simulation in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='2 We observe that for small a, the attacker more often divides the agents into more groups, indicated by more dashed red lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' As a result, the attacker fails to prevent consensus among the agents (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' 5), despite the condition in Proposition 3 being satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' On the other hand, with large a, the attacker is more focused to make the difference among agents’ states larger while separating the agents into fewer groups compared to the case with small a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' These features can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' 8, where there are no black lines in the edge e34, and thus no consensus among the agents in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' We next present a comparison in the optimal state dif- ference zk(E A∗ k , EA∗ k , ED∗ k ) and agent-group index c(GD k ) across different a and b = 1 − a in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' We observe that with larger a, the attacker successfully prevents con- sensus among agents (shown with larger value of zk) at time k = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' On the other hand, with smaller a (corre- sponding to larger b), the attacker obtains higher c(GD k ) at the cost of low zk, implying that the attacker fails to prevent consensus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' It is interesting that the values of zk and � c(GD k ) remain almost constant for some different a, implying that there is a critical value of weights a and b that determine the consensus and the number of clus- ters at infinite time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' in this case, the critical value of a is located in 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='4 < a < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='2 Varying Initial States x0 We also observe how the initial states x0 affect the agent- group index of the agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' We consider the graph shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' 10, which consists of 10 agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' All parameters other than the initial states are set to be the same and satisfy the conditions in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Specifically, we set βA = βD = 1, β A = 2, κA = ρA = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='1, κD = ρD = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='7, and a = 1 − b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' The state trajectories of the agents with varying x0 are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' 11–13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Here we consider three cases of initial states x0: 13 (1) x0 = [1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='9, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='8, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='44, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='35, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='48, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='19, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='28]T, (2) x0 = [1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='9, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='8, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='44, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='35, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='48, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='5, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='1, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='2]T, (3) x0 = [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='8, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='44, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='35, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='48, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='58, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='8, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='75]T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Note that in Case (1), agents 1–3 have closer initial states and are far from the other agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Similarly, in Case (2), agents 8–10 have initial states that are different from the other agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' However, in Case (3), agent states are distributed approximately evenly in the range [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='35, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='8] so that it is hard for the attacker to divide them into clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' 11, we can see that in Case (1), agents 1–3, which have weak connection to other agents (only con- nected by one edge), are grouped together and converge to the same state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' This occurs by attacking the edge con- necting agents 3 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' On the other hand, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' 12 for Case (2), agents 8–10 are separated from the others because the edge connecting agents 5 and 8 is attacked continuously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Clearly, in Cases (1) and (2) it is easier for the attacker to separate agents since their initial states form clusters matching the network topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' In Case (3), however, the initial state values do not ex- hibit such properties and as a result, the states converge towards the same value as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' In this sim- ulation, the attacker is not able to effectively attack cer- tain edges at all times;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' as a consequence, the agents are not divided into clusters and thus consensus happens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' The attacker may be able to prevent consensus with higher weight a, as discussed in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='1 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' For obtaining Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' 11–13, we solve combinatorial opti- mization problems to find optimal strategies of the play- ers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' We remark that the computational complexity of this problem depends on the number of edges E of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' We have reduced the complexity by disregarding some com- binations of edges that are clearly not optimal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' for ex- ample, attacking only the edge connecting agents 4 and 7 does not disconnect the graph, and thus cannot be the best move for the attacker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='3 Varying Energy and Cost Parameters We continue by discussing the effect of the attacker’s recharge rate ρA and unit costs of attacks βA and β A on the consensus and cluster forming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Recall that in the theoretical results in Sections 5 and 6, the ratios of ρA to β A and ρA to βA are used to derive the necessary con- ditions and sufficient conditions for preventing consen- sus as well as the upper bound of the number of clusters formed at infinite time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Assuming that b = 0, the number of clusters is dictated by ρA/β A as discussed in Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' We show the number of clusters over different topologies of the un- derlying graph G in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' We consider networks with 0 5 10 15 20 25 30 Time 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='8 1 State Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Agent states in Case 1 0 5 10 15 20 25 30 Time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='5 1 State Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Agent states in Case 2 0 5 10 15 20 25 30 Time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='8 State Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Agent states in Case 3 n = 5, with the edges positioned to yield the most con- nected topology, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=', maximum λ, given the same num- ber of edges |E|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Note that, with n = 5, there are at most n(n−1)/2 = 10 number of edges in the underlying graph G (which happens for the complete graph G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' We observe that with ρA/β A ≥ |E|, the agents are divided into 5 clusters (all agents are separated) as shown in the upper left area of the figure indicated by “5” as derived in Proposition 6 whereas in the lower right area indi- cated by “1” the agents converge to the same cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' It is clear that in a more connected graph, the agents are more likely to converge to a fewer number of clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='2 Players’ Performance Under Varying Horizon Length and Game Period In this subsection, we evaluate the players’ performance under varying horizon length h and game period T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' To evaluate the performance of the players, we introduce the applied utilities ˆU A k := azk(E A∗ k , EA∗ k , ED∗ k )−bc(GD∗ k ) and ˆU D k := −azk(E A∗ k , EA∗ k , ED∗ k ) + bc(GD∗ k ), with GD∗ k = (V, ((E \\ (E A∗ k ∪ EA∗ k )) ∪ ED∗ k ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' These are el- 14 PSfrag replacements |E| ρA β A Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Number of clusters at k = 50 with b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' The un- derlying graphs used are those with 5 agents with maximum 10 edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' 0 2 4 6 8 10 12 14 16 18 Time 0 5 10 15 20 25 30 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' � k ˆU A k in the path graph (solid lines) and the com- plete graph (dashed lines) for varying value of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' The ap- plied utility for h = 2 and h = 3 in the path graph is almost identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' ements of utility functions U A l and U D l correspond- ing to the αth step, α = k mod T + 1, of the game with index l = ⌊k/T ⌋ + 1, where the obtained strategies (E A∗ (l−1)T +α−1, EA∗ (l−1)T +α−1, ED∗ (l−1)T +α−1) = (E A∗ l,α, EA∗ l,α, ED∗ l,α) are applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Since U A l = −U D l , having higher applied utility for the attacker implies lower ap- plied utility for the defender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Note that the values of h and T are uniform among the players.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' In this subsection, we consider the weight aij = ˆa, ˆa < 1/n in (2) which implies that different agents have dif- ferent convergence speeds depending on the number of their neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Furthermore, we consider various ini- tial states x0 for the agents in order to more accurately evaluate the attacker’s performance and the pattern of applied utilities ˆU A k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' We use up to 1000 randomly gener- ated initial states in this simulation for each agent rang- ing from −1 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Throughout this subsection, we use parameters n = 3, ρA = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='1, κA = 7, β A = 2βA = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='1 Players’ Performance Under Varying Horizon Length We now consider the case of varying value of horizon length h when the network is a path graph and a com- plete graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Note that the value of h is still uniform among the attacker and the defender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' The evolutions of the attacker’s applied utility ˆU A k with varying h (with T = 1 for every h) are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Table 3 Difference in the optimal actions and the resulting utilities in the path graph G between h = 2 and h = 3 Initial states |E A∗ 0 | �19 k=0 ˆU A k h = 2 h = 3 h = 2 h = 3 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='824, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='798, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='413]T 2 2 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='74 [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='983, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='649, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='535]T 2 2 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='89 [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='787, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='786, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='265]T 2 1 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='41 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='00 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='624, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='629, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='821]T 2 1 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='92 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='45 0 2 4 6 8 10 12 14 16 18 Time 0 5 10 15 20 25 30 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' � k ˆU A k in the path graph (solid lines) and the com- plete graph (dashed lines) for varying T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' The applied utility for T = 1 and T = 2 in the path graph is almost identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Since the path graph is the least connected graph, the attacker will be able to make multiple groups of agents relatively easily compared to more connected graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' As a result, the attacker may not need to have a very long horizon length h to improve its utility since it does not need to save energy as much compared to the case of the complete graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' This is shown with the overlapping red and yellow solid lines in the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' 15, implying that the horizon length h = 3 is already as good as the case of h = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' On the other hand, the blue solid line is far below the red and the yellow ones, implying that having h being too short can result in a worse utility for the attacker over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' The differences of the attacker’s strategies for some no- table cases in the path graph G between h = 2 and h = 3 are shown in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Here, we see the difference in the optimal actions between the attacker with h = 2 and h = 3 in the path graph G even though the plots of applied utilities in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' 15 are very similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' We ob- serve that when the initial states of some agents are suf- ficiently close, the attacker with h = 2 keeps attacking both edges at k = 0, whereas the attacker with h = 3 chooses to save its energy by attacking fewer edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' At k = 19 the attacker with h = 3 obtains higher applied utility, indicating that it is able to better use its energy than the attacker with h = 2 by attacking later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' On the other hand, since the complete graph is the most connected graph, here the attacker will need more energy to disconnect the graph and obtain some utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Conse- quently, even with longer h, the difference of � ˆU A k is smaller compared to the path graph case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' The difference 15 5 5 5 5 5 4 5 4 3 4 3 310 5 5 5 5 5 5 5 9 5 5 5 5 5 5 5 8 5 5 5 5 5 5 5 7 5 5 5 5 5 5 53 3 2 3 2 2 2 2 2 2 2 1 1 1 1 1 1 1 8 6 109 5 5 5 5 5 5 4 a 5 5 5 5 5 5 4 3 4 5 5 5 5 4 3 3 3 5 5 5 4 3 3 2 2 5 5 4 3 2 2 2 1 5 4 3 2 1 1 1 1 2 3 4 5 9 7 ed6between the red and the yellow dashed lines is clearer however,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' suggesting that the attacker still benefits by having h = 3 (compared to the very little difference in the path graph case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' The attacker’s different behavior for the path graph and the complete graph G suggests that in a less connected graph, the effectiveness of longer h may saturate from a lower value compared to the one in a more connected graph G, given the attacker’s energy parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' In general, we observe that having a longer h may re- sult in a better applied utility for the attacker over time due to its role as a leader of the game, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=', the attacker moves first and is able to choose its strategy that min- imizes the defender’s best response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Additionally, there is also a clear pattern on when � ˆU A k increases;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' this im- plies that the variation of initial states may not affect the attacker’s optimal strategy, except in some cases as explained above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' We also remark that the effect of different values of h is also influenced by the underlying graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Specifically, in a less connected graph G, having a very short horizon may even be more harmful compared to the case with a more connected G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' For example, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' 16, the difference of � ˆU A k in the path graph between h = 1 and h = 2 is much more apparent than in the complete graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' The possible reason is that in the path graph, it is easier for the attacker to disconnect all agents and make n groups at some time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Thus, with large enough h, the at- tacker can save enough energy to make n groups more often.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' On the other hand, we also observe that increasing horizon length from h = 2 to h = 3 has minimal effect on the attacker’s utility for the path graph, indicating that increasing horizon length past a certain value may not be beneficial anymore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' As we see later, the similar phenomenon also happens for varying values of T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='2 Players’ Performance Under Varying Game Pe- riod We then continue by simulating the case of varying value of game period T (value of h is set to be h = 3 for both players so that the assumption T ≤ h is always satisfied).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' The average value of � ˆU A k over time is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' 16, where in general, the attacker with shorter game period T has higher applied utility especially at later time for both the path graph and the complete graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' The attacker with shorter T will be more adaptive to the changes of the agents’ and players’ conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' In the context of this game, the attacker with shorter T may delay the attack further to maximize its utility later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' This in turn increases the attacker’s utility at later time, similar to the case of longer h discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Note that the yellow dashed and solid lines are the same as the yellow lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' 15, and we observe that the green and the purple lines do not differ as much as the red and Table 4 Average total number of edges attacked in the path graph G h T �k m=0|E A∗ m | (Normal) �k m=0|E A∗ m | (Strong) k = 9 k = 19 k = 9 k = 19 1 1 7 16 5 6 2 0 0 8 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='959 3 0 0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='993 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='971 2 0 0 8 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='970 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='970 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='970 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='003 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content='015 the blue lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' 15, indicating that for the attacker, having a large value of T may not be as disadvantageous as having short h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Table 4 shows the average number of edges attacked by normal and strong jamming signals given different values of h and T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' It is interesting to note that for h > T , the attacker never attacks any edge with normal signals, indicating that it prefers to save its energy to use it later for more powerful attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Consequently, the number of edges attacked strongly with h > T becomes more than those in the case of h = T , which results in the larger applied utilities as described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' We can also observe that in the case of h = 3 and T = 1, the attacker is able to strongly attack more edges than the other cases in Table 4 in average at k = 19, even though at k = 9 it attacks slightly fewer edges than the case of closer values of h and T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' This suggests that the attacker tends to save its energy more in the case of larger value of h and smaller T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' 9 Conclusion We have formulated a two-player game in a cluster form- ing of resilient multiagent systems played over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' The players consider the impact of their actions on future communication topology and agent states, and adjust their strategies according to a rolling horizon approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Necessary conditions and sufficient conditions for form- ing clusters among agents have been derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' We have discussed the effect of the weights of the utility func- tions and different initial states on cluster forming, and evaluated the effects of varying horizon length and game period on the players’ performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Possible future extensions include the case where the players’ utility functions are not zero-sum, the case where the players do not have perfect knowledge, and the setting where each agent is capable to decide its own strategies in a decentralized way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' We have also consid- ered in [22] the case where the players’ horizon lengths and game periods are not uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' This case can be fur- ther generalized to decentralized settings where agents decide their own strategies in an asynchronous way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' 16 Furthermore, it is also interesting to consider a case where the players may not have a complete knowledge of the other players.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' This incomplete version of the game is considered in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' References [1] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Altafini.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Consensus problems on networks with antagonistic interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Autom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfmwTN/content/2301.03302v1.pdf'} +page_content=' Control, 58(4):935–946, 2013.' metadata={'source': 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more specifically the direct and adjoint state methods. For highly complex parametric problems, these +two approaches may become too costly. +To reduce computational times, Proper Orthogonal Decomposition +(POD) and Reduced Basis Methods (RBMs) have already been investigated. The majority of these algorithms +are however intrusive in the sense that the High-Fidelity (HF) code must be modified. To address this issue, +non-intrusive strategies are employed. The NIRB two-grid method uses the HF code solely as a “black-box”, +requiring no code modification. Like other RBMs, it is based on an offline-online decomposition. The offline +stage is time-consuming, but it is only executed once, whereas the online stage is significantly less expensive +than an HF evaluation. +In this paper, we propose new NIRB two-grid algorithms for both the direct and adjoint state methods. +On a classical model problem, the heat equation, we prove that HF evaluations of sensitivities reach an optimal +convergence rate in L∞(0, T; H1(Ω)), and then establish that these rates are recovered by the proposed NIRB +approximations. These results are supported by numerical simulations. We then numerically demonstrate that +a further deterministic post-treatment can be applied to the direct method. This further reduces computational +costs of the online step while only computing a coarse solution of the initial problem. All numerical results are +run with the model problem as well as a more complex problem, namely the Brusselator system. +1 +Introduction. +Sensitivity analysis is a critical step in optimizing the parameters of a parametric model. The goal is to see how +sensitive its results are to small changes of its input parameters. It is especially useful in the biomedical field +when experiments are extremely complex or prohibitively expensive. Indeed, conducting several experiments to +determine the impact of all parameters involved in biological processes may be difficult, if not impossible. +Several methods have been developed for computing sensitivities, see [4] for an overview. We focus here +on two differential-based sensitivity analysis approaches in connection with models given as reaction-diffusion +equations. +• “The direct method”, also known as the ”forward method”, which may be used when dealing with dis- +cretized solutions of parametric Partial Differential Equations (PDEs). The sensitivities (of the solution or +other outputs of interest) are computed directly from the original problem. One drawback is that it necessi- +tates solving a new system for each parameter of interest, i.e., for P parameters of interest, P + 1 problems +have to be solved. +• “The adjoint state method”, also known as the ”backward method”. It may be a viable option [44] when the +direct method becomes prohibitively expensive. In this setting, the goal is to compute the sensitivities of an +objective function that one aims at minimizing. The associated Lagrangian is formulated, and by choosing +appropriate multipliers, a new system known as ”the adjoint” is derived. This approach is preferred in +many situations since it avoids calculating the sensitivities with respect to the solutions. For example, +in the framework of inverse problems, one can determine the ”true” parameter from several measures +1Felix-Klein-Institut f¨ur Mathematik, Kaiserslautern TU, 67657, Deutschland +1 +arXiv:2301.00761v1 [math.NA] 2 Jan 2023 + +(which are frequently provided by multiple sensors) while combining it with a gradient-type optimization +algorithm. As a result, we get the ”integrated effects” on the outputs over a time interval. The advantage is +that it only requires two systems to solve regardless of the number of parameters of interest. +Thus, the direct method is appealing when there are relatively few parameters or a large number of objective +functions, whereas the adjoint state method is preferred when there are many parameters and few objective +functions. +Earlier works. +For extremely complex simulations, both methods may still be impractical. Several reduction +techniques have thus been investigated in order to reduce the complexity of the sensitivity computation. Among +them, Reduced Basis Methods (RBMs) are a well-developed field [36, 40, 3]. They use an offline-online decompo- +sition, in which the offline step is time-consuming but is only performed once, and the online step is significantly +less expensive than a High-Fidelity (HF) evaluation. In the context of sensitivity analysis, the majority of these +studies rely on a Galerkin projection onto the adjoint state system in the online part. In what follows, we present +a brief review of previous works on RBMs combined with both sensitivity methods. +• Let us begin with the direct method. It has been employed and studied with RB spaces in various applica- +tions, e. g., [39, 47, 13]). The sensitivities may also be useful to enhance the reduced state approximation. +Unlike the other studies cited below, the sensitivities in [25] are computed to improve RB methods (see +also [24] with a Lagrangian formulation or [23] with a finite difference approach [23]). Still to improve an +approximation, in [31], a combined method is proposed (based on local and global approximations with +series expansion and a RB expression), which was first developed in [30]. +Note also that variance-based sensitivity analysis has been investigated using RBMs [28] and non-intrusive +RB [34]. +• The adjoint state formulation can be thought of as a PDE-constrained optimization. The first applications +of this method in conjunction with computational reduction approaches can be found in [27] in the context +of RBMs, where several RB sub-spaces are compared or in [42] with the POD method, with an affine +parameter dependence. Currently, particular emphasis is being placed on developing accurate a-posteriori +error estimates in order to improve basis generation [41, 46, 11, 12] with Proper Orthogonal Decomposition +(POD) and/or RBMs. +RBMs and POD have also been investigated in the context of optimal control under uncertainty [9]. In +recent studies, the case of infinite-dimensional control function is considered with RB approximations on +the state, adjoint, and control variables [29, 1]. Even if the adjoint state method is frequently preferred, +writing its associated reduced problem can be difficult when the adjoint formulation is not straightforward. +It may also be reformulated to take advantage of previously developed RB theory. For example, in [38], it +is rewritten as a saddle-point problem for Stokes-type problems. +To conclude this brief overview of RBMs applied to sensitivity analysis, we add that non-intrusive methods have +been developed, in the framework of the inverse problem, without computing the sensitivities (see the PBDW +method [37, 22, 10] with a direct formulation). +Motivation. +Even though the Galerkin projection is prevalent in the literature, its main disadvantage lies in its +intrusiveness. Indeed, in order to approximate the solution of a PDE, the matrices computed from its variational +formulation must be changed in the HF code. This may be difficult if the HF is very complex or even impossible +if it has been purchased, as is often the case in an industrial context. From an engineering standpoint, Non- +Intrusive Reduced Basis (NIRB) methods are more practical to implement than intrusive RBMs because they only +require the execution of the HF code as a ”black-box” solver. Apparently, NIRB methods have not yet been used +to approximate sensitivities except for statistical approaches such as variance-based sensitivity analysis. +In this paper, we aim at computing the sensitivities with respect to some parameters of interest µ ∈ G, +with the direct and adjoint methods combined with NIRB techniques. We focus on the NIRB two-grid method +[7, 17, 8, 43] (see also different NIRB methods [6, 2, 15] from the two-grid method). Like most RBMs, the NIRB +two-grid method relies on the assumption that the manifold of all solutions S = {u(µ), µ ∈ G} has a small +Kolmogorov width [33] (in what follows, uh(µ) will refer to the HF solution for the parameter µ). +The two-grid algorithm can be employed for a variety of PDEs and is simple to implement. It has been +studied with FEM in the context of elliptic equations [7] and parabolic equations [19] (see also [17] for finite +volume schemes). Furthermore, because it is non-intrusive, it is suitable for a wide range of problems. The +2 + +effectiveness of this method relies on its offline/online decomposition (as most RBMs). The offline part is time- +consuming but it is only performed once. On the contrary, the specific feature of the NIRB approach is to solve +the parametric problem on a coarse mesh only during the online step, and then to rapidly improve the precision +of the coarse solution. It makes this portion of the algorithm much cheaper than a HF evaluation. +In this paper, we combine the two-grids framework with both sensitivity analysis methods. Then, drawing in- +spiration from recent works [18], we efficiently apply a deterministic process to further reduce the computational +cost of its online stage with the direct method, in the context of parabolic equations. During the online stage, +this additional step allows us to solve only the initial problem on the coarse mesh, regardless of the number of +parameters of interest, making this novel approach very appealing. We highlight the fact that because the direct +approach requires a new system to be solved for each parameter, the adjoint method is preferred in many studies +(as cited above), despite the fact that its formulation is more complex and yields integrated sensitivities over +time. +Outline of the paper. +This article is about extending the NIRB two-grid method to the computation of sensitiv- +ities and performing the associated numerical analysis. We present and illustrate the NIRB algorithms applied +to both sensitivity analysis methods with several numerical results. With the direct method, we have carried out +a thorough theoretical analysis of the heat equation as model problem. In this setting, we have optimal conver- +gence rates in L∞(0, T; H1(Ω)) for the spatial HF semi-discretized sensitivity solution and for its fully-discretized +form. It turns out that we obtain theoretically and numerically these optimal rates also for the NIRB sensitivity +approximations. Our main theoretical result is given by Theorem 4.1. +The rest paper is organized as follows. Section 2 describes both sensitivity methods along with established +convergence results and the NIRB two-grid algorithm for parabolic equations. In Section 3, we present the al- +gorithms for the direct and adjoint methods with the NIRB two-grid approach, as well as the new version of +the algorithm for the direct method. Section 4 is devoted to the theoretical results on the rate of convergence +for the NIRB sensitivity approximation. In the last section 5, several numerical results are presented and illus- +trate the theoretical ones. The implementation and the use of Automatic Differentiation (AD) is discussed as well. +2 +Mathematical Background. +Let Ω be a bounded domain in Rd, with d ≤ 3 and a smooth enough boundary ∂Ω, and consider a parametric +problem P on Ω. For the NIRB two-grid method, we consider two spatial ”grids” of Ω: +• one fine mesh, denoted Th, where its size h is defined as +h = max +K∈Mh +hK, +(1) +• and on coarse mesh, denoted TH, with its size defined as +H = max +K∈MH +HK >> h, +(2) +where the diameter hK (or HK) of any element K in a mesh is equal to sup +x,y∈K +|x − y|. +In this section, we first introduce our model problem, that of the heat equation, in a continuous setting, and then +its spatial (over the two meshes) and time discretizations. Then, we recall the NIRB algorithm in the context of +parabolic equations, and finally, we detail the sensitivity problems for this model problem. +In the next sections, C will denote various positive constants independent of the size of the meshes h and +H and of the parameter µ, and C(µ) will denote constants independent of the sizes of the meshes h and H but +dependent of µ. +2.1 +A model problem: The heat equation. +2.1.1 +The continuous problem. +We consider the following heat equation on the domain Ω with homogeneous Dirichlet conditions, which takes +the form +3 + +� +� +� +� +� +ut − ∇ · (A(µ)∇u) = f, in Ω×]0, T], +u(·, 0) = u0(·), in Ω, +(3) +u(·, t) = 0, on ∂Ω×]0, T], +where +f ∈ L2(Ω × [0, T]), while u0 ∈ H1 +0(Ω) and µ = (µ1, · · · , µP) ∈ G ⊂ RP is the parameter, such that +A : Ω × G → Md(R) is measurable, bounded, and uniformly elliptic. +(4) +For any t > 0, the solution u(·, t) ∈ H1 +0(Ω), and ut(·, t) ∈ L2(Ω) stands for the derivative of u with respect to +time. +We use the conventional notations for space-time dependent Sobolev spaces [35] +Lp(0, T; V) := {u(x, t) | ∥u∥Lp(0,T;V) := +� � T +0 +��u(·, t) +��p +V dt +�1/p +< ∞}, 1 ≤ p < ∞, +L∞(0, T; V) := {u(x, t) | ∥u∥L∞(0,T;V) := ess sup +0≤t≤T +��u(·, t) +�� +V < ∞}, +where V is a real Banach space with norm∥·∥V . The variational form of (3) is given by: +� +� +� +� +� +� +� +Find u ∈ L2(0, T; H1 +0(Ω)) with ut ∈ L2(0, T; H−1(Ω)) such that +(ut(t, ·), v) + a(u(t, ·), v; µ) = ( f (t, ·), v), ∀v ∈ H1 +0(Ω) and t ∈ (0, T), +(5) +u(·, 0) = u0(·), in Ω, +where a is given by +a(w, v; µ) = +� +Ω A(x; µ)∇w(x) · ∇v(x) dx, +∀w, v ∈ H1 +0(Ω). +(6) +We remind that (5) is well posed (see [14] for the existence and the uniqueness of solutions to problem (5)) and +we refer to the notations of [14]. Note that we will use the notation (·, ·) to denote the classical L2-inner product +on Ω. +2.1.2 +The various discretizations. +For the NIRB algorithm, we use the two spatial grids on the variational formulation (5) of our problem (3). We +employed P1 finite elements to discretize in space. Thus, we introduce Vh and VH, the continuous piecewise +linear finite element functions (on fine and coarse meshes, respectively) that vanish on the boundary ∂Ω. We +consider the so-called Ritz projection operator P1 +h : H1 +0(Ω) → Vh (P1 +H on VH is defined similarly) which is given +by +(∇P1 +hu, ∇v) = (∇u, ∇v), +∀v ∈ Vh, for u ∈ H1 +0(Ω). +(7) +In the context of time-dependent problems, a time stepping method of finite difference type is used to get a fully +discrete approximation of the solution of (3). As for the spatial domain, we consider two different time grids: +• One time grid, denoted F, is associated to fine solutions (for the generation of the snapshots). To avoid +making notations more cumbersome, we will consider a uniform time step ∆tF. The time levels can be +written tn = n ∆tF, where n ∈ N∗. +• Another time grid, denoted G, is used for coarse solutions. By analogy with the fine grid, we consider a +uniform grid with time step ∆tG. Now, the time levels are written �tm = m ∆tG, where m ∈ N∗. +As in the elliptic context [7], the NIRB algorithm is designed to recover the optimal estimate in space. Yet, +since there is no such argument as the Aubin-Nitsche argument for time stepping methods, we must consider +time discretizations that provide the same precision with larger time steps. Thus, we consider a higher order +time scheme for the coarse solution. As in [19], we used an Euler scheme (first order approximation) for the +fine solution and a Crank-Nicolson scheme (second order approximation) for the coarse solution on our model +problem. +Thus, we deal with two kind of notations for the discretized solutions: +4 + +• uh(x, t) and uH(x, t) that respectively denote the fine and coarse solutions of the spatially semi-discrete +solution, at time t ≥ 0. +• un +h(x) and um +H(x) that respectively denote the fine and coarse full-discretized solutions at time tn = n × ∆tF +and �tm = m × ∆tG. +Remark 2.1. To simplify the notations, we consider that both time grids end at time T here, +T = NT ∆tF = MT ∆tG. +The semi-discrete form of the variational problem (5) writes for the fine mesh (similarly for the coarse mesh): +� +� +� +� +� +� +� +Find uh(t) = uh(·, t) ∈ Vh for t ∈ [0, T] such that +(uh,t(t), vh) + a(uh(t), vh; µ) = ( f (t), vh), ∀vh ∈ Vh and t ∈]0, T], +(8) +uh(·, 0) = u0 +h(·) = P1 +h(u0)(·). +From the definition of P1 +h (7), the initial condition u0 +h (and similarly for the coarse mesh) is such that +(∇u0 +h, ∇vh) = (∇u0, ∇vh), ∀vh ∈ Vh, +(9) +and hence, it corresponds to the finite element solution of the corresponding elliptic problem of (3) with A(1) = Id +(that of the Poisson’s equation) and whose exact solution is u0. +The full discrete form of the variational problem (5) for the fine mesh with an implicit Euler scheme writes: +� +� +� +� +� +� +� +Find un +h ∈ Vh for n = 0, . . . , NT such that +(∂un +h, vh) + a(un +h, vh; µ) = ( f (tn), vh), ∀vh ∈ Vh and n = 1, . . . , NT, +(10) +uh(·, 0) = u0 +h(·), +where the time derivative in the variational form of the problem (8) has been replaced by a backward difference +quotient, ∂un +h = +un +h−un−1 +h +∆tF +. +For the coarse mesh with a Crank-Nicolson scheme, and with the notation ∂um +H = um +H−um−1 +H +∆tG +, it becomes: +� +� +� +� +� +� +� +Find um +H ∈ VH for m = 0, . . . , MT, such that +(∂um +H, vH) + a( um +H+um−1 +H +2 +, vH; µ) = ( f (�tm− 1 +2 ), vH), ∀vH ∈ VH and m = 1, . . . MT, +uH(·, 0) = u0 +H(·), +(11) +where �tm− 1 +2 = �tm+�tm−1 +2 +. +For the NIRB approximation, we will need to interpolate in space and in time the coarse solution. So let us +introduce the quadratic interpolation in time of a coarse solution at time tn ∈ Im = [�tm−1,�tm] defined on [�tm−2,�tm] +from the coarse approximations at times �tm−2,�tm−1, and �tm, for all m = 2, . . . , MT. To this purpose, we employ +the following parabola on [�tm−2,�tm]: +For m ≥ 2, ∀n ∈ Im = [�tm−1,�tm], +I2 +n[um +H](µ) := +um−2 +H +(µ) +(�tm − �tm−2)(�tm−2 − �tm−1) +� +− (tn)2 + (�tm−1 + �tm)tn − tm−1tm� ++ +um−1 +H +(µ) +(�tm−2 − �tm−1)(�tm−1 − �tm) +� +− (tn)2 + (�tm + �tm−2)tn − tmtm−2� ++ +um +H(µ) +(�tm−1 − �tm)(�tm − �tm−2) +� +− (tn)2 + (�tm−2 + �tm−1)tn − tm−2tm−1� +. +(12) +For tn ∈ I1 = [�t0,�t1], we use the same parabola defined by the coarse approximations at times �t0, �t1, �t2 as the +one used over [�t1,�t2]. We denote by � +uH +n(µ) = I2 +n[um +H](µ) the quadratic interpolation of um +H at a time n. Note that +we choose this interpolation in order to keep an approximation of order 2 in time ∆tG (it works also with other +quadratic interpolations). +In the next section, we recall the NIRB algorithm in the context of parabolic equations. +5 + +2.2 +Reminders on the Non-Intrusive Reduced Basis method (NIRB) in the context of +parabolic equations. +Let u(µ) be the exact solution of problem (3) for a parameter µ ∈ G. With the NIRB algorithm, we aim at +quickly approximating this solution by using a reduced space, denoted XN +h , constructed from N fully discretized +solutions of (10), namely the so-called snapshots. Since each snapshot is a HF finite element approximation in +space at a time tn, n = 0, ..., NT (NT being potentially very high), not all of the time steps may be required for the +construction of the reduced space. Here, for each parameter µi, i = 1, . . . , Nµ, selected for the basis construction, +the number of time steps employed (which depends on i) is denoted Ni. Thus, the reduced basis is defined as +XN +h := Span{u +(nj)i +h +(µi)| i = 1, . . . , Nµ, j = 1, . . . , Ni, (nj)i ⊂ {1, · · · , NT}}, +(13) +with N := +Nµ +∑ +i=1 +Ni. +We recall the offline/online decomposition of the NIRB procedure with parabolic equations: +• “Offline step” +The offline part of the algorithm allows us to construct the reduced space XN +h . +1. From training parameters (µi)i∈{1,...,Ntrain}, we define Gtrain = +∪ +i∈{1,...,Ntrain}µi. Then, we employ a greedy +procedure to adequately choose the parameters (µi)i=1,...,Nµ within Gtrain to construct the RB. For this +procedure, we refer to algorithm 1 (described for the setting Nµ = N in order to simplify notations). +Note that a POD-greedy algorithm may also be employed [19, 21, 20, 32]. +Algorithm 1 Greedy algorithm +Input: tol, {un +h(µ1), · · · , un +h(µNtrain) with µi ∈ Gtrain, n = 0, . . . , NT}. +Output: Reduced basis {Φh +1, · · · , Φh +N}. +Choose µ1, n1 = +arg max +µ∈Gtrain, n∈{0,...,NT} +���un +h(µ) +��� +L2(Ω) , +Set Φh +1 = +un1 +h (µ1) +���un1 +h (µ1) +��� +L2 +Set G1 = {µ1, n1} and X1 +h = span{Φh +1}. +for k = 2 to N do: +µk, nk = arg +max +(µ, n)∈(Gtrain×{0,...,NT})\Gk−1 +���un +h(µ) − Pk−1(un +h(µ)) +��� +L2, with Pk−1(un +h(µ)) := +k−1 +∑ +i=1 +(un +h(µ), Φh +i ) Φk +i . +Compute � +Φh +k = unk +h (µk) − Pk−1(unk +h (µk)) and set Φh +k = +� +Φh +k +���� +� +Φh +k +���� +L2(Ω) +Set Gk = Gk−1 ∪ {µk} and Xk +h = Xk−1 +h +⊕ span{Φh +k} +Stop when +���un +h(µ) − Pk−1(un +h(µ)) +��� +L2 ≤ tol, ∀µ ∈ Gtrain, ∀n = 0, . . . , NT. +end for +The greedy algorithm is usually less expensive than the POD-greedy (thanks to a-posteriori error +estimates for stationary problems). Although for time dependent problems, the latter is more rea- +sonable when the snapshots are computed for all time steps, our choice of using a greedy procedure +is motivated by the fact that it is more efficient with the post-treatment introduced below. The RB +functions (time-independent), denoted (Φh +i )i=1,...,N, are generated at the end of this step, from fine +fully-discretized solutions {un +h(µi)}i∈{1,...,Nµ}, n={0,...,NT} (solving problem (10) with HF solver). Note +that even if all the time steps are computed, only Ni are used for each i ∈ {1, . . . , Nµ} in the RB +construction. Since at each step k, all sets added in the basis are in the orthogonal complement of +Xk−1 +h +, it yields an L2 orthogonal basis without further processing. +Hence, XN +h +can be defined as +XN +h = Span{Φh +1, . . . , Φh +N}. +6 + +Remark 2.2. In practice, the algorithm is halted with a stopping criterion such as an error threshold or a +maximum number of basis functions to generate. +2. Then, we solve the following eigenvalue problem: +� +� +� +� +� +Find Φh ∈ XN +h , and λ ∈ R such that: +∀v ∈ XN +h , +� +Ω ∇Φh · ∇v dx = λ +� +Ω Φh · v dx, +(14) +We get an increasing sequence of eigenvalues λi, and orthogonal eigenfunctions (Φh +i )i=1,··· ,N, which +do not depend on time, orthonormalized in L2(Ω) and orthogonalized in H1(Ω). Note that with +Gram-Schmidt procedure, we only obtain an L2-orthonormalized RB. +3. For any parameter µk, k = 1, . . . , Nµ, the classical NIRB approximation differs from the HF uh(µk) +computed in the offline stage [19]. Thus, as proposed in [7], to improve NIRB accuracy, we use a +”rectification post-processing”. To this purpose, we need a rectification matrix for each fine time step, +denoted Rn, and constructed from coarse snapshots, generated by solving (11) and whose parameters +are the same as for the fine snapshots. +Thus, for all n = 1, . . . , NT, we compute the vectors +Rn +u,i = ((An)TAn + δIN)−1(An)TBn +i , +i = 1, · · · , N, +(15) +where +∀i = 1, · · · , N, +and +∀µk ∈ Gtrain, +An +k,i = +� +Ω +� +uH +n(µk) · Φh +i dx, +(16) +Bn +k,i = +� +Ω un +h(µk) · Φh +i dx, +(17) +and where IN refers to the identity matrix and δ is a regularization parameter. +Remark 2.3. Note that since every time step has its own rectification matrix, the matrix An is a “flat” rectan- +gular matrix (Ntrain ≤ N), and thus the parameter δ is required for the inversion of (An)TAn. +We also remark that with the rectification post-treatment, the standard greedy algorithm 1 may leads to more +accurate approximations, compared to the POD-greedy algorithm. It comes from the fact that the coefficients of +the matrix are directly derived from the snapshots in that case. +• “Online step” +The online part of the algorithm is much faster than a HF evaluation. +4. We solve the problem (3) on the coarse mesh TH for a new parameter µ ∈ G at each time step +m = 0, . . . , MT. +5. We quadratically interpolate in time the coarse solution on the fine time grid with (12). +6. Then, we linearly interpolate � +uH +n(µ) on the fine mesh in order to compute the L2-inner product with +the RB functions. The approximation used in the two-grid method is +For n = 0, . . . , NT, +uN,n +Hh (µ) := +N +∑ +i=1 +( � +uH +n(µ), Φh +i ) Φh +i , +(18) +and with the rectification post-treatment step, it becomes +Rn +u[uN +Hh](µ) := +N +∑ +i,j=1 +Rn +u,ij ( � +uH +n(µ), Φh +j ) Φh +i , +(19) +where Rn +u is the rectification matrix at time tn, given by (15). +7 + +In [19], we have proven the following estimate on the heat equation +for n = 0, . . . , NT, +���u(tn)(µ) − uN,n +Hh (µ) +��� +H1(Ω) ≤ ε(N) + C1(µ)h + C2(N)H2 + C3(µ)∆tF + C4(N)∆t2 +G, +(20) +where C1, C2, C3 and C4 are constants independent of h and H, ∆tF and ∆tG. The term ε(N) depends on a proper +choice of the RB space as a surrogate for the best approximation space associated to the Kolmogorov N-width. +It decreases when N increases and it is linked to the error between the fine solution and its projection on the +reduced space XN +h , given by +�����un +h(µ) − +N +∑ +i=1 +(un +h(µ), Φh +i ) Φh +i +����� +H1(Ω) +. +(21) +The constant C2 increases with N and thus, a trade-off needs to be done between increasing N to obtain a more +accurate manifold, and keeping a constant C2 as low as possible. +If H is such as H2 ∼ h, ∆t2 +G ∼ ∆tF, and ε(N) is small enough, with C2(N) and C4(N) not too large, the estimate +(20) entails an error estimate in O(h + ∆tF), and thus, we recover an optimal error estimate in L∞(0, T; H1(Ω)). +Before adapting NIRB to the sensitivity analysis context, we first recall how to derive the sensitivities functions +in the next section. +2.3 +Sensitivity analysis: The direct problem. +In this section, we recall the sensitivity systems (continuous and discretized versions) for P parameters of interest. +Then, we prove the numerical results of the direct method on the model problem. To not make the notations too +cumbersome, we will consider A(µ) = µ Id, with µ ∈ R+∗ for the analysis theorems. +2.3.1 +The continuous setting. +In this setting, we consider P parameters of interest, denoted µp = 1, . . . , µP, and we want to approximate the +exact derivatives +Ψp(t, x; µ) := ∂u +∂µp +(t, x; µ). +(22) +In order to seek these sensitivities, we solve P new systems, which can directly be obtained by differentiating the +initial problem with respect to µp. The continuous initial problem (5) may be rewritten +� +� +� +� +� +Find u(t) ∈ V for t ∈ [0, T] such that +(ut(t), v) = F(u(t), v; µ) := −a(u(t), v; µ) + ( f (t), v), ∀v ∈ V, t > 0, +u(·, 0) = u0(·), +where the bilinear form a is defined by (6). Using the chain rule and since the time and the parameter derivatives +can commute, +(Ψp,t(t), v) = ∂F +∂u (u(t), v; µ) · Ψp(t) + ∂F +∂µ(u(t), v). +Since the initial condition here does not depend on µ, we obtain the following problem +� +� +� +� +� +� +� +� +� +Find Ψp(t) ∈ V for t ∈ [0, T] such that +(Ψp,t(t), v) + a(Ψp(t), v; µ) = −( ∂A +∂µp (µ)∇u(t), ∇v), for v ∈ V, for t > 0, +Ψ0 +p = 0, +(23) +which is well-posed since u ∈ L2(0, T; H1 +0(Ω)), and under the assumptions (4), the so-called ”parabolic regularity +estimate” implies that u ∈ L2(0, T; H2(Ω)) ∩ L∞(0, T; H1 +0(Ω)) [14, 45]. +8 + +2.3.2 +The spatially semi-discretized version. +As previously for the state solution, we discretize in space and in time the sensitivity problems (23). +The corresponding spatially semi-discretized formulations (on Th) read +� +� +� +� +� +� +� +� +� +Find Ψp,h(t) ∈ Vh for t ∈ [0, . . . , T] such that +(Ψp,h,t(t), vh) + a(Ψp,h(t), vh; µ) = −( ∂A +∂µp (µ)∇uh(t; µ), ∇vh), for vh ∈ Vh, for t ∈]0, T], +Ψ0 +p,h(·) = P1 +h(Ψ0 +p)(·), +(24) +where P1 +h is given by (7). Before proceeding with the proof of Theorem (4.1), we need several results that can +be deduced from [45], but require some precisions. Indeed, first, in [45], the estimates are proven on the heat +equation with a non-varying diffusion coefficient. Secondly, the right-hand side function f vanishes when seek- +ing the error estimates, whereas in our case, the right-hand side function depends on u and necessitates precised +estimates. +On the semi-discretized formulation, the following estimate holds. +Theorem 2.4. Let Ω be a convex polyhedron. Let A(µ) = µ Id, with µ ∈ R+∗ . Consider u ∈ H1(0, T; H2(Ω)) be the +solution of (3) with u0 ∈ H2(Ω) and uh be the semi-discretized variational form (8). Let Ψ and Ψh be the corresponding +sensitivities , respectively given by (23) and (24). Then +∀t ∈ [0, T], +��Ψh(t) − Ψ(t) +�� +L2(Ω) ≤ Ch2����Ψ0��� +H2(Ω) + +� T +0 ∥Ψt∥H2(Ω) ds +� ++ C(µ)h2� � T +0 ∥ut∥2 +H2(Ω) ds +�1/2 +. +Proof. As in [45], we first decompose the error with two components θ and ρ such that +∀t ∈ [0, T], e(t) := Ψh(t) − Ψ(t) = (Ψh(t) − P1 +hΨ(t)) + (P1 +hΨ(t) − Ψ(t)), += θ(t) + ρ(t). +(25) +• For the estimate on ρ(t), a classical FEM estimate [45, 5] is +���P1 +hv − v +��� +L2(Ω) + h +���∇(P1 +hv − v) +��� +L2(Ω) ≤ Ch2∥v∥H2(Ω) , +∀v ∈ H2 ∩ H1 +0, +(26) +which leads to +��ρ(t) +�� +L2(Ω) ≤ Ch2��Ψ(t) +�� +H2(Ω) , ∀t ∈ [0, T], +≤ Ch2����Ψ0��� +H2(Ω) + +� T +0 ∥Ψt∥H2(Ω) ds +� +, ∀t ∈ [0, T]. +(27) +• For the estimate on θ(t), let us consider v ∈ Vh, +∀t ∈]0, T], (θt(t), v) + µ(∇θ(t), ∇v) = (Ψh,t(t), v) + µ(∇Ψh(t), ∇v) − (P1 +hΨt(t), v) − µ(∇P1 +hΨ(t), ∇v). +Since v ∈ H1 +0, by definition of P1 +h (7), the semi-discretized weak formulations (24) implies +(θt(t), v) + µ(∇θ(t), ∇v) = −(∇uh(t), ∇v) − (P1 +hΨt(t), v) − µ(∇P1 +hΨ(t), ∇v), += −(∇uh(t), ∇v) − (P1 +hΨt(t), v) − µ(∇Ψ(t), ∇v). +Thanks to the continuous weak formulation (23), and since the operator P1 +h and the time derivative com- +mute, it can be rewritten +(θt(t), v) + µ(∇θ(t), ∇v) = (∇u(t) − ∇uh(t), ∇v) + (Ψt(t) − (P1 +hΨ)t(t), v), += (∇u(t) − ∇uh(t), ∇v) − (ρt(t), v). +Choosing v = θ(t), it yields +(θt(t), θ(t)) + µ +��∇θ(t) +��2 +L2(Ω) = (∇u(t) − ∇uh(t), ∇θ(t)) − (ρt(t), θ(t)), +9 + +and using the continuous and semi-discretized weak formulations on the state variable u(t) ((5) and (8) +respectively), we obtain +(θt(t), θ(t)) + µ +��∇θ(t) +��2 +L2(Ω) = 1 +µ(uh,t(t) − ut(t), θ(t)) − (ρt(t), θ(t)), +(28) +where the first term of the right-hand side is a new contribution (compared to the proof of Theorem 1.2 +[45]). Since +(θt(t), θ(t)) = 1 +2 +d +dt( +��θ(t) +��2 +L2(Ω)) = +��θ(t) +�� +L2(Ω) +d +dt +��θ(t) +�� +L2(Ω) , +(29) +and, since the second term in (28) is positive, it becomes with Cauchy-Schwarz inequality (the case where +θ(t) = 0 for some t may easily be handled) +d +dt +��θ(t) +�� +L2(Ω) ≤ 1 +µ +��uh,t(t) − ut(t) +�� +L2(Ω) + +��ρt(t) +�� +L2(Ω) . +Integrating over time, it follows that +��θ(t) +�� +L2(Ω) ≤ +��θ(0) +�� +L2(Ω) +� +�� +� +T1 ++ 1 +µ +� T +0 +��uh,t − ut +�� +L2(Ω) ds +� +�� +� +T2 ++ +� T +0 +��ρt +�� +L2(Ω) ds +� +�� +� +T3 +. +(30) +– From the initial conditions, since u0 +h = P1 +hu0, T1 = 0. +Note that other optimal order choices of +discretized initial conditions (such as the L2 orthogonal projection onto Vh) lead to an estimate in +Ch2���Ψ0��� +H2(Ω) for T1. +– To estimate T2, in analogy with θ and ρ, let us introduce θu and ρu, such that +∀t ∈ [0, T], uh(t) − u(t) = (uh(t) − P1 +hu(t)) + (P1 +hu(t) − u(t)), += θu(t) + ρu(t). +(31) +We remark that the term T2 can also be written +T2 = 1 +µ +� T +0 +��θu,t + ρu,t +�� +L2(Ω) ds ≤ 1 +µ +� T +0 ∥θu,t∥L2(Ω) + +��ρu,t +�� +L2(Ω) ds. +Then, by Cauchy-Schwarz inequality, +T2 ≤ +√ +T +µ +�� � T +0 ∥θu,t∥2 +L2(Ω) ds +�1/2 ++ +� � T +0 +��ρu,t +��2 +L2(Ω) ds +�1/2� +, +(32) +We can bound � T +0 ∥θu,t∥2 +L2(Ω), using the variational formulations (5) and (8). We first write for t ∈]0, T]: +(θu,t(t), v) + µ(∇θu(t), ∇v) = (uh,t(t), v) + µ(∇uh(t), ∇v) − (P1 +hut(t), v) − µ(∇P1 +hu(t), ∇v), += ( f (t), v) − (P1 +hut(t), v) − µ(∇u(t), ∇v), += −(ρu,t(t), v). +Formally by using v = θu,t(t) and (29), it entails +��θu,t(t) +��2 +L2(Ω) + µ +2 +d +dt +��∇θu(t) +��2 +L2(Ω) = −(ρu,t(t), θu,t(t)), +such that (with Young’s inequality) +��θu,t(t) +��2 +L2(Ω) + µ d +dt +��∇θu(t) +��2 +L2(Ω) ≤ +��ρu,t(t) +��2 +L2(Ω) . +Integrating over time, we obtain +� T +0 ∥θu,t∥2 +L2(Ω) ds + µ +��∇θu(t) +��2 +L2(Ω) ≤ µ +��∇θu(0) +��2 +L2(Ω) + +� T +0 +��ρu,t +��2 +L2(Ω) ds, +10 + +and since the second term is always positive and that we have chosen u0 +h = P1 +hu0, it yields +� T +0 ∥θu,t∥2 +L2(Ω) ≤ +� T +0 +��ρu,t +��2 +L2(Ω) . +(33) +Remark 2.5. Note that with another choice of discretized initial solution, we would have +��∇θu(0) +��2 +L2(Ω) ≤ +���∇u0 +h − ∇u0��� +2 +L2(Ω) + Ch2���u0��� +2 +H2(Ω) , +which would have lead to an estimate in O(h) on the L2(Ω) error estimate of Ψ(t). In practice, this is not an +issue since the effect of the initial data exponentially decreases [45]. +Therefore, from (32), we obtain +T2 ≤ 2 +√ +T +µ +� � T +0 +��ρu,t +��2 +L2(Ω) ds +�1/2 +. +(34) +By definition of P1 +h (7), we have +��ρu,t(t) +�� +L2(Ω) = +���P1 +hut(t) − ut(t) +��� +L2(Ω) ≤ Ch2��ut(t) +�� +H2(Ω) , +(35) +and thus, (34) yields +T2 ≤ C2 +√ +T +µ +h2� � T +0 ∥ut∥2 +H2(Ω) ds +�1/2 +. +(36) +– Finally, for T3, we only need to use (35) again, but with Ψ instead of u. Therefore +T3 = +� T +0 +��ρt +�� +L2(Ω) ds ≤ Ch2 +� T +0 ∥Ψt∥H2(Ω) ds . +(37) +Combining (27), (30), (36), and (37) concludes the proof. +We can derive a similar result for the H1 +0 norm. +Theorem 2.6. Let Ω be a convex polyhedron. Let A(µ) = µ Id, with µ ∈ R+∗ . Consider u ∈ H1(0, T; H2(Ω)) be the +solution of (3) with u0 ∈ H2(Ω) and uh be the semi-discretized variational form (8). Let Ψ and Ψh be the corresponding +sensitivities , respectively given by (23) and (24). +∀t ∈ [0, T], +��Ψ(t) − Ψh(t) +�� +H1(Ω) ≤ Ch +����Ψ0��� +H2(Ω) + +� T +0 ∥Ψt∥H2(Ω) ds +� ++ C(µ)h2 +�� � T +0 ∥ut∥2 +H2 ds +�1/2 ++ +� � T +0 ∥Ψt∥2 +H2(Ω) ds +�1/2� +. +Proof. Using the same notation as before (25), we first decompose the error with the two components θ and ρ +such that +∀t ∈ [0, T], ∇Ψh(t) − ∇Ψ(t) = ∇θ(t) + ∇ρ(t). +(38) +• For the estimate on ρ(t), we use (26) to obtain +��∇ρ(t) +�� +L2(Ω) ≤ Ch +��Ψ(t) +�� +H2(Ω) , ∀t ∈ [0, T], +which leads to +��∇ρ(t) +�� +L2(Ω) ≤ Ch +����Ψ0��� +H2(Ω) + +� T +0 ∥Ψt∥H2(Ω) +ds +� +, ∀t ∈ [0, T]. +(39) +• For the estimate on θ(t), let us consider v ∈ Vh. As in the previous proof, ∀t ∈ [0, T], we write +(θt(t), v) + µ(∇θ(t), ∇v) = (Ψh,t(t), v) + µ(∇Ψh(t), ∇v) − (P1 +hΨt(t), v) − µ(∇P1 +hΨ(t), ∇v). +11 + +Instead of replacing v by θ(t) as in the L2 estimate, here we formally replace v by θt(t), thus +∀t ∈]0, T], +��θt(t) +��2 +L2(Ω) + µ(∇θ(t), ∇θt(t)) = (∇u(t) − ∇uh(t), ∇θt(t)) − (ρt(t), θt(t)). +Thanks to the variational formulations on the state solution u ((5) and (8) respectively) +��θt(t) +��2 +L2(Ω) + µ(∇θ(t), ∇θt(t)) = ( 1 +µ(uh,t(t) − ut(t)), θt(t)) − (ρt(t), θt(t)), +and thus (with Young’s inequality), +��θt(t) +��2 +L2(Ω) + µ(∇θ(t), ∇θt(t)) ≤ 1 +2 +����� +1 +µ(uh,t(t) − ut(t)) +����� +2 +L2(Ω) ++ 1 +2 +��θt(t) +��2 +L2(Ω) + 1 +2 +��ρt(t) +��2 +L2(Ω) + 1 +2 +��θt(t) +��2 +L2(Ω) , +≤ +1 +2µ2 +��uh,t(t) − ut(t) +��2 +L2(Ω) + 1 +2 +��ρt(t) +��2 +L2(Ω) + +��θt(t) +��2 +L2(Ω) . +Thus, +µ(∇θ(t), ∇θt(t)) ≤ +1 +2µ2 +��uh,t(t) − ut(t) +��2 +L2(Ω) + 1 +2 +��ρt(t) +��2 +L2(Ω) , +(40) +and by (29), we have +d +dt +��∇θ(t) +��2 +L2(Ω) ≤ 1 +µ3 +��uh,t(t) − ut(t) +��2 +L2(Ω) + 1 +µ +��ρt(t) +��2 +L2(Ω) . +Integrating over time, it entails +��∇θ(t) +��2 +L2(Ω) ≤ +��∇θ(0) +��2 +L2(Ω) +� +�� +� +T′ +1 ++ 1 +µ3 +� T +0 +��uh,t − ut +��2 +L2(Ω) ds +� +�� +� +T′ +2 ++ 1 +µ +� T +0 +��ρt +��2 +L2(Ω) ds +� +�� +� +T′ +3 +. +(41) +– From the initial conditions, T′ +1 = 0. +– We can also write T′ +2 as +T′ +2 = 1 +µ3 +� T +0 +��θu,t + ρu,t +��2 +L2(Ω) ds . +Therefore using (33), +T′ +2 ≤ 2 +µ3 +� T +0 ∥θu,t∥2 +L2(Ω) + +��ρu,t +��2 +L2(Ω) ds ≤ 4 +µ3 +� T +0 +��ρu,t +��2 +L2(Ω) ds ≤ Ch4 +µ3 +� T +0 ∥ut∥2 +H2(Ω) ds . +(42) +– Similarly, +T′ +3 ≤ Ch4 +µ +� T +0 ∥Ψt∥2 +H2(Ω) ds . +(43) +Hence, combining (38) with (39), (41), (42) and (43) concludes the proof. +2.3.3 +The fully-discretized versions. +From (24), we can derive the fully-discretized systems for the fine and coarse grids. +The direct sensitivity problems with respect to the parameter µp on the fine mesh Th with an Euler scheme read +� +� +� +� +� +� +� +� +� +Find Ψn +p,h ∈ Vh for n ∈ {0, . . . , NT} such that +(∂Ψn +p,h, vh) + a(Ψn +p,h, vh; µ) = −( ∂A +∂µp (µ)∇un +h(µ), ∇vh), for vh ∈ Vh, for n = {1, . . . , NT}, +(44) +Ψ0 +p,h(·) = P1 +hΨ0 +p(·), +where, as before, the time derivative in the variational form of the problem (23) has been replaced by a backward +difference quotient, ∂Ψn +h = +Ψn +h−Ψn−1 +h +∆tF +. +With the fully-discretized version (44), the following estimate holds. +12 + +Theorem 2.7. Let Ω be a convex polyhedron. Let A(µ) = µ Id, with µ ∈ R+∗ . +Consider u ∈ H1(0, T; H2(Ω)) ∩ H2(0, T; L2(Ω)) be the solution of (3) with u0 ∈ H2(Ω) and un +h be the fully-discretized +variational form (10). Let Ψ and Ψn +h be the corresponding sensitivities , respectively given by (23) and (44). Then +∀n = 0, . . . , NT, +��Ψn +h − Ψ(t) +�� +L2(Ω) ≤ Ch2���Ψ0��� +H2(Ω) + h2� +C +� tn +0 ∥Ψt∥H2(Ω) ds + C(µ) +� � tn +0 ∥ut∥2 +H2(Ω) ds +�1/2� ++ ∆tF +� +C +� tn +0 ∥Ψtt∥L2(Ω) ds + C(µ) +� � tn +0 ∥utt∥2 +L2(Ω) ds +�1/2� +. +Proof. Now, we define θn and ρn on the discretized time grid (tn)n=0,...,NT. +∀n = 0, . . . , NT, en := Ψn +h − Ψ(tn) = (Ψn +h − P1 +hΨ(tn)) + (P1 +hΨ(tn) − Ψ(tn)), += θn + ρn. +(45) +• In analogy with (26) the estimate on ρn is +��ρn�� +L2(Ω) ≤ Ch2��Ψ(tn) +�� +H2(Ω) ≤ Ch2����Ψ0��� +H2(Ω) + +� tn +0 ∥Ψt∥H2(Ω) ds +� +, ∀n ∈ {0, . . . , NT}. +(46) +• For θn, the equation (28) becomes +(∂θn, θn) + µ +��∇θn��2 +L2(Ω) = 1 +µ(∂un +h − ut(tn), θn) − (wn +1 + wn +2, θn), += 1 +µ(∂un +h − ut(tn), θn) − (wn, θn), +(47) +where wn +1 and wn +2 are defined by +wn +1 := (P1 +h − I)∂Ψ(tn), +wn +2 := ∂Ψ(tn) − Ψt(tn), +and +wn := wn +1 + wn +2. +(48) +By definition of ∂ and by Cauchy-Schwarz inequality (and since the second term of the left-hand side of +(47) is always positive), +��θn��2 +L2(Ω) ≤ +����θn−1��� +L2(Ω) + ∆tF +� 1 +µ +���∂un +h − ut(tn) +��� +L2(Ω) + +��wn�� +L2(Ω) +����θn�� +L2(Ω) , +and by repeated application, and since +���θ0��� +L2(Ω) = 0 (again, the case where some θn are equal to 0 can be +easily handled), it entails +��θn�� +L2(Ω) ≤ ∆tF +n +∑ +j=1 +1 +µ +���∂uj +h − ut(tj) +��� +L2(Ω) +� +�� +� +T2,n ++ ∆tF +n +∑ +j=1 +���wj��� +L2(Ω) +� +�� +� +T3,n +, +(49) +– We first decompose T2,n in two contributions +∆tF +µ +n +∑ +j=1 +���∂uj +h − ut(tj) +��� +L2(Ω) ≤ ∆tF +µ +n +∑ +j=1 +����∂θj +u +��� +L2(Ω) + +���wj +u +��� +L2(Ω) +� +, +where +wj +u := wj +1,u + wj +2,u with wj +1,u := (P1 +h − I)∂u(tj), +and +wj +2,u := ∂u(tj) − ut(tj). +(50) +Then by using Cauchy-Schwarz inequality (as in the semi-discretized case (32)), +∆tF +µ +n +∑ +j=1 +���∂uj +h − ut(tj) +��� +L2(Ω) ≤ +√ +tn +µ +�� n +∑ +j=1 +∆tF +���∂θj +u +��� +2 +L2(Ω) +� +�� +� +Tθ +�1/2 ++ +� n +∑ +j=1 +∆tF +���wj +u +��� +2 +L2(Ω) +� +�� +� +Tw +�1/2� +. +(51) +13 + +* Let us begin by the estimate on Tθ. On the state solution u, by choosing v = ∂θn +u, from (10) (the +operator ∂ and the spatial derivative can commute), we have +���∂θn +u +��� +2 +L2(Ω) + µ(∇θn +u, ∂∇θn +u) = −(wn +u, ∂θn +u), +(52) +where θn +u is the discrete version of (31). By definition of ∂ (and with Young’s inequality), +���∂θn +u +��� +2 +L2(Ω) + µ +∆tF +��∇θn +u +��2 +L2(Ω) ≤ +µ +2∆tF +���∇θn +u +��2 +L2(Ω) + +���∇θn−1 +u +��� +2 +L2(Ω) +� ++ 1 +2 +���wn +u +��2 +L2(Ω) + +���∂θn��� +2 +L2(Ω) +� +, +which entails +���∂θn +u +��� +2 +L2(Ω) ≤ +µ +∆tF +���∇θn−1 +u +��� +2 +L2(Ω) − +µ +∆tF +��∇θn +u +��2 +L2(Ω) + +��wn +u +��2 +L2(Ω) , ∀n = 1, . . . , NT. +(53) +Summing over the time steps, we get +n +∑ +j=1 +���∂θj +u +��� +2 +L2(Ω) ≤ +��� +n +∑ +j=1 +µ +∆tF +����∇θj−1 +u +��� +2 +L2(Ω) − +���∇θj +u +��� +2 +L2(Ω) +� ++ +���wj +u +��� +2 +L2(Ω) +���, +and we obtain +n +∑ +j=1 +���∂θn +u +��� +2 +L2(Ω) ≤ +��� µ +∆tF +����∇θ0 +u +��� +2 +L2(Ω) − +��∇θn +u +��2 +L2(Ω) +���� + +n +∑ +j=1 +���wj +u +��� +2 +L2(Ω) . +From the initial condition, θ0 +u = 0, +n +∑ +j=1 +���∂θn +u +��� +2 +L2(Ω) ≤ +µ +∆tF +��∇θn +u +��2 +L2(Ω) + +n +∑ +j=1 +���wj +u +��� +2 +L2(Ω) . +(54) +From (53) and by repeated application, we find for the first right-hand side term that +��∇θn +u +��2 +L2(Ω) ≤ ∆tF +µ +n +∑ +j=1 +���wj +u +��� +2 +L2(Ω) , +which gives for (54), multiplying by ∆tF to recover Tθ, +n +∑ +j=1 +∆tF +���∂θj +u +��� +2 +L2(Ω) ≤ 2 +n +∑ +j=1 +∆tF +���wj +u +��� +2 +L2(Ω) . +(55) +Now, going back to (51), we obtain +∆tF +µ +n +∑ +j=1 +���∂uj +h − ut(tj) +��� +L2(Ω) ≤ C +µ +� n +∑ +j=1 +∆tF +���wj +u +��� +2 +L2(Ω) +� +�� +� +Tw +�1/2 +≤ C +µ +� n +∑ +j=1 +∆tF +����wj +1,u +��� +2 +L2(Ω) + +���wj +2,u +��� +2 +L2(Ω) +��1/2 +. +(56) +* It remains to estimate Tw. +· Let us first consider the construction for w1,u +wj +1,u = (P1 +h − I)∂u(tj) = +1 +∆tF +(P1 +h − I) +� tj +tj−1 ut ds = +1 +∆tF +� tj +tj−1(P1 +h − I)ut ds , +14 + +since P1 +h and the time integral commute. Thus, from Cauchy-Schwarz inequality, +∆tF +n +∑ +j=1 +���wj +1,u +��� +2 +L2(Ω) ≤ ∆tF +n +∑ +j=1 +� +Ω +� 1 +∆t2 +F +� tj +tj−1((P1 +h − I)ut)2 ds ∆tF +� +≤ +n +∑ +j=1 +� tj +tj−1 +���(P1 +h − I)ut +��� +2 +L2(Ω) +ds , +≤ Ch4 +n +∑ +j=1 +� tj +tj−1∥ut∥2 +H2(Ω) , by definition of P1 +h, +≤ Ch4 +� tn +0 ∥ut∥2 +H2(Ω) +ds. +(57) +· To estimate the L2 norm of w2,u, we write +wj +2,u = +1 +∆tF +(u(tj) − u(tj−1)) − ut(tj) = − 1 +∆tF +� tj +tj−1(s − tj−1)utt(s) ds, +such that we end up with +∆tF +n +∑ +j=1 +���wj +2,u +��� +2 +L2(Ω) ≤ +n +∑ +j=1 +����� +� tj +tj−1(s − tj−1)utt(s) ds +����� +2 +L2(Ω) +≤ ∆t2 +F +� tn +0 ∥utt∥2 +L2(Ω) ds, +(58) +– We still have to find a bound for T3,n, defined in (49). +* For the estimates on wj +1, +wj +1 = +1 +∆tF +� tj +tj−1(P1 +h − I)Ψt ds , +and thus, +∆tF +n +∑ +j=1 +���wj +1 +��� +L2(Ω) ≤ Ch2 +� tn +0 ∥Ψt∥H2(Ω) ds . +* For wj +2, we have +∆tFwj +2 = Ψ(tj) − Ψ(tj−1) − ∆tFΨt(tj) = − +� tj +tj−1(s − tj−1)Ψtt(s) ds , +and therefore +∆tF +n +∑ +j=1 +���wj +2 +��� +L2(Ω) ≤ +n +∑ +j=1 +����� +� tj +tj−1(s − tj−1)Ψtt(s) ds +����� +L2(Ω) +≤ ∆tF +� tn +0 ∥Ψtt∥L2(Ω) ds . +Altogether, +T3,n ≤ Ch2 +� tn +0 ∥Ψt∥H2(Ω) ds + ∆tF +� tn +0 ∥Ψtt∥L2(Ω) ds , +(59) +and the proof ends by using (46), (49), (56), (57), (58), and (59). +With the fully-discretized version (44), the following estimate holds with H1 norm. +Theorem 2.8. Let Ω be a convex polyhedron. Let A(µ) = µ Id, with µ ∈ R+∗ . +Consider u ∈ H1(0, T; H2(Ω)) ∩ H2(0, T; L2(Ω)) be the solution of (3) with u0 ∈ H2(Ω) and un +h be the fully-discretized +variational form (10). Let Ψ and Ψn +h be the corresponding sensitivities , respectively given by (23) and (44). Then +∀n = 0, . . . , NT, +��∇Ψn +h − ∇Ψ(t) +�� +L2(Ω) ≤ h +� +C +���Ψ0��� +H2(Ω) + C(µ) +� tn +0 ∥Ψt∥H2(Ω) ds + C(µ) +� � tn +0 ∥ut∥2 +H2(Ω) ds +�1/2� ++ C(µ)∆tF +� � tn +0 ∥Ψtt∥L2(Ω) ds + +� � tn +0 ∥utt∥2 +L2(Ω) ds +�1/2�� +. +15 + +Proof. The proof combines the ideas of the two previous ones, since we seek the estimate in the H1 norm (as in +the semi-discretized problem) but with the fully-discretized version. +• In analogy with (26), the estimate on ρn is now given by +��∇ρn�� +L2(Ω) ≤ Ch +��Ψ(tn) +�� +H2(Ω) ≤ Ch +����Ψ0��� +H2(Ω) + +� tn +0 ∥Ψt∥H2(Ω) ds +� +, ∀n = 0, . . . , NT. +(60) +• For θn, instead of choosing v = θn as in (47), we take v = ∂θn +���∂θn��� +2 +L2(Ω) + µ(∇θn, ∇∂θn) = 1 +µ(∂un +h − ut(tn), ∂θn) − (wn, ∂θn), +(61) +and we obtain (as before with the semi-discretized version (40)) +µ(∇θn, ∇∂θn) ≤ +1 +2µ2 +���∂un +h − ut(tn) +��� +2 +L2(Ω) + 1 +2 +��wn��2 +L2(Ω) . +By definition of ∂ +µ +��∇θn��2 +L2(Ω) ≤ (√µ∇θn, √µ∇θn−1) + ∆tF +2µ2 +���∂un +h − ut(tn) +��� +2 +L2(Ω) + ∆tF +2 +��wn��2 +L2(Ω) , +which entails (by Young’s inequality) +µ +��∇θn��2 +L2(Ω) ≤ µ +���∇θn−1��� +2 +L2(Ω) + ∆tF +µ2 +���∂un +h − ut(tn) +��� +2 +L2(Ω) + ∆tF +��wn��2 +L2(Ω) , +and, by recursion (as in (49)) +µ +��∇θn��2 +L2(Ω) ≤ ∆tF +µ2 +n +∑ +j=1 +���∂uj +h − ut(tj) +��� +2 +L2(Ω) +� +�� +� +T′ +2,n ++ ∆tF +n +∑ +j=1 +���wj��� +2 +L2(Ω) +� +�� +� +T′ +3,n +. +(62) +– To estimate T′ +2,n, we write +T′ +2,n ≤ 2 +µ2 +� n +∑ +j=1 +∆tF +���∂θj +u +��� +2 +L2(Ω) +� +�� +� +Tθ ++ +n +∑ +j=1 +∆tF +���wj +u +��� +2 +L2(Ω) +� +�� +� +Tw +� +, +and thanks to the previous estimate on Tθ (55), we find that +T′ +2,n ≤ 6 +µ2 +� n +∑ +j=1 +∆tF +���wj +u +��� +2 +L2(Ω) +� +�� +� +Tw +� +, +which, by (57) and (58), yields +T′ +2,n ≤ C +µ2 +� +h4 +� tn +0 ∥ut∥2 +H2(Ω) ds + ∆t2 +F +� tn +0 ∥utt∥2 +L2(Ω) ds +� +. +– To find a bound for T′ +3,n, we simply use (57) and (58) again but with the sensitivity function Ψ instead +of u. +Combining the estimates on T′ +2,n and T′ +3,n with (62), and (60) concludes the proof. +16 + +With ∂Ψm +H = Ψm +H−Ψm−1 +H +∆tG +, on the coarse mesh TH with the Crank-Nicolson scheme, the fully-discretized system +(11) yields +� +� +� +� +� +� +� +� +� +� +� +Find Ψm +p,H ∈ VH for m ∈ {0, . . . , MT} such that +(∂Ψm +p,H, vH) + a( +Ψm +p,H+Ψm−1 +p,H +2 +, vH; µ) = −( ∂A +∂µp (µ) ∇um +H(µ)+∇um−1 +H +(µ) +2 +, ∇vH), for vH ∈ VH, for m = {1, . . . , MT}, (63) +Ψ0 +p,H(·) = P1 +HΨ0 +p(·). +We have the following result in the L2 norm with the Crank-Nicolson scheme on the coarse mesh TH. +Theorem 2.9. Let Ω be a convex polyhedron. Let A(µ) = µ Id, with µ ∈ R+∗ . +Consider u ∈ H2(0, T; H2(Ω)) ∩ H3(0, T; L2(Ω)) be the solution of (3) with u0 ∈ H2(Ω) and um +H be the fully-discretized +variational form (11) (on the coarse mesh TH). Let Ψ and Ψm +H be the corresponding sensitivities , respectively given by (23) +and (44). Then +∀m = 0, . . . , MT, +���Ψm +h − Ψ(�tm) +��� +L2(Ω) ≤ CH2����Ψ0��� +H2(Ω) + +� �tm +0 +∥Ψt∥H2(Ω) ds + C(µ) +� � �tm +0 +∥ut∥2 +H2(Ω) ds +�1/2� ++ C∆t2 +G +� � �tm +0 +∥Ψttt∥L2(Ω) ds + +� � �tm +0 +∥∆utt∥2 +L2(Ω) ds +�1/2 ++ C(µ) +�� � �tm +0 +∥uttt∥2 +L2(Ω) ds]1/2 + +� �tm +0 +∥∆Ψtt∥L2(Ω) ds +�� +. +Proof. +• For ρm we have the same estimate as before (46) (but with the coarse size H). +• We introduce the following notation +� +um +H = 1 +2(um +H + um−1 +H +). +(64) +Thanks to the Crank-Nicolson formulation on Ψm +H (63) and um +H (11) on the coarse mesh TH (and on the weak +formulation on u (5) and by definition of P1 +H (7)), +(∂θm, v) + µ(∇� +θm, ∇v) = (∂Ψm +H, v) − (∂P1 +H(Ψ(�tm)), v) + µ(∇� +Ψm +H, ∇v) − µ +2 +� +(∇P1 +HΨ(�tm), ∇v) + (∇P1 +HΨ(�tm−1), ∇v) +� +, += −(∇ � +um +H, ∇v) − (∂P1 +H(Ψ(�tm)), v) − µ +2 +� +(∇Ψ(�tm), ∇v) + (∇Ψ(�tm−1), ∇v) +� +, += −(∇ � +um +H, ∇v) −(∂P1 +H(Ψ(�tm)), v) + (∂Ψ(�tm), v) +� +�� +� +−wm +I +−(∂Ψ(�tm), v) + (Ψt(�tm− 1 +2 ), v) +� +�� +� +−wm +II +− (Ψt(�tm− 1 +2 ), v) − µ +2 +� +(∇Ψ(�tm), ∇v) + (∇Ψ(�tm−1), ∇v) +� += −(∇ � +um +H, ∇v) − wm +I − wm +II + (∇u(�tm− 1 +2 ), ∇v) + µ(∇Ψ(�tm− 1 +2 ), ∇v) +− µ +2 +� +(∇Ψ(�tm), ∇v) + (∇Ψ(�tm−1), ∇v) +� += (∇u(�tm− 1 +2 ) − � +∇um +H, ∇v) − (wm +I + wm +II + µwm +III, v), +where wm +I , wm +II and wm +III are defined by +wm +I := (P1 +H − I)∂Ψ(�tm), wm +II := ∂Ψ(�tm) − Ψt(�tm− 1 +2 ), and wm +III := ∆ψ(�tm− 1 +2 ) − 1 +2(∆Ψ(�tm) + ∆Ψ(�tm−1)). (65) +Thus, (47) with a Crank-Nicolson scheme and with v = � +θm becomes +(∂θm, � +θm) + µ(∇� +θm, ∇� +θm) = 1 +µ(∂um +H − ut(�tm− 1 +2 ), � +θm) − (wm +I + wm +II + µwm +III, � +θm), += 1 +µ(∂um +H − ut(�tm− 1 +2 ), � +θm) − (wm +T , � +θm), +(66) +where wm +T = wm +I + wm +II + µwm +III. By definition of ∂ (with the coarse time grid), and since the second term in +(66) is always positive, we get +(θm, � +θm) − (θm−1, � +θm) ≤ ∆tG +� 1 +µ +���∂um +H − ut(�tm− 1 +2 ) +��� +L2(Ω) + +��wm +T +�� +L2(Ω) +����� +θm +��� +L2(Ω) , +17 + +and by definition of � +θm (64), +��θm��2 +L2(Ω) − +���θm−1��� +2 +L2(Ω) ≤ ∆tG +� 1 +µ +���∂um +h − ut(�tm− 1 +2 ) +��� +L2(Ω) + +��wm +T +�� +L2(Ω) +����θm + θm−1��� +L2(Ω) , +so that, after cancellation of a common factor, +��θm�� +L2(Ω) − +���θm−1��� +L2(Ω) ≤ ∆tG +� 1 +µ +���∂um +H − ut(�tm− 1 +2 ) +��� +L2(Ω) + +��wm +T +�� +L2(Ω) +� +, +and by recursive application, it entails +��θm�� +L2(Ω) ≤ ∆tG +µ +m +∑ +j=1 +���∂uj +H − ut(�tj− 1 +2 ) +��� +L2(Ω) +� +�� +� +T′′ +2,n ++ ∆tG +m +∑ +j=1 +���wj +T +��� +L2(Ω) +� +�� +� +T′′ +3,n +. +(67) +– To estimate T′′ +2,n, we use the same tricks as before (51). First, we can decompose T′′ +2,n in 2 contributions, +such that +∆tG +µ +m +∑ +j=1 +���∂uj +H − ut(�tj− 1 +2 ) +��� +L2(Ω) ≤ ∆tG +µ +m +∑ +j=1 +���∂uj +H − ∂Ph +1 u(�tj) +��� +L2(Ω) +� +�� +� +���∂θj +u +��� +L2(Ω) ++ +���∂Ph +1 u(�tj) − ut(�tj− 1 +2 ) +��� +L2(Ω) +� +�� +� +���wj +I,u+wj +II,u +��� +L2(Ω) +, +where we denote by wm +I,u, wm +II,u the same terms respectively defined by wm +I , wm +II (65) but with u instead +of Ψ +wm +I,u := (P1 +h − I)∂u(�tm), +wm +II,u := ∂u(�tm) − ut(�tm− 1 +2 ). +(68) +We now apply Cauchy-Schwarz inequality +∆tG +µ +m +∑ +j=1 +���∂uj +H − ut(�tj− 1 +2 ) +��� +L2(Ω) ≤ +√�tm +µ +�� m +∑ +j=1 +∆tG +���∂θj��� +2 +L2(Ω) +�1/2 ++ +� m +∑ +j=1 +∆tG +���wj +I,u + wj +II,u +��� +2 +L2(Ω) +�1/2� +, +(69) +≤ +√�tm +µ +�� m +∑ +j=1 +∆tG +���∂θj��� +2 +L2(Ω) +�1/2 ++ +� m +∑ +j=1 +∆tG +���wj +I,u + wj +II,u +��� +2 +L2(Ω) + +���wj +III,u +��� +2 +L2(Ω) +�1/2� +* To estimate the first term of (69), we use v = ∂θm +u , and we now have (from the Crank-Nicolson +scheme on u (11)) +���∂θm +u +��� +2 +L2(Ω) + µ(∇� +θm +u , ∇∂θm +u ) = −(wm +I,u + wm +II,u + µwm +III,u, ∂θm +u ), +where +wm +III,u := ∆u(�tm− 1 +2 ) − 1 +2(∆u(�tm) + ∆u(�tm−1)), +(70) +and θm +u is the discrete version of (31). By definitions of ∂ and � +θm +u , it can be rewritten +���∂θm +u +��� +2 +L2(Ω) + +µ +2∆tG +��∇θm +u +��2 +L2(Ω) − +µ +2∆tF +���∇θm−1 +u +��� +2 +L2(Ω) = −(wm +I,u + wm +II,u + µwm +III,u, ∂θm +u ), +and it leads to (using Young’s inequality) +���∂θm +u +��� +2 +L2(Ω) + +µ +∆tG +��∇θm +u +��2 +L2(Ω) − +µ +∆tG +���∇θm−1 +u +��� +2 +L2(Ω) ≤ +���wm +I,u + wm +II,u + µwm +III,u +��� +2 +L2(Ω) . +(71) +Now we find, as in (54) (by summing over all time steps in order to obtain a telescoping sum) +m +∑ +j=1 +���∂θj +u +��� +2 +L2(Ω) ≤ +µ +∆tG +���∇θj +u +��� +2 +L2(Ω) + +m +∑ +j=1 +���wj +I,u + wj +II,u + wj +III,u +��� +2 +L2(Ω) , +(72) +18 + +The term∥∇θm +u ∥2 +L2(Ω) can easily be bounded by repeated application using (71). We find that (since +the first term of (71) is positive) +µ +∆tG +��∇θm +u +��2 +L2(Ω) ≤ +m +∑ +j=1 +���wj +I,u + wj +II,u + µwj +III,u +��� +2 +L2(Ω) , +and thus (72) gives +m +∑ +j=1 +���∂θj +u +��� +2 +L2(Ω) ≤ 2 +m +∑ +j=1 +���wj +I,u + wj +II,u + µwj +III,u +��� +2 +L2(Ω) ≤ 4 +m +∑ +j=1 +���wj +I,u + wj +II,u +��� +2 +L2(Ω) + +���µwj +III,u +��� +2 +L2(Ω) , +(73) +therefore we obtain for (69) +∆tG +µ +m +∑ +j=1 +���∂uj +H − ut(�tj− 1 +2 ) +��� +L2(Ω) ≤ 3 +� +�tm +��∆tG +µ2 +m +∑ +j=1 +���wj +I,u + wj +II,u +��� +2 +L2(Ω) + +m +∑ +j=1 +∆tG +���wj +III,u +��� +2 +L2(Ω) +�1/2� +, +which yields +∆tG +µ +m +∑ +j=1 +���∂uj +H − ut(�tj− 1 +2 ) +��� +L2(Ω) ≤ 3 +� +�tm +�� 2 +µ2 +m +∑ +j=1 +∆tG +���wj +I,u +��� +2 +L2(Ω) + 2 +µ2 +m +∑ +j=1 +∆tG +���wj +II,u +��� +2 +L2(Ω) ++ +m +∑ +j=1 +∆tG +���wj +III,u +��� +2 +L2(Ω) +�1/2� +. +(74) +* Now, we can estimate the right-hand side terms of (74) as done in [45]. We first remark that +wj +I,u = wj +1,u (and the estimate is given by (57) but with the coarse spatial and time grids), so it +remains to seek bounds for wj +II,u and wj +III,u. +· For wj +II,u, +∆tG +m +∑ +j=1 +���wj +II,u +��� +2 +L2(Ω) = +1 +∆tG +m +∑ +j=1 +���u(�tj) − u(�tj−1) − ∆tG ut(�tj− 1 +2 ) +��� +2 +L2(Ω) , += +1 +4∆tG +m +∑ +j=1 +������ +� �tj− 1 +2 +�tj−1 (s − �tj−1)2uttt(s) + +� �tj +�tj− 1 +2 (s − �tj)2uttt(s) ds +������ +2 +L2(Ω) +≤ C∆t3 +G +m +∑ +j=1 +����� +� �tj +�tj−1 uttt(s) ds +����� +2 +L2(Ω) +, +≤ C∆t4 +G +m +∑ +j=1 +� �tj +�tj−1∥uttt∥2 +L2(Ω) ds, with Cauchy-Schwarz inequality, +≤ C∆t4 +G +� �tm +�t0 ∥uttt∥2 +L2(Ω) ds. +(75) +· For wj +III,u, +∆tG +m +∑ +j=1 +���wj +III,u +��� +2 +L2(Ω) = ∆tG +m +∑ +j=1 +����∆u(�tj− 1 +2 ) − 1 +2(∆u(�tj) + ∆u(�tj−1)) +���� +2 +L2(Ω) +, += ∆tG +4 +m +∑ +j=1 +������ +� �tj− 1 +2 +�tj−1 (tj−1 − s)∆utt(s) ds + +� �tj +�tj− 1 +2 (s − �tj)∆utt(s) ds +������ +2 +L2(Ω) +, +≤ C∆t3 +G +m +∑ +j=1 +����� +� �tj +�tj−1 ∆utt ds +����� +2 +L2(Ω) +, +≤ C∆t4 +G +� �tm +�t0 ∥∆utt∥2 +L2(Ω) ds, by Cauchy-Schwarz inequality. +(76) +19 + +Altogether, +T′′ +2,n ≤ C +� H2 +µ +� � �tm +0 +∥ut∥2 +H2(Ω) ds +�1/2 ++ ∆t2 +G +�� 1 +µ +� �tm +0 +∥uttt∥2 +L2(Ω) +�1/2 ++ +� � �tm +0 +∥∆utt∥2 +L2(Ω) +�1/2 +ds +�� +. +(77) +– To estimate T′′ +3,n defined in (67), we remark that wj +I = wj +1 (but with the coarse spatial and time grids), +and +* for wj +II, +∆tG +���wj +II +��� +L2(Ω) ≤ +���Ψ(�tj) − Ψ(�tj−1) − ∆tGΨt(�tj− 1 +2 ) +��� +L2(Ω) , += 1 +2 +������ +� �tj− 1 +2 +�tj−1 (s − �tj−1)2Ψttt(s) + +� �tj +�tj− 1 +2 (s − �tj)2Ψttt(s) ds +������ +L2(Ω) +≤ C∆t2 +G +� �tj +�tj−1∥Ψttt∥L2(Ω) ds. +(78) +* Finally, for wj +III, +∆tG +���wj +III +��� +L2(Ω) = ∆tG +����Ψ(�tj− 1 +2 ) − 1 +2(Ψ(�tj) + Ψ(�tj−1)) +���� +L2(Ω) +≤ C∆t2 +G +� �tj +�tj−1∥∆Ψtt∥L2(Ω) ds. +(79) +Altogether, +T′′ +3,n ≤ CH2 +� �tm +0 +∥Ψt∥H2(Ω) ds + C∆t2 +G +� �tm +0 +� +∥Ψttt∥L2(Ω) + µ∥∆Ψtt∥L2(Ω) +� +ds, +(80) +which concludes the proof (combining (67), (77) and (80)). +In analogy with the previous work on parabolic equations, we define +� +ΨH +n = I2 +n[Ψm +H](µ), +for n = 1, . . . , NT, +(81) +with I2 +n defined by (12) as the quadratic interpolation in time of the coarse solution at time tn ∈ Im = [�tm−1,�tm] +defined on [�tm−2,�tm] from the values Ψm−2 +H +, Ψm−1 +H +, and Ψm +H, for all m = 2, . . . , MT. For tn ∈ I1 = [�t0,�t1], we use +the same parabola defined by the values Ψ0 +H, Ψ1 +H, ψ2 +H as the one used over [�t1,�t2]. Note that, as before, we could +have chosen another quadratic interpolation. +Corollary 2.10 (of Theorem 2.9). Under the assumptions of Theorem 2.9, let um +H be the fully-discretized solution (11) on +the coarse mesh TH. Let Ψ and Ψm +H be the corresponding sensitivities, respectively given by (23) and by (63). Let � +ΨH +n be +the quadratic interpolation of the coarse solution Ψm +H given by (81). Then, +∀n = 0, . . . , NT, +���� +Ψh +n − Ψ(tn) +��� +L2(Ω) ≤ CH2����Ψ0��� +H2(Ω) + +� tn +0 ∥Ψt∥H2(Ω) ds + C(µ) +� � tn +0 ∥ut∥2 +H2(Ω) ds +�1/2� ++ C∆t2 +G +� � tn +0 ∥Ψttt∥L2(Ω) ds + +� � tn +0 ∥∆utt∥2 +L2(Ω) ds +�1/2 ++ C(µ) +�� � tn +0 ∥uttt∥2 +L2(Ω) ds ]1/2 + +� tn +0 ∥∆Ψtt∥L2(Ω) ds +�� +. +In the next section, we proceed with the adjoint state formulation. +2.4 +Sensitivity analysis: The adjoint problem. +The adjoint method may be seen as an inverse method, where the goal is to retrieve the optimal parameter of +an objective function F. The objective function will have a different meaning whether the goal is to retrieve the +parameters from several measures (for parameter identification) or if we want to optimize a function depending +20 + +on the variables (PDE-constrained optimization). In the first case, F will have the following form (in its fully- +discretized form) +F(µ) = 1 +2 +NT +∑ +n=1 +��un +h(µ) − un��2 +L2(Ω) +� +�� +� +∥err(tn; µ)∥ +2 +L2(Ω) +, +(82) +where un refer to the measures, which may be noisy (here for simplicity we consider the case of measures on the +variables although it may be given by other outputs). In the second setting, it will be written +F(µ) = +NT +∑ +n=1 +gnun +h(µ), +(83) +with gn some suitable weights. Note that by differentiating F with respect to the parameters µp, p = 1, . . . , P, we +can observe the influence on the objective function of the input parameters through the normalized sensitivity +coefficients (also called elasticity of P) [4] +Sk = ∂F +∂µk +(µ) × +µk +F(µ), k = 1, . . . , P. +(84) +2.4.1 +The continuous setting. +• Let us for instance consider the first case outlined above, given in the continuous version by +F(µ) = 1 +2 +� T +0 +��err(t; µ) +��2 +L2(Ω) . +(85) +• To minimize F under the constraint that u is the solution of our model problem (3), we consider the +following Lagrangian with (χ, ϕ) the Lagrangian multipliers +L(u, χ, ϕ; µ) = F(µ) + +� T +0 (χ, (∇ · (A(µ)∇u) + f − ut)) ds + +� T +0 (ϕ, u)L2(∂Ω) ds, +(86) +where +– χ ∈ V is the multiplier associated to the constraint “u is a solution of (3)”, +– ϕ ∈ R is the multiplier associated to the constraint of the Dirichlet boundary condition on ∂Ω. Since +here we consider homogeneous condition, we just impose ϕ = 0. +• Differentiating L with respect to the parameter µp, for p = 1, . . . , P, we obtain the following adjoint system +in its variational form (see A for more details) +� +� +� +� +� +Find χ(t) ∈ V for t ∈ [0, T] such that +(χt(t), v) = −( ∂err +∂u (t; µ), v) + (A(µ)∇χ(t), ∇v), ∀v ∈ V, t < T, +χ(·, T) = 0, in Ω. +(87) +• After solving (89) with the parameter µ, one can compute dF +dµp by noticing that +dF +dµp += dL +dµp += +� T +0 +� +χ, ∇ · ( ∂A +∂µp +(µ)∇u) +� +ds, from (124). +(88) +Remark 2.11. For a stable implementation, one may have to add a regularization term depending of the parameter to +the cost function F(p). +21 + +2.4.2 +Discretized setting. +In analogy with the direct method, we first discretize the system in space, and then we apply an Euler scheme +with the fine grids and a Crank-Nicolson scheme with the coarse ones. The semi-discretized version on Th writes +� +� +� +� +� +� +� +Find χh(t) ∈ Vh for t ∈ [0, T] such that +(χh,t(t), vh) − a(χh, vh; µ) = −( ∂errh +∂uh (t; µ), vh), ∀vh ∈ Vh, t < T, +χh(·, T) = 0, in Ω. +(89) +With the fully-discretized version, on the fine grids, the adjoint system becomes in its variational formulation +� +� +� +� +� +� +� +Find χn +h ∈ Vh for n ∈ {0, . . . , NT} such that +(∂χn +h, vh) − a(χh, vh; µ) = −(un +h − un, vh), ∀n = 0, . . . , NT − 1, +χNT +h (·) = 0. +(90) +Note that to compute ∂errn +h +∂uh , we need the fine solutions un +h and the measures. As for the state variable (11), we also +compute the adjoint on the coarse mesh with the Crank-Nicolson scheme, +� +� +� +� +� +� +� +� +� +Find χm +H ∈ VH for m ∈ {0, . . . , MT} such that +(∂χm +H, vH) − a( 1 +2(χm +H + χm−1 +H +), vH; µ) = − 1 +2 +� +(um +H − um, vH) + (um−1 +H +− um−1, vH) +� +), ∀vH ∈ VH, ∀m = 0, . . . , MT − 1, (91) +χMT +H (·) = 0. +Finally, note that the problems (90) and (92) are well-posed, since they are solved backward in time (see [16] for +precisions in the general setting of time-dependent PDEs). +The next section adapts the NIRB two-grid algorithm in the context of sensitivity analysis. +3 +NIRB algorithms applied to sensitivity analysis +3.1 +On the direct problem. +3.1.1 +NIRB algorithm. +Let u(µ) be the exact solution of problem (3) for a parameter µ ∈ G and Ψp(µ) its sensitivity with respect to +the parameter µp, p = 1, . . . , P. We consider P parameters of interest. In this context, we use the following +offline/online decomposition for the NIRB procedure: +• “Offline part” +1. For a set of training parameters (�µi)i=1,··· ,Np,train, we define Gp,train = +∪ +i∈{1,...,Np,train}�µi. Then, through +a greedy algorithm 1, we adequately choose the parameters of the RB. During this procedure, we +compute fine fully-discretized solutions {Ψn +p,h(�µi)}i∈{1,...Nµ,p}, n={0,...,NT} (Nµ,p ≤ Np,train) with the HF +solver, by solving either (44) or the following problem (where un +h in (44) has been replaced by its NIRB +approximation uN,n +Hh or by its rectified version Rn +u[uN,n +Hh ] obtained from the algorithm of section 2.2) +� +� +� +� +� +� +� +� +� +Find Ψn +p,h ∈ Vh for n ∈ {0, . . . , NT} such that +(∂Ψn +p,h, vh) + a(Ψn +p,h, vh; �µ) = −( ∂A +∂µp (µ)∇uN,n +Hh (µ), ∇vh) for n = {1, . . . , NT}, +(92) +Ψ0 +p,h(·) = P1 +hΨ0 +p(·). +The term −( ∂A +∂µp (µ)∇uN,n +Hh (µ), ∇vh) in (92) is replaced by −( ∂A +∂µp (µ)∇Rn +u[uN,n +Hh ](µ), ∇vh) in case of the +rectification post-treatment. Note that un +h can directly be used (as in (44)) since this step belongs to the +offline part of the NIRB algorithm. However, if the number of parameters required for the initial RB is +22 + +lower than the number of parameters needed for the sensitivities RB or if one combine the sensitivities +with an optimization algorithm, it may be convenient to employ (92) instead of (44). +In analogy to section 2.2, a few time steps may be selected for each parameter of the RB, and thus, +we obtain Np L2 orthogonal RB (time-independent) functions, denoted (ζh +p,i)i=1,...,Np, and the reduced +spaces X +Np +p,h := Span{ζh +p,1, . . . , ζh +p,Np} for p = 1, . . . , P. +2. Then, for each p, we solve the eigenvalue problem (14) on X +Np +p,h: +� +� +� +� +� +Find ζh ∈ X +Np +p,h, and λ ∈ R such that: +∀v ∈ X +Np +p,h, +� +Ω ∇ζh · ∇v dx = λ +� +Ω ζh · v dx. +(93) +For each parameter p ∈ {1, . . . , P}, we get an increasing sequence of eigenvalues λp +i , and eigenfunc- +tions (ζh +p,i)i=1,··· ,Np, orthonormalized in L2(Ω) and orthogonalized in H1(Ω). +3. As in the offline step 3 from section 2.2, we enhance the NIRB approximation with a rectification post- +processing. Thus, we introduce the rectification matrices, denoted Rp,n +Ψ . They are associated to the +sensitivities problem (44), defined for each p ∈ {1, . . . , P} and each fine time step n ∈ {1, . . . , NT}, and +constructed from coarse snapshots, generated by solving (63) and whose parameters are the same as +for the fine snapshots. +Thus, for all n = 1, . . . , NT and all p = 1, . . . , P, we compute the vectors +Rp,n +Ψ,i = ((Ap,n)TAp,n + δpINp)−1(Ap,n)TBp,n +i +, +i = 1, · · · , Np, +(94) +where +∀i = 1, · · · , Np, +and +∀�µk ∈ Gp, +Ap,n +k,i = +� +Ω +� +Ψp,H +n(�µk) · ζh +p,i dx, +(95) +Bp,n +k,i = +� +Ω Ψn +p,h(�µk) · ζh +p,i dx, +(96) +and where INp refers to the identity matrix and δp is a regularization term (note that we used (81) for +� +Ψp,H +n(�µk)). +Remark 3.1. In general, Np,train < Np and the parameter δp is required for the inversion of (Ap,n)TAp,n. +• “Online part” +The online part of the algorithm is much faster than a double HF evaluation (to seek the sensitivity Ψn +p,h, +we also need the solution un +h with a HF evaluation). +4. Indeed, we first solve the problem (3) on the coarse mesh TH for a new parameter µ ∈ G at each time +step m = 0, . . . , MT using (11). +5. Then, for each p = 1, . . . , P, we solve the coarse associated sensitivity problems (63) with the same +parameter µ, at each time step m = 0, . . . , MT. +6. We quadratically interpolate in time the coarse solution Ψm +p,H on the fine time grid with (81). +7. Then, we linearly interpolate � +Ψp,H +n(µ) on the fine mesh in order to compute the L2-inner product with +the basis functions. The approximation used in the two-grid method is +For n = 0, . . . , NT, +Ψ +Np,n +p,Hh(µ) := +Np +∑ +i=1 +(� +Ψp,H +n(µ), ζh +p,i) ζh +p,i, +(97) +and with the rectification post-treatment step, it becomes +Rp,n +Ψ [ΨN +p,Hh](µ) := +Np +∑ +i,j=1 +Rp,n +ij +(� +Ψp,H +n(µ), ζh +p,j) ζh +p,i, +(98) +where Rp,n +Ψ +is the rectification matrix at time tn, given by (94). +23 + +In the next section, we propose an adaptation of this algorithm with a new post-treatment which reduces the +online computational time. +3.1.2 +New NIRB algorithm for the direct problem. +The main drawback of the algorithm described in the previous section is that it requires 1 + P coarse systems in +the online part (see the steps 4 and 5 in section 3.1.1). The online portion of the new algorithm described below +only requires the resolution of two coarse problems, regardless the number of parameters of interest. We refer to +the following offline/online decomposition: +• “Offline part” +1. For a parameter training set Gtrain, we compute the RB functions of the initial problem, denoted +(Φh +i )i=1,...,N and generates XN +h by the steps 1-2 of 2.2 (see algorithm 1). +2. As before, from the training sets Gp,train, we generate the reduced spaces X +Np +p,h, for p = 1, . . . , P using +steps 1 and 2 of section 3.1.1, and at the end of this part, we obtain Np RB functions (time-independent), +denoted (ζh +p,i)i=1,...,Np for each p = 1, . . . , P. We introduce GT defined by +GT := Gtrain ∩ Gp,train, +(99) +and Nµ,T the number of parameters in GT. +3. We use the fact that the sensitivities are directly derived from the initial solutions, and we consider +new rectification matrices, denoted �Rp,n and defined for each p ∈ {1, . . . , P} and each fine time step +n ∈ {1, . . . , NT}. In this new post-treatment, they are constructed from coarse snapshots of the initial +solution, generated by solving (11) and whose parameters are the same as for the fine sensitivities, +generated by solving (63). +Thus, for all n = 1, . . . , NT and all p = 1, . . . , P, we compute the vectors +�Rp,n +i += ((An)TAn + δIN)−1(An)TBp,n +i +, +i = 1, · · · , Np, +(100) +where this time +∀�µk ∈ GT, +An +k,i = +� +Ω +� +uH +n(�µk) · Φh +i dx, ∀i = 1, · · · , N, +(101) +Bp,n +k,i = +� +Ω Ψn +p,h(�µk) · ζh +p,i dx, ∀i = 1, · · · , Np, +(102) +and where IN refers to the identity matrix and δ is a regularization term (required for the inversion of +(An)TAn). Note that � +uH +n(�µk) is the quadratic interpolation given by (12). We highlight the fact that +this step requires that GT ̸= ∅ (99). +• “Online step” +4. We solve the problem (3) on the coarse mesh TH for a new parameter µ ∈ G at each time step m = +0, . . . , MT using (11). +5. We quadratically interpolate in time the coarse solution um +H on the fine time grid with (12). +6. Then, we linearly interpolate � +uH +n(µ) on the fine mesh in order to compute the L2-inner product with +the basis functions. The new NIRB approximation is given by +�Rp,n[ΨN +p,Hh](µ) := +Np +∑ +i=1 +N +∑ +j=1 +�Rp,n +ij +( � +uH +n(µ), Φh +j ) ζh +p,i, +(103) +where �Rp,n is the rectification matrix at time tn, given by (100). +24 + +3.2 +On the adjoint formulation. +The adjoint formulation requires some modifications of the NIRB algorithm compared to section 3.1.1. Since in +(88), for all n ∈ {0, . . . , NT}, the fine solution un +h(µ) is required to obtain the sensitivities on F, it follows that +here we have to compute two reductions: one for the initial solution u and one for the adjoint χ. As a matter of +fact, in (3.1.1), the RB generation for u was optional. +So let u(µ) be the exact solution of problem (3) for a parameter µ ∈ G and χ(µ) its adjoint given by (89). In this +setting, we use the following offline/online decomposition for the NIRB procedure: +• “Offline part” +1. During the offline stage, we first construct the reduced space XN +h and the RB function (Φh +1, . . . , Φh +N) +with the steps 1-2 of section 2.2. +2. Then, we use steps 1-2 of section 3.1.1, but instead of solving (44) on the sensitivities, we generate the +reduced space XN1 +1 +by solving the adjoint problem on the fine mesh (90). +Thus, for a set of training parameters (�µi)i=1,··· ,N1,train, we define G1,train = +∪ +i∈{1,...,N1,train}�µi. +Then, +through a greedy procedure 1, we adequately choose the parameters of the RB. During this proce- +dure, we compute fine fully-discretized solutions {χn +h(�µi)}i∈{1,...Nµ,1}, n={0,...,NT} (Nµ,1 ≤ N1,train) with +the HF solver, by solving either (90) or the following problem (where un +h in (90) has been replaced by +its NIRB approximation uN,n +Hh or by its rectified version Rn +u[uN,n +Hh ] obtained from the algorithm of section +2.2) +� +� +� +� +� +� +� +Find χn +h ∈ Vh for n ∈ {0, . . . , NT} such that +(∂χn +h, vh) − a(χh, vh; µ) = −(uN,n +Hh − un, vh), ∀n = 0, . . . , NT − 1, +χNT +h (·) = 0. +(104) +The term −(uN,n +Hh (µ) − un, vh) in (104) is replaced by −(Rn +u[uN,n +Hh ](µ) − un, vh) in case of the rectification +post-treatment. In practice, since in step 1 a RB for un +h has already been generated, it is more convenient +to employ (104) instead of (90). +In analogy to section 2.2, a few time steps may be selected for each parameter of the RB, and thus, +we obtain N1 L2 orthogonal RB (time-independent) functions, denoted (ξh +i )i=1,...,N1, and the reduced +space XN1 +h +:= Span{ξh +1, . . . , ξh +N1}. +3. Then, we solve the eigenvalue problem (14) on XN1 +h : +� +� +� +� +� +Find ξh ∈ XN1 +h , and λ ∈ R such that: +∀v ∈ XN1 +h , +� +Ω ∇ξh · ∇v dx = λ +� +Ω ξh · v dx. +(105) +We get an increasing sequence of eigenvalues λi, and eigenfunctions (ξh +i )i=1,··· ,N1, orthonormalized in +L2(Ω) and orthogonalized in H1(Ω). +4. As in the offline step 3 from section 3.1.1, we enhance the NIRB approximation with a rectifica- +tion post-processing. Thus, we introduce a rectification matrix, denoted Rn +χ for each fine time step +n ∈ {1, . . . , NT}. It is associated to the adjoint problem (90) and constructed from coarse snapshots, +generated by solving (92) and whose parameters are the same as for the fine snapshots. +Thus, for all n = 1, . . . , NT, we compute the vectors +Rn +χ,i = ((An)TAn + δIN1)−1(An)TBn +i , +i = 1, · · · , N1, +(106) +where +∀i = 1, · · · , N1, +and +∀�µk ∈ Gp, +An +k,i = +� +Ω +� +χH +n(�µk) · ξh +i dx, +(107) +Bn +k,i = +� +Ω χn +h(�µk) · ξh +i dx, +(108) +25 + +and where IN1 refers to the identity matrix and δp is a regularization term required for the inversion +of (An)TAn (note that we used (81) for � +χH +n(�µk)). +• “Online part” +4. We first solve the problem (3) on the coarse mesh TH for a new parameter µ ∈ G at each time step +m = 0, . . . , MT using (11). +5. Then, we solve the coarse associated adjoint problem (92) with the same parameter µ, at each time +step m = 0, . . . , MT. +6. We quadratically interpolate in time the coarse solution χm +H on the fine time grid with (81). +7. Then, we linearly interpolate � +χH +n(µ) on the fine mesh in order to compute the L2-inner product with +the basis functions. The approximation used for the adjoint in the two-grid method is +For n = 0, . . . , NT, +χN1,n +Hh (µ) := +N1 +∑ +i=1 +( � +χH +n(µ), ξh +i ) ξh +i , +(109) +and with the rectification post-treatment step, it becomes +Rn +χ[χN1 +Hh](µ) := +N1 +∑ +i,j=1 +Rn +χ,ij ( � +χH +n(µ), ξh +j ) ξh +i , +(110) +where Rn +χ is the rectification matrix at time tn, given by (106). +8. Then, we use the steps 5 and 6 of section 2.2 in order to obtain a NIRB approximation for u(µ) from +the coarse solution um +H given by step 4 of this online part. +9. Finally, the sensitivities NIRB approximations of F are given by +for p = 1, . . . , P, [ ∂F +∂µp +]N1 +Hh(µ) := +tn +∑ +j=1 +∆tF +� +χN1,j +Hh , ∇ · ( ∂A +∂µp +(µ)∇uN,j +Hh) +� +, from (88), +(111) +and with the rectification post-treatment step, it becomes +for p = 1, . . . , P, Rχ[[ ∂F +∂µp +]N1 +Hh](µ) := +tn +∑ +j=1 +∆tF +� +Rj +χ[χN1,j +Hh ](µ), ∇ · ( ∂A +∂µp +(µ)∇Rj +u[uN,j +Hh](µ)) +� +. +(112) +The next section gives our main result on the NIRB two-grid method error estimate in the context of sensitivity +analysis. +4 +NIRB error estimate on the sensitivities +Main result +Our main result is the following theorem. +Theorem 4.1. (NIRB error estimate for the sensitivities.) Let A(µ) = µ Id, with µ ∈ R+∗ , and let us consider the problem +3 with its exact solution u(x, t; µ), and the full discretized solution un +h(x; µ) to the problem 10. Let Ψ(x, t; µ) and Ψn +h(x; µ) +respectively by the corresponding sensitivities , given by (23) and (44). Let (ζh +i )i=1,...,N1 be the L2-orthonormalized and +H1-orthogonalized RB generated with the greedy algorithm 1 through the NIRB algorithm 3.1.1. Let us consider the NIRB +approximation, +For n = 0, . . . , NT, +ΨN,n +Hh (µ) := +N1 +∑ +i=1 +(� +ΨH +n(µ), ζh +i ) ζh +i , +(113) +where � +ΨH +n(µ) is given by (81). Then, the following estimate holds +∀n = 0, . . . , NT, +���Ψ(tn)(µ) − ΨN,n +Hh (µ) +��� +H1(Ω) ≤ ε(N) + C1(µ)h + C2(µ, N)H2 + C3(µ)∆tF + C4(µ, N)∆t2 +G, +(114) +where C1, C2, C3 and C4 are constants independent of h and H, ∆tF and ∆tG. The term ε depends on the Kolmogorov +N-width and measures the error given by (21). +26 + +If H is such as H2 ∼ h, ∆t2 +G ∼ ∆tF, and ε(N) is small enough, with C2(µ, N) and C4(µ, N) not too large, it +results in an error estimate in O(h + ∆tF). Theorem 4.1 then states that we recover optimal error estimates in +L∞(0, T; H1(Ω)). We now go on with the proof of Theorem 4.1. +Proof. The NIRB approximation at time step n = 0, . . . , NT, for a new parameter µ ∈ G is defined by (97). Thus, +the triangle inequality gives +���Ψ(tn)(µ) − ΨN,n +Hh (µ) +��� +H1(Ω) ≤ +��Ψ(tn)(µ) − Ψn +h(µ) +�� +H1(Ω) + +���Ψn +h(µ) − ΨN,n +hh (µ) +��� +H1(Ω) + +���ΨN,n +hh (µ) − ΨN,n +Hh (µ) +��� +H1(Ω) +=: T1 + T2 + T3, +(115) +where ΨN1,n +hh +(µ) = +N1 +∑ +i=1 +(Ψn +h(µ), ζh +i ) ζh +i . +• The first term T1 may be estimated using the inequality given by Theorem 2.8, such that +��Ψ(tn)(µ) − Ψn +h(µ) +�� +H1(Ω) ≤ C(µ) (h + ∆tF). +(116) +• We then denote by S′ +h = {Ψn +h(µ, t), µ ∈ G, n = 0, . . . NT} the set of all the sensitivities . For our model +problem, this manifold has a low complexity. +It means that for an accuracy ε = ε(N) related to the +Kolmogorov N-width of the manifold S′ +h, for any µ ∈ G, and any n ∈ 0, . . . , NT, T2 is bounded by ε which +depends on the Kolmogorov N-width. +T2 = +������ +Ψn +h(µ) − +N1 +∑ +i=1 +(Ψn +h(µ), ζh +i ) ζh +i +������ +H1(Ω) +≤ ε(N). +(117) +• Since (ζh +i )i=1,...,N1 is a family of L2 and H1 orthogonalized RB functions (see [19] for only L2 orthonormalized +RB functions) +���ΨN,n +hh − ΨN,n +Hh +��� +2 +H1(Ω) = +N1 +∑ +i=1 +|(Ψn +h(µ) − � +ΨH +n(µ), ζh +i )|2���ζh +i +��� +2 +H1(Ω) , +(118) +where � +ΨH +n(µ) is the quadratic interpolation of the coarse snapshots on time tn, ∀n = 0, . . . , NT, defined by +(81). From the RB orthonormalization in L2, the equation (105) yields +���ζh +i +��� +2 +H1 := +���∇ζh +i +��� +2 +L2(Ω) = λi +���ζh +i +��� +2 +L2(Ω) = λi ≤ max +i=1,··· ,Nλi = λN, +(119) +such that the equation (118) leads to +���ΨN,n +hh − ΨN,n +Hh +��� +2 +H1(Ω) ≤ CλN +���Ψn +h(µ) − � +ΨH +n(µ) +��� +2 +L2(Ω) . +(120) +Now by definition of � +ΨH +n(µ) and by corollary 2.10 and Theorem 2.7, for tn ∈ Im, +���Ψn +h(µ) − � +ΨH +n(µ) +��� +L2(Ω) ≤ C(µ)(H2 + ∆t2 +G + h2 + ∆tF), +(121) +and we end up for equation (120) with +���ΨN,n +hh − ΨN,n +Hh +��� +H1(Ω) ≤ C(µ) +� +λN(H2 + ∆t2 +G + h2 + ∆tF), +(122) +where C(µ) does not depend on N. Combining these estimates (116), (117) and (122) concludes the proof +and yields the estimate (114). +27 + +Figure 1: H1 +0 NIRB errors +5 +Numerical results. +In this section, we have applied the NIRB algorithms on several numerical tests. We have implemented both +schemes (Euler and RK2) using FreeFem++ (version 4.9) [26] to compute the fine and coarse snapshots, and the +solutions have been stored in VTK format. +Then we have applied the plain NIRB and the NIRB rectified algorithms with python, in order to highlight +the non-intrusive side of the two-grid method (as in [19]). After saving the NIRB approximations with Paraview +module on Python, the errors have been computed with FreeFem++. +5.1 +On the heat equation. +We have solved (3) and (23) on the parameter set G = [0.5, 9.5], with u0 solution of Poisson’s equation −∆u0 = f +and Φ0 = 0. We have retrieved several snapshots on t = [0, 2] (note that the coarse time grid must belong to the +interval of the fine one), and tried our algorithms on several size of meshes, always with ∆tF ≃ h and ∆tG ≃ H +(both schemes are stables), and such that h = H2. +• We have taken 18 parameters in G for the RB construction such that µi = 0.5i, i = 1, . . . , 19, i ̸= 2 and a +reference solution to problem (92), with µ = 1 and its mesh and time step such that hre f ≃ ∆tF,re f = 0.001. +In figure Figure 1, we present the errors of the FEM solutions and compare them to the one obtained with +the NIRB algorithm with the rectification to observe the convergence rate. +A +Derivation of the adjoint for the heat equation. +In this appendix, we recall the main steps to derive the adjoint of our model problem, in order to compute +( ∂F +∂µk )k=1,...,P. For a more general problem, we refer to [44] in case of FEM. +28 + +FEM H relative errors +NIRB H relative error with H = V h +0.80- +0.80- +h +h +FEM coarse error +NiRB+rectification +0.50 - +0.50- +FEM fine error +0.20 - +0.20 +Error (log +Error (log +0.05- +0.05 +10-2 +10-1 +10-2 +10-1 +h (size of the fine mesh) +h (size of the fine mesh)• We consider the Lagrangian formulation (86), denoted by L. +• Differentiating L with respect to the parameter µp, for p = 1, . . . , P, we obtain +dL +dµp +(u, χ, ζ; µ) = +� T +0 +� � +Ω +derr +dµp +(µ) dx + +� +χ, d[∇ · (A(µ)∇u) + f − ut] +dµp +�� +ds. +(123) +In our setting, the objective does not depend directly on the parameter µp. The time and the parameter +derivatives can commute ( d +dt +� +du +dµp +� += +d +dµp +� +du +dt +� +), and since f is independent of µ, the term linked to f +vanishes. Therefore, using the chain rule, it may be rewritten +dL +dµp +(u, χ, ζ; µ) = +� T +0 +��∂err +∂u , Ψp +� ++ +� +χ, ∇ · ( ∂A +∂µp +(µ)∇u) +� ++ +� +χ, ∂[∇ · (A(µ)∇u)] +∂u +Ψp +� +− +� +χ, Ψp,t +� +� +�� +� +TIBP +� +ds, +(124) +where +Ψp(t, x; µ) := ∂u +∂µp +(t, x; µ). +As we saw before, a classical forward sensitivity computation would require P + 1 systems of PDEs to +solve. Here, we want to avoid calculating the sensitivities of the state variables. To do so, the strategy of the +adjoint method is to factorize all the terms depending on Ψp, and to impose them to vanish by adequately +choosing χ (which is arbitrary). By IBP on TIBP, +� T +0 +� +Ω χ · Ψp,t dx ds = +� +Ω +� +χ(T) · Ψp(T) − χ(0) · Ψp(0) +� +dx − +� T +0 +� +Ω χt · Ψp dx ds , +and choosing χ(T) = 0, and since in our example, u0 does not depend on µ, it yields +dL +dµp +(u, χ, ζ; µ) = +� T +0 +��∂err +∂u , Ψp +� ++ +� +χ, ∇ · ( ∂A +∂µp +(µ)∇u) +� ++ +� +χ, ∂[∇ · (A(µ)∇u)] +∂u +Ψp +� ++ +� +χt, Ψp +�� +ds. +Thus, we want the following term to vanish +� T +0 +� +Ω +�∂err +∂u · Ψp + χ∂[∇ · (A(µ)∇u)] +∂u +· Ψp +� +�� +� +TGF ++χt · Ψp +� +dx ds. +(125) +Now, applying Green’s formula twice, we have +� T +0 +� +Ω TGF dx ds = +� T +0 +� +Ω χ∇ · (A(µ)∇Ψp) dxds = +� T +0 +� +− +� +Ω A(µ)∇χ · ∇Ψp dx + +� +∂Ω A(µ)χ · ∇nΨp dσ +� +ds , += +� T +0 +� � +Ω ∇ · (A(µ)∇χ) · Ψp dx − +� +∂Ω A(µ)∇nχ · Ψp dσ + +� +∂Ω A(µ)χ · ∇nΨp dσ +� +ds , +with ∇n(·) the normal derivative. Therefore, from the initial boundary conditions, since ∀t ≥ 0, ∀µ ∈ G, +u(t) = 0 on ∂Ω, we also have Ψp(t) = 0 on ∂Ω and by imposing χ = 0 on ∂Ω, (125) becomes +� T +0 +� +Ω +��∂err +∂u + ∇ · (A(µ)∇χ) + χt +� +· Ψp +� +dxds = 0, +and this equation leads us to the following adjoint state problem +� +� +� +� +� +� +� +� +� +� +� +Find χ ∈ V such that +χt = − ∂err +∂u − ∇ · (A(µ)∇χ), in Ω × [0, T[, +χ(x, T) = 0, in Ω, +χ(x, t) = 0, on ∂Ω × [0, T[. +(126) +29 + +Acknowledgment +This work is supported by the SPP2311 program. We would like to give special thanks to Ole Burghardt for his +precious help on Automatic Differentiation. +References +[1] E. Bader, M. K¨archer, M. A Grepl, and K. Veroy. Certified reduced basis methods for parametrized dis- +tributed elliptic optimal control problems with control constraints. SIAM Journal on Scientific Computing, +38(6):A3921–A3946, 2016. +[2] M. Barrault, C. Nguyen, A. Patera, and Y. Maday. 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Computer methods in applied mechanics and engineering, 129(4):393–409, 1996. +32 + diff --git a/8NAyT4oBgHgl3EQf2_mV/content/tmp_files/load_file.txt b/8NAyT4oBgHgl3EQf2_mV/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e2f0a741dfde3c53b4d8646da393e0385d0f000c --- /dev/null +++ b/8NAyT4oBgHgl3EQf2_mV/content/tmp_files/load_file.txt @@ -0,0 +1,1307 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf,len=1306 +page_content='The non-intrusive reduced basis two-grid method applied to sensitivity analysis January 3, 2023 Elise Grosjean 1, Bernd Simeon 1 Abstract This paper deals with the derivation of Non-Intrusive Reduced Basis (NIRB) techniques for sensitivity anal- ysis, more specifically the direct and adjoint state methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' For highly complex parametric problems, these two approaches may become too costly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' To reduce computational times, Proper Orthogonal Decomposition (POD) and Reduced Basis Methods (RBMs) have already been investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' The majority of these algorithms are however intrusive in the sense that the High-Fidelity (HF) code must be modified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' To address this issue, non-intrusive strategies are employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' The NIRB two-grid method uses the HF code solely as a “black-box”, requiring no code modification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Like other RBMs, it is based on an offline-online decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' The offline stage is time-consuming, but it is only executed once, whereas the online stage is significantly less expensive than an HF evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' In this paper, we propose new NIRB two-grid algorithms for both the direct and adjoint state methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' On a classical model problem, the heat equation, we prove that HF evaluations of sensitivities reach an optimal convergence rate in L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' H1(Ω)), and then establish that these rates are recovered by the proposed NIRB approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' These results are supported by numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' We then numerically demonstrate that a further deterministic post-treatment can be applied to the direct method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' This further reduces computational costs of the online step while only computing a coarse solution of the initial problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' All numerical results are run with the model problem as well as a more complex problem, namely the Brusselator system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 1 Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Sensitivity analysis is a critical step in optimizing the parameters of a parametric model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' The goal is to see how sensitive its results are to small changes of its input parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' It is especially useful in the biomedical field when experiments are extremely complex or prohibitively expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Indeed, conducting several experiments to determine the impact of all parameters involved in biological processes may be difficult, if not impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Several methods have been developed for computing sensitivities, see [4] for an overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' We focus here on two differential-based sensitivity analysis approaches in connection with models given as reaction-diffusion equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' “The direct method”, also known as the ”forward method”, which may be used when dealing with dis- cretized solutions of parametric Partial Differential Equations (PDEs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' The sensitivities (of the solution or other outputs of interest) are computed directly from the original problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' One drawback is that it necessi- tates solving a new system for each parameter of interest, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=', for P parameters of interest, P + 1 problems have to be solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' “The adjoint state method”, also known as the ”backward method”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' It may be a viable option [44] when the direct method becomes prohibitively expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' In this setting, the goal is to compute the sensitivities of an objective function that one aims at minimizing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' The associated Lagrangian is formulated, and by choosing appropriate multipliers, a new system known as ”the adjoint” is derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' This approach is preferred in many situations since it avoids calculating the sensitivities with respect to the solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' For example, in the framework of inverse problems, one can determine the ”true” parameter from several measures 1Felix-Klein-Institut f¨ur Mathematik, Kaiserslautern TU, 67657, Deutschland 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='00761v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='NA] 2 Jan 2023 (which are frequently provided by multiple sensors) while combining it with a gradient-type optimization algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' As a result, we get the ”integrated effects” on the outputs over a time interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' The advantage is that it only requires two systems to solve regardless of the number of parameters of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Thus, the direct method is appealing when there are relatively few parameters or a large number of objective functions, whereas the adjoint state method is preferred when there are many parameters and few objective functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Earlier works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' For extremely complex simulations, both methods may still be impractical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Several reduction techniques have thus been investigated in order to reduce the complexity of the sensitivity computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Among them, Reduced Basis Methods (RBMs) are a well-developed field [36, 40, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' They use an offline-online decompo- sition, in which the offline step is time-consuming but is only performed once, and the online step is significantly less expensive than a High-Fidelity (HF) evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' In the context of sensitivity analysis, the majority of these studies rely on a Galerkin projection onto the adjoint state system in the online part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' In what follows, we present a brief review of previous works on RBMs combined with both sensitivity methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Let us begin with the direct method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' It has been employed and studied with RB spaces in various applica- tions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=', [39, 47, 13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' The sensitivities may also be useful to enhance the reduced state approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Unlike the other studies cited below, the sensitivities in [25] are computed to improve RB methods (see also [24] with a Lagrangian formulation or [23] with a finite difference approach [23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Still to improve an approximation, in [31], a combined method is proposed (based on local and global approximations with series expansion and a RB expression), which was first developed in [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Note also that variance-based sensitivity analysis has been investigated using RBMs [28] and non-intrusive RB [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' The adjoint state formulation can be thought of as a PDE-constrained optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' The first applications of this method in conjunction with computational reduction approaches can be found in [27] in the context of RBMs, where several RB sub-spaces are compared or in [42] with the POD method, with an affine parameter dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Currently, particular emphasis is being placed on developing accurate a-posteriori error estimates in order to improve basis generation [41, 46, 11, 12] with Proper Orthogonal Decomposition (POD) and/or RBMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' RBMs and POD have also been investigated in the context of optimal control under uncertainty [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' In recent studies, the case of infinite-dimensional control function is considered with RB approximations on the state, adjoint, and control variables [29, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Even if the adjoint state method is frequently preferred, writing its associated reduced problem can be difficult when the adjoint formulation is not straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' It may also be reformulated to take advantage of previously developed RB theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' For example, in [38], it is rewritten as a saddle-point problem for Stokes-type problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' To conclude this brief overview of RBMs applied to sensitivity analysis, we add that non-intrusive methods have been developed, in the framework of the inverse problem, without computing the sensitivities (see the PBDW method [37, 22, 10] with a direct formulation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Motivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Even though the Galerkin projection is prevalent in the literature, its main disadvantage lies in its intrusiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Indeed, in order to approximate the solution of a PDE, the matrices computed from its variational formulation must be changed in the HF code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' This may be difficult if the HF is very complex or even impossible if it has been purchased, as is often the case in an industrial context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' From an engineering standpoint, Non- Intrusive Reduced Basis (NIRB) methods are more practical to implement than intrusive RBMs because they only require the execution of the HF code as a ”black-box” solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Apparently, NIRB methods have not yet been used to approximate sensitivities except for statistical approaches such as variance-based sensitivity analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' In this paper, we aim at computing the sensitivities with respect to some parameters of interest µ ∈ G, with the direct and adjoint methods combined with NIRB techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' We focus on the NIRB two-grid method [7, 17, 8, 43] (see also different NIRB methods [6, 2, 15] from the two-grid method).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Like most RBMs, the NIRB two-grid method relies on the assumption that the manifold of all solutions S = {u(µ), µ ∈ G} has a small Kolmogorov width [33] (in what follows, uh(µ) will refer to the HF solution for the parameter µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' The two-grid algorithm can be employed for a variety of PDEs and is simple to implement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' It has been studied with FEM in the context of elliptic equations [7] and parabolic equations [19] (see also [17] for finite volume schemes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Furthermore, because it is non-intrusive, it is suitable for a wide range of problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' The 2 effectiveness of this method relies on its offline/online decomposition (as most RBMs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' The offline part is time- consuming but it is only performed once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' On the contrary, the specific feature of the NIRB approach is to solve the parametric problem on a coarse mesh only during the online step, and then to rapidly improve the precision of the coarse solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' It makes this portion of the algorithm much cheaper than a HF evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' In this paper, we combine the two-grids framework with both sensitivity analysis methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Then, drawing in- spiration from recent works [18], we efficiently apply a deterministic process to further reduce the computational cost of its online stage with the direct method, in the context of parabolic equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' During the online stage, this additional step allows us to solve only the initial problem on the coarse mesh, regardless of the number of parameters of interest, making this novel approach very appealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' We highlight the fact that because the direct approach requires a new system to be solved for each parameter, the adjoint method is preferred in many studies (as cited above), despite the fact that its formulation is more complex and yields integrated sensitivities over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Outline of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' This article is about extending the NIRB two-grid method to the computation of sensitiv- ities and performing the associated numerical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' We present and illustrate the NIRB algorithms applied to both sensitivity analysis methods with several numerical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' With the direct method, we have carried out a thorough theoretical analysis of the heat equation as model problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' In this setting, we have optimal conver- gence rates in L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' H1(Ω)) for the spatial HF semi-discretized sensitivity solution and for its fully-discretized form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' It turns out that we obtain theoretically and numerically these optimal rates also for the NIRB sensitivity approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Our main theoretical result is given by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' The rest paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Section 2 describes both sensitivity methods along with established convergence results and the NIRB two-grid algorithm for parabolic equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' In Section 3, we present the al- gorithms for the direct and adjoint methods with the NIRB two-grid approach, as well as the new version of the algorithm for the direct method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Section 4 is devoted to the theoretical results on the rate of convergence for the NIRB sensitivity approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' In the last section 5, several numerical results are presented and illus- trate the theoretical ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' The implementation and the use of Automatic Differentiation (AD) is discussed as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 2 Mathematical Background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Let Ω be a bounded domain in Rd, with d ≤ 3 and a smooth enough boundary ∂Ω, and consider a parametric problem P on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' For the NIRB two-grid method, we consider two spatial ”grids” of Ω: one fine mesh, denoted Th, where its size h is defined as h = max K∈Mh hK, (1) and on coarse mesh, denoted TH, with its size defined as H = max K∈MH HK >> h, (2) where the diameter hK (or HK) of any element K in a mesh is equal to sup x,y∈K |x − y|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' In this section, we first introduce our model problem, that of the heat equation, in a continuous setting, and then its spatial (over the two meshes) and time discretizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Then, we recall the NIRB algorithm in the context of parabolic equations, and finally, we detail the sensitivity problems for this model problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' In the next sections, C will denote various positive constants independent of the size of the meshes h and H and of the parameter µ, and C(µ) will denote constants independent of the sizes of the meshes h and H but dependent of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='1 A model problem: The heat equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='1 The continuous problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' We consider the following heat equation on the domain Ω with homogeneous Dirichlet conditions, which takes the form 3 � � � � � ut − ∇ · (A(µ)∇u) = f, in Ω×]0, T], u(·, 0) = u0(·), in Ω, (3) u(·, t) = 0, on ∂Ω×]0, T], where f ∈ L2(Ω × [0, T]), while u0 ∈ H1 0(Ω) and µ = (µ1, · · · , µP) ∈ G ⊂ RP is the parameter, such that A : Ω × G → Md(R) is measurable, bounded, and uniformly elliptic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (4) For any t > 0, the solution u(·, t) ∈ H1 0(Ω), and ut(·, t) ∈ L2(Ω) stands for the derivative of u with respect to time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' We use the conventional notations for space-time dependent Sobolev spaces [35] Lp(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' V) := {u(x, t) | ∥u∥Lp(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='V) := � � T 0 ��u(·, t) ��p V dt �1/p < ∞}, 1 ≤ p < ∞, L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' V) := {u(x, t) | ∥u∥L∞(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='V) := ess sup 0≤t≤T ��u(·, t) �� V < ∞}, where V is a real Banach space with norm∥·∥V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' The variational form of (3) is given by: � � � � � � � Find u ∈ L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' H1 0(Ω)) with ut ∈ L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' H−1(Ω)) such that (ut(t, ·), v) + a(u(t, ·), v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' µ) = ( f (t, ·), v), ∀v ∈ H1 0(Ω) and t ∈ (0, T), (5) u(·, 0) = u0(·), in Ω, where a is given by a(w, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' µ) = � Ω A(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' µ)∇w(x) · ∇v(x) dx, ∀w, v ∈ H1 0(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (6) We remind that (5) is well posed (see [14] for the existence and the uniqueness of solutions to problem (5)) and we refer to the notations of [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Note that we will use the notation (·, ·) to denote the classical L2-inner product on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='2 The various discretizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' For the NIRB algorithm, we use the two spatial grids on the variational formulation (5) of our problem (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' We employed P1 finite elements to discretize in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Thus, we introduce Vh and VH, the continuous piecewise linear finite element functions (on fine and coarse meshes, respectively) that vanish on the boundary ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' We consider the so-called Ritz projection operator P1 h : H1 0(Ω) → Vh (P1 H on VH is defined similarly) which is given by (∇P1 hu, ∇v) = (∇u, ∇v), ∀v ∈ Vh, for u ∈ H1 0(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (7) In the context of time-dependent problems, a time stepping method of finite difference type is used to get a fully discrete approximation of the solution of (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' As for the spatial domain, we consider two different time grids: One time grid, denoted F, is associated to fine solutions (for the generation of the snapshots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' To avoid making notations more cumbersome, we will consider a uniform time step ∆tF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' The time levels can be written tn = n ∆tF, where n ∈ N∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Another time grid, denoted G, is used for coarse solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' By analogy with the fine grid, we consider a uniform grid with time step ∆tG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Now, the time levels are written �tm = m ∆tG, where m ∈ N∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' As in the elliptic context [7], the NIRB algorithm is designed to recover the optimal estimate in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Yet, since there is no such argument as the Aubin-Nitsche argument for time stepping methods, we must consider time discretizations that provide the same precision with larger time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Thus, we consider a higher order time scheme for the coarse solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' As in [19], we used an Euler scheme (first order approximation) for the fine solution and a Crank-Nicolson scheme (second order approximation) for the coarse solution on our model problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Thus, we deal with two kind of notations for the discretized solutions: 4 uh(x, t) and uH(x, t) that respectively denote the fine and coarse solutions of the spatially semi-discrete solution, at time t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' un h(x) and um H(x) that respectively denote the fine and coarse full-discretized solutions at time tn = n × ∆tF and �tm = m × ∆tG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' To simplify the notations, we consider that both time grids end at time T here, T = NT ∆tF = MT ∆tG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' The semi-discrete form of the variational problem (5) writes for the fine mesh (similarly for the coarse mesh): � � � � � � � Find uh(t) = uh(·, t) ∈ Vh for t ∈ [0, T] such that (uh,t(t), vh) + a(uh(t), vh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' µ) = ( f (t), vh), ∀vh ∈ Vh and t ∈]0, T], (8) uh(·, 0) = u0 h(·) = P1 h(u0)(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' From the definition of P1 h (7), the initial condition u0 h (and similarly for the coarse mesh) is such that (∇u0 h, ∇vh) = (∇u0, ∇vh), ∀vh ∈ Vh, (9) and hence, it corresponds to the finite element solution of the corresponding elliptic problem of (3) with A(1) = Id (that of the Poisson’s equation) and whose exact solution is u0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' The full discrete form of the variational problem (5) for the fine mesh with an implicit Euler scheme writes: � � � � � � � Find un h ∈ Vh for n = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , NT such that (∂un h, vh) + a(un h, vh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' µ) = ( f (tn), vh), ∀vh ∈ Vh and n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , NT, (10) uh(·, 0) = u0 h(·), where the time derivative in the variational form of the problem (8) has been replaced by a backward difference quotient, ∂un h = un h−un−1 h ∆tF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' For the coarse mesh with a Crank-Nicolson scheme, and with the notation ∂um H = um H−um−1 H ∆tG , it becomes: � � � � � � � Find um H ∈ VH for m = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , MT, such that (∂um H, vH) + a( um H+um−1 H 2 , vH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' µ) = ( f (�tm− 1 2 ), vH), ∀vH ∈ VH and m = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' MT, uH(·, 0) = u0 H(·), (11) where �tm− 1 2 = �tm+�tm−1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' For the NIRB approximation, we will need to interpolate in space and in time the coarse solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' So let us introduce the quadratic interpolation in time of a coarse solution at time tn ∈ Im = [�tm−1,�tm] defined on [�tm−2,�tm] from the coarse approximations at times �tm−2,�tm−1, and �tm, for all m = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , MT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' To this purpose, we employ the following parabola on [�tm−2,�tm]: For m ≥ 2, ∀n ∈ Im = [�tm−1,�tm], I2 n[um H](µ) := um−2 H (µ) (�tm − �tm−2)(�tm−2 − �tm−1) � − (tn)2 + (�tm−1 + �tm)tn − tm−1tm� + um−1 H (µ) (�tm−2 − �tm−1)(�tm−1 − �tm) � − (tn)2 + (�tm + �tm−2)tn − tmtm−2� + um H(µ) (�tm−1 − �tm)(�tm − �tm−2) � − (tn)2 + (�tm−2 + �tm−1)tn − tm−2tm−1� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (12) For tn ∈ I1 = [�t0,�t1], we use the same parabola defined by the coarse approximations at times �t0, �t1, �t2 as the one used over [�t1,�t2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' We denote by � uH n(µ) = I2 n[um H](µ) the quadratic interpolation of um H at a time n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Note that we choose this interpolation in order to keep an approximation of order 2 in time ∆tG (it works also with other quadratic interpolations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' In the next section, we recall the NIRB algorithm in the context of parabolic equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='2 Reminders on the Non-Intrusive Reduced Basis method (NIRB) in the context of parabolic equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Let u(µ) be the exact solution of problem (3) for a parameter µ ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' With the NIRB algorithm, we aim at quickly approximating this solution by using a reduced space, denoted XN h , constructed from N fully discretized solutions of (10), namely the so-called snapshots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Since each snapshot is a HF finite element approximation in space at a time tn, n = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=', NT (NT being potentially very high), not all of the time steps may be required for the construction of the reduced space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Here, for each parameter µi, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , Nµ, selected for the basis construction, the number of time steps employed (which depends on i) is denoted Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Thus, the reduced basis is defined as XN h := Span{u (nj)i h (µi)| i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , Nµ, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , Ni, (nj)i ⊂ {1, · · · , NT}}, (13) with N := Nµ ∑ i=1 Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' We recall the offline/online decomposition of the NIRB procedure with parabolic equations: “Offline step” The offline part of the algorithm allows us to construct the reduced space XN h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' From training parameters (µi)i∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=',Ntrain}, we define Gtrain = ∪ i∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=',Ntrain}µi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Then, we employ a greedy procedure to adequately choose the parameters (µi)i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=',Nµ within Gtrain to construct the RB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' For this procedure, we refer to algorithm 1 (described for the setting Nµ = N in order to simplify notations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Note that a POD-greedy algorithm may also be employed [19, 21, 20, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Algorithm 1 Greedy algorithm Input: tol, {un h(µ1), · · · , un h(µNtrain) with µi ∈ Gtrain, n = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , NT}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Output: Reduced basis {Φh 1, · · · , Φh N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Choose µ1, n1 = arg max µ∈Gtrain, n∈{0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=',NT} ���un h(µ) ��� L2(Ω) , Set Φh 1 = un1 h (µ1) ���un1 h (µ1) ��� L2 Set G1 = {µ1, n1} and X1 h = span{Φh 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' for k = 2 to N do: µk, nk = arg max (µ, n)∈(Gtrain×{0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=',NT})\\Gk−1 ���un h(µ) − Pk−1(un h(µ)) ��� L2, with Pk−1(un h(µ)) := k−1 ∑ i=1 (un h(µ), Φh i ) Φk i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Compute � Φh k = unk h (µk) − Pk−1(unk h (µk)) and set Φh k = � Φh k ���� � Φh k ���� L2(Ω) Set Gk = Gk−1 ∪ {µk} and Xk h = Xk−1 h ⊕ span{Φh k} Stop when ���un h(µ) − Pk−1(un h(µ)) ��� L2 ≤ tol, ∀µ ∈ Gtrain, ∀n = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , NT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' end for The greedy algorithm is usually less expensive than the POD-greedy (thanks to a-posteriori error estimates for stationary problems).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Although for time dependent problems, the latter is more rea- sonable when the snapshots are computed for all time steps, our choice of using a greedy procedure is motivated by the fact that it is more efficient with the post-treatment introduced below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' The RB functions (time-independent), denoted (Φh i )i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=',N, are generated at the end of this step, from fine fully-discretized solutions {un h(µi)}i∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=',Nµ}, n={0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=',NT} (solving problem (10) with HF solver).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Note that even if all the time steps are computed, only Ni are used for each i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , Nµ} in the RB construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Since at each step k, all sets added in the basis are in the orthogonal complement of Xk−1 h , it yields an L2 orthogonal basis without further processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Hence, XN h can be defined as XN h = Span{Φh 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , Φh N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 6 Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' In practice, the algorithm is halted with a stopping criterion such as an error threshold or a maximum number of basis functions to generate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Then, we solve the following eigenvalue problem: � � � � � Find Φh ∈ XN h , and λ ∈ R such that: ∀v ∈ XN h , � Ω ∇Φh · ∇v dx = λ � Ω Φh · v dx, (14) We get an increasing sequence of eigenvalues λi, and orthogonal eigenfunctions (Φh i )i=1,··· ,N, which do not depend on time, orthonormalized in L2(Ω) and orthogonalized in H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Note that with Gram-Schmidt procedure, we only obtain an L2-orthonormalized RB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' For any parameter µk, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , Nµ, the classical NIRB approximation differs from the HF uh(µk) computed in the offline stage [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Thus, as proposed in [7], to improve NIRB accuracy, we use a ”rectification post-processing”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' To this purpose, we need a rectification matrix for each fine time step, denoted Rn, and constructed from coarse snapshots, generated by solving (11) and whose parameters are the same as for the fine snapshots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Thus, for all n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , NT, we compute the vectors Rn u,i = ((An)TAn + δIN)−1(An)TBn i , i = 1, · · · , N, (15) where ∀i = 1, · · · , N, and ∀µk ∈ Gtrain, An k,i = � Ω � uH n(µk) · Φh i dx, (16) Bn k,i = � Ω un h(µk) · Φh i dx, (17) and where IN refers to the identity matrix and δ is a regularization parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Note that since every time step has its own rectification matrix, the matrix An is a “flat” rectan- gular matrix (Ntrain ≤ N), and thus the parameter δ is required for the inversion of (An)TAn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' We also remark that with the rectification post-treatment, the standard greedy algorithm 1 may leads to more accurate approximations, compared to the POD-greedy algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' It comes from the fact that the coefficients of the matrix are directly derived from the snapshots in that case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' “Online step” The online part of the algorithm is much faster than a HF evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' We solve the problem (3) on the coarse mesh TH for a new parameter µ ∈ G at each time step m = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , MT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' We quadratically interpolate in time the coarse solution on the fine time grid with (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Then, we linearly interpolate � uH n(µ) on the fine mesh in order to compute the L2-inner product with the RB functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' The approximation used in the two-grid method is For n = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , NT, uN,n Hh (µ) := N ∑ i=1 ( � uH n(µ), Φh i ) Φh i , (18) and with the rectification post-treatment step, it becomes Rn u[uN Hh](µ) := N ∑ i,j=1 Rn u,ij ( � uH n(µ), Φh j ) Φh i , (19) where Rn u is the rectification matrix at time tn, given by (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 7 In [19], we have proven the following estimate on the heat equation for n = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , NT, ���u(tn)(µ) − uN,n Hh (µ) ��� H1(Ω) ≤ ε(N) + C1(µ)h + C2(N)H2 + C3(µ)∆tF + C4(N)∆t2 G, (20) where C1, C2, C3 and C4 are constants independent of h and H, ∆tF and ∆tG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' The term ε(N) depends on a proper choice of the RB space as a surrogate for the best approximation space associated to the Kolmogorov N-width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' It decreases when N increases and it is linked to the error between the fine solution and its projection on the reduced space XN h , given by �����un h(µ) − N ∑ i=1 (un h(µ), Φh i ) Φh i ����� H1(Ω) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (21) The constant C2 increases with N and thus, a trade-off needs to be done between increasing N to obtain a more accurate manifold, and keeping a constant C2 as low as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' If H is such as H2 ∼ h, ∆t2 G ∼ ∆tF, and ε(N) is small enough, with C2(N) and C4(N) not too large, the estimate (20) entails an error estimate in O(h + ∆tF), and thus, we recover an optimal error estimate in L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' H1(Ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Before adapting NIRB to the sensitivity analysis context, we first recall how to derive the sensitivities functions in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='3 Sensitivity analysis: The direct problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' In this section, we recall the sensitivity systems (continuous and discretized versions) for P parameters of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Then, we prove the numerical results of the direct method on the model problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' To not make the notations too cumbersome, we will consider A(µ) = µ Id, with µ ∈ R+∗ for the analysis theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='1 The continuous setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' In this setting, we consider P parameters of interest, denoted µp = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , µP, and we want to approximate the exact derivatives Ψp(t, x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' µ) := ∂u ∂µp (t, x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (22) In order to seek these sensitivities, we solve P new systems, which can directly be obtained by differentiating the initial problem with respect to µp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' The continuous initial problem (5) may be rewritten � � � � � Find u(t) ∈ V for t ∈ [0, T] such that (ut(t), v) = F(u(t), v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' µ) := −a(u(t), v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' µ) + ( f (t), v), ∀v ∈ V, t > 0, u(·, 0) = u0(·), where the bilinear form a is defined by (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Using the chain rule and since the time and the parameter derivatives can commute, (Ψp,t(t), v) = ∂F ∂u (u(t), v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' µ) · Ψp(t) + ∂F ∂µ(u(t), v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Since the initial condition here does not depend on µ, we obtain the following problem � � � � � � � � � Find Ψp(t) ∈ V for t ∈ [0, T] such that (Ψp,t(t), v) + a(Ψp(t), v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' µ) = −( ∂A ∂µp (µ)∇u(t), ∇v), for v ∈ V, for t > 0, Ψ0 p = 0, (23) which is well-posed since u ∈ L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' H1 0(Ω)), and under the assumptions (4), the so-called ”parabolic regularity estimate” implies that u ∈ L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' H2(Ω)) ∩ L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' H1 0(Ω)) [14, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='2 The spatially semi-discretized version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' As previously for the state solution, we discretize in space and in time the sensitivity problems (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' The corresponding spatially semi-discretized formulations (on Th) read � � � � � � � � � Find Ψp,h(t) ∈ Vh for t ∈ [0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , T] such that (Ψp,h,t(t), vh) + a(Ψp,h(t), vh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' µ) = −( ∂A ∂µp (µ)∇uh(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' µ), ∇vh), for vh ∈ Vh, for t ∈]0, T], Ψ0 p,h(·) = P1 h(Ψ0 p)(·), (24) where P1 h is given by (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Before proceeding with the proof of Theorem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='1), we need several results that can be deduced from [45], but require some precisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Indeed, first, in [45], the estimates are proven on the heat equation with a non-varying diffusion coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Secondly, the right-hand side function f vanishes when seek- ing the error estimates, whereas in our case, the right-hand side function depends on u and necessitates precised estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' On the semi-discretized formulation, the following estimate holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Let Ω be a convex polyhedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Let A(µ) = µ Id, with µ ∈ R+∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Consider u ∈ H1(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' H2(Ω)) be the solution of (3) with u0 ∈ H2(Ω) and uh be the semi-discretized variational form (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Let Ψ and Ψh be the corresponding sensitivities , respectively given by (23) and (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Then ∀t ∈ [0, T], ��Ψh(t) − Ψ(t) �� L2(Ω) ≤ Ch2����Ψ0��� H2(Ω) + � T 0 ∥Ψt∥H2(Ω) ds � + C(µ)h2� � T 0 ∥ut∥2 H2(Ω) ds �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' As in [45], we first decompose the error with two components θ and ρ such that ∀t ∈ [0, T], e(t) := Ψh(t) − Ψ(t) = (Ψh(t) − P1 hΨ(t)) + (P1 hΨ(t) − Ψ(t)), = θ(t) + ρ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (25) For the estimate on ρ(t), a classical FEM estimate [45, 5] is ���P1 hv − v ��� L2(Ω) + h ���∇(P1 hv − v) ��� L2(Ω) ≤ Ch2∥v∥H2(Ω) , ∀v ∈ H2 ∩ H1 0, (26) which leads to ��ρ(t) �� L2(Ω) ≤ Ch2��Ψ(t) �� H2(Ω) , ∀t ∈ [0, T], ≤ Ch2����Ψ0��� H2(Ω) + � T 0 ∥Ψt∥H2(Ω) ds � , ∀t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (27) For the estimate on θ(t), let us consider v ∈ Vh, ∀t ∈]0, T], (θt(t), v) + µ(∇θ(t), ∇v) = (Ψh,t(t), v) + µ(∇Ψh(t), ∇v) − (P1 hΨt(t), v) − µ(∇P1 hΨ(t), ∇v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Since v ∈ H1 0, by definition of P1 h (7), the semi-discretized weak formulations (24) implies (θt(t), v) + µ(∇θ(t), ∇v) = −(∇uh(t), ∇v) − (P1 hΨt(t), v) − µ(∇P1 hΨ(t), ∇v), = −(∇uh(t), ∇v) − (P1 hΨt(t), v) − µ(∇Ψ(t), ∇v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Thanks to the continuous weak formulation (23), and since the operator P1 h and the time derivative com- mute, it can be rewritten (θt(t), v) + µ(∇θ(t), ∇v) = (∇u(t) − ∇uh(t), ∇v) + (Ψt(t) − (P1 hΨ)t(t), v), = (∇u(t) − ∇uh(t), ∇v) − (ρt(t), v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Choosing v = θ(t), it yields (θt(t), θ(t)) + µ ��∇θ(t) ��2 L2(Ω) = (∇u(t) − ∇uh(t), ∇θ(t)) − (ρt(t), θ(t)), 9 and using the continuous and semi-discretized weak formulations on the state variable u(t) ((5) and (8) respectively), we obtain (θt(t), θ(t)) + µ ��∇θ(t) ��2 L2(Ω) = 1 µ(uh,t(t) − ut(t), θ(t)) − (ρt(t), θ(t)), (28) where the first term of the right-hand side is a new contribution (compared to the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='2 [45]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Since (θt(t), θ(t)) = 1 2 d dt( ��θ(t) ��2 L2(Ω)) = ��θ(t) �� L2(Ω) d dt ��θ(t) �� L2(Ω) , (29) and, since the second term in (28) is positive, it becomes with Cauchy-Schwarz inequality (the case where θ(t) = 0 for some t may easily be handled) d dt ��θ(t) �� L2(Ω) ≤ 1 µ ��uh,t(t) − ut(t) �� L2(Ω) + ��ρt(t) �� L2(Ω) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Integrating over time, it follows that ��θ(t) �� L2(Ω) ≤ ��θ(0) �� L2(Ω) � �� � T1 + 1 µ � T 0 ��uh,t − ut �� L2(Ω) ds � �� � T2 + � T 0 ��ρt �� L2(Ω) ds � �� � T3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (30) – From the initial conditions, since u0 h = P1 hu0, T1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Note that other optimal order choices of discretized initial conditions (such as the L2 orthogonal projection onto Vh) lead to an estimate in Ch2���Ψ0��� H2(Ω) for T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' – To estimate T2, in analogy with θ and ρ, let us introduce θu and ρu, such that ∀t ∈ [0, T], uh(t) − u(t) = (uh(t) − P1 hu(t)) + (P1 hu(t) − u(t)), = θu(t) + ρu(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (31) We remark that the term T2 can also be written T2 = 1 µ � T 0 ��θu,t + ρu,t �� L2(Ω) ds ≤ 1 µ � T 0 ∥θu,t∥L2(Ω) + ��ρu,t �� L2(Ω) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Then, by Cauchy-Schwarz inequality, T2 ≤ √ T µ �� � T 0 ∥θu,t∥2 L2(Ω) ds �1/2 + � � T 0 ��ρu,t ��2 L2(Ω) ds �1/2� , (32) We can bound � T 0 ∥θu,t∥2 L2(Ω), using the variational formulations (5) and (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' We first write for t ∈]0, T]: (θu,t(t), v) + µ(∇θu(t), ∇v) = (uh,t(t), v) + µ(∇uh(t), ∇v) − (P1 hut(t), v) − µ(∇P1 hu(t), ∇v), = ( f (t), v) − (P1 hut(t), v) − µ(∇u(t), ∇v), = −(ρu,t(t), v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Formally by using v = θu,t(t) and (29), it entails ��θu,t(t) ��2 L2(Ω) + µ 2 d dt ��∇θu(t) ��2 L2(Ω) = −(ρu,t(t), θu,t(t)), such that (with Young’s inequality) ��θu,t(t) ��2 L2(Ω) + µ d dt ��∇θu(t) ��2 L2(Ω) ≤ ��ρu,t(t) ��2 L2(Ω) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Integrating over time, we obtain � T 0 ∥θu,t∥2 L2(Ω) ds + µ ��∇θu(t) ��2 L2(Ω) ≤ µ ��∇θu(0) ��2 L2(Ω) + � T 0 ��ρu,t ��2 L2(Ω) ds, 10 and since the second term is always positive and that we have chosen u0 h = P1 hu0, it yields � T 0 ∥θu,t∥2 L2(Ω) ≤ � T 0 ��ρu,t ��2 L2(Ω) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (33) Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Note that with another choice of discretized initial solution, we would have ��∇θu(0) ��2 L2(Ω) ≤ ���∇u0 h − ∇u0��� 2 L2(Ω) + Ch2���u0��� 2 H2(Ω) , which would have lead to an estimate in O(h) on the L2(Ω) error estimate of Ψ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' In practice, this is not an issue since the effect of the initial data exponentially decreases [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Therefore, from (32), we obtain T2 ≤ 2 √ T µ � � T 0 ��ρu,t ��2 L2(Ω) ds �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (34) By definition of P1 h (7), we have ��ρu,t(t) �� L2(Ω) = ���P1 hut(t) − ut(t) ��� L2(Ω) ≤ Ch2��ut(t) �� H2(Ω) , (35) and thus, (34) yields T2 ≤ C2 √ T µ h2� � T 0 ∥ut∥2 H2(Ω) ds �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (36) – Finally, for T3, we only need to use (35) again, but with Ψ instead of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Therefore T3 = � T 0 ��ρt �� L2(Ω) ds ≤ Ch2 � T 0 ∥Ψt∥H2(Ω) ds .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (37) Combining (27), (30), (36), and (37) concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' We can derive a similar result for the H1 0 norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Let Ω be a convex polyhedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Let A(µ) = µ Id, with µ ∈ R+∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Consider u ∈ H1(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' H2(Ω)) be the solution of (3) with u0 ∈ H2(Ω) and uh be the semi-discretized variational form (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Let Ψ and Ψh be the corresponding sensitivities , respectively given by (23) and (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' ∀t ∈ [0, T], ��Ψ(t) − Ψh(t) �� H1(Ω) ≤ Ch ����Ψ0��� H2(Ω) + � T 0 ∥Ψt∥H2(Ω) ds � + C(µ)h2 �� � T 0 ∥ut∥2 H2 ds �1/2 + � � T 0 ∥Ψt∥2 H2(Ω) ds �1/2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Using the same notation as before (25), we first decompose the error with the two components θ and ρ such that ∀t ∈ [0, T], ∇Ψh(t) − ∇Ψ(t) = ∇θ(t) + ∇ρ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (38) For the estimate on ρ(t), we use (26) to obtain ��∇ρ(t) �� L2(Ω) ≤ Ch ��Ψ(t) �� H2(Ω) , ∀t ∈ [0, T], which leads to ��∇ρ(t) �� L2(Ω) ≤ Ch ����Ψ0��� H2(Ω) + � T 0 ∥Ψt∥H2(Ω) ds � , ∀t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (39) For the estimate on θ(t), let us consider v ∈ Vh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' As in the previous proof, ∀t ∈ [0, T], we write (θt(t), v) + µ(∇θ(t), ∇v) = (Ψh,t(t), v) + µ(∇Ψh(t), ∇v) − (P1 hΨt(t), v) − µ(∇P1 hΨ(t), ∇v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 11 Instead of replacing v by θ(t) as in the L2 estimate, here we formally replace v by θt(t), thus ∀t ∈]0, T], ��θt(t) ��2 L2(Ω) + µ(∇θ(t), ∇θt(t)) = (∇u(t) − ∇uh(t), ∇θt(t)) − (ρt(t), θt(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Thanks to the variational formulations on the state solution u ((5) and (8) respectively) ��θt(t) ��2 L2(Ω) + µ(∇θ(t), ∇θt(t)) = ( 1 µ(uh,t(t) − ut(t)), θt(t)) − (ρt(t), θt(t)), and thus (with Young’s inequality), ��θt(t) ��2 L2(Ω) + µ(∇θ(t), ∇θt(t)) ≤ 1 2 ����� 1 µ(uh,t(t) − ut(t)) ����� 2 L2(Ω) + 1 2 ��θt(t) ��2 L2(Ω) + 1 2 ��ρt(t) ��2 L2(Ω) + 1 2 ��θt(t) ��2 L2(Ω) , ≤ 1 2µ2 ��uh,t(t) − ut(t) ��2 L2(Ω) + 1 2 ��ρt(t) ��2 L2(Ω) + ��θt(t) ��2 L2(Ω) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Thus, µ(∇θ(t), ∇θt(t)) ≤ 1 2µ2 ��uh,t(t) − ut(t) ��2 L2(Ω) + 1 2 ��ρt(t) ��2 L2(Ω) , (40) and by (29), we have d dt ��∇θ(t) ��2 L2(Ω) ≤ 1 µ3 ��uh,t(t) − ut(t) ��2 L2(Ω) + 1 µ ��ρt(t) ��2 L2(Ω) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Integrating over time, it entails ��∇θ(t) ��2 L2(Ω) ≤ ��∇θ(0) ��2 L2(Ω) � �� � T′ 1 + 1 µ3 � T 0 ��uh,t − ut ��2 L2(Ω) ds � �� � T′ 2 + 1 µ � T 0 ��ρt ��2 L2(Ω) ds � �� � T′ 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (41) – From the initial conditions, T′ 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' – We can also write T′ 2 as T′ 2 = 1 µ3 � T 0 ��θu,t + ρu,t ��2 L2(Ω) ds .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Therefore using (33), T′ 2 ≤ 2 µ3 � T 0 ∥θu,t∥2 L2(Ω) + ��ρu,t ��2 L2(Ω) ds ≤ 4 µ3 � T 0 ��ρu,t ��2 L2(Ω) ds ≤ Ch4 µ3 � T 0 ∥ut∥2 H2(Ω) ds .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (42) – Similarly, T′ 3 ≤ Ch4 µ � T 0 ∥Ψt∥2 H2(Ω) ds .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (43) Hence, combining (38) with (39), (41), (42) and (43) concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='3 The fully-discretized versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' From (24), we can derive the fully-discretized systems for the fine and coarse grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' The direct sensitivity problems with respect to the parameter µp on the fine mesh Th with an Euler scheme read � � � � � � � � � Find Ψn p,h ∈ Vh for n ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , NT} such that (∂Ψn p,h, vh) + a(Ψn p,h, vh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' µ) = −( ∂A ∂µp (µ)∇un h(µ), ∇vh), for vh ∈ Vh, for n = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , NT}, (44) Ψ0 p,h(·) = P1 hΨ0 p(·), where, as before, the time derivative in the variational form of the problem (23) has been replaced by a backward difference quotient, ∂Ψn h = Ψn h−Ψn−1 h ∆tF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' With the fully-discretized version (44), the following estimate holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 12 Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Let Ω be a convex polyhedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Let A(µ) = µ Id, with µ ∈ R+∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Consider u ∈ H1(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' H2(Ω)) ∩ H2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' L2(Ω)) be the solution of (3) with u0 ∈ H2(Ω) and un h be the fully-discretized variational form (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Let Ψ and Ψn h be the corresponding sensitivities , respectively given by (23) and (44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Then ∀n = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , NT, ��Ψn h − Ψ(t) �� L2(Ω) ≤ Ch2���Ψ0��� H2(Ω) + h2� C � tn 0 ∥Ψt∥H2(Ω) ds + C(µ) � � tn 0 ∥ut∥2 H2(Ω) ds �1/2� + ∆tF � C � tn 0 ∥Ψtt∥L2(Ω) ds + C(µ) � � tn 0 ∥utt∥2 L2(Ω) ds �1/2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Now, we define θn and ρn on the discretized time grid (tn)n=0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=',NT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' ∀n = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , NT, en := Ψn h − Ψ(tn) = (Ψn h − P1 hΨ(tn)) + (P1 hΨ(tn) − Ψ(tn)), = θn + ρn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (45) In analogy with (26) the estimate on ρn is ��ρn�� L2(Ω) ≤ Ch2��Ψ(tn) �� H2(Ω) ≤ Ch2����Ψ0��� H2(Ω) + � tn 0 ∥Ψt∥H2(Ω) ds � , ∀n ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , NT}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (46) For θn, the equation (28) becomes (∂θn, θn) + µ ��∇θn��2 L2(Ω) = 1 µ(∂un h − ut(tn), θn) − (wn 1 + wn 2, θn), = 1 µ(∂un h − ut(tn), θn) − (wn, θn), (47) where wn 1 and wn 2 are defined by wn 1 := (P1 h − I)∂Ψ(tn), wn 2 := ∂Ψ(tn) − Ψt(tn), and wn := wn 1 + wn 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (48) By definition of ∂ and by Cauchy-Schwarz inequality (and since the second term of the left-hand side of (47) is always positive),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' ��θn��2 L2(Ω) ≤ ����θn−1��� L2(Ω) + ∆tF � 1 µ ���∂un h − ut(tn) ��� L2(Ω) + ��wn�� L2(Ω) ����θn�� L2(Ω) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' and by repeated application,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' and since ���θ0��� L2(Ω) = 0 (again,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' the case where some θn are equal to 0 can be easily handled),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' it entails ��θn�� L2(Ω) ≤ ∆tF n ∑ j=1 1 µ ���∂uj h − ut(tj) ��� L2(Ω) � �� � T2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='n + ∆tF n ∑ j=1 ���wj��� L2(Ω) � �� � T3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='n ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (49) – We first decompose T2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='n in two contributions ∆tF µ n ∑ j=1 ���∂uj h − ut(tj) ��� L2(Ω) ≤ ∆tF µ n ∑ j=1 ����∂θj u ��� L2(Ω) + ���wj u ��� L2(Ω) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' where wj u := wj 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='u + wj 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='u with wj 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='u := (P1 h − I)∂u(tj),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' and wj 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='u := ∂u(tj) − ut(tj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (50) Then by using Cauchy-Schwarz inequality (as in the semi-discretized case (32)), ∆tF µ n ∑ j=1 ���∂uj h − ut(tj) ��� L2(Ω) ≤ √ tn µ �� n ∑ j=1 ∆tF ���∂θj u ��� 2 L2(Ω) � �� � Tθ �1/2 + � n ∑ j=1 ∆tF ���wj u ��� 2 L2(Ω) � �� � Tw �1/2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (51) 13 Let us begin by the estimate on Tθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' On the state solution u, by choosing v = ∂θn u, from (10) (the operator ∂ and the spatial derivative can commute), we have ���∂θn u ��� 2 L2(Ω) + µ(∇θn u, ∂∇θn u) = −(wn u, ∂θn u), (52) where θn u is the discrete version of (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' By definition of ∂ (and with Young’s inequality), ���∂θn u ��� 2 L2(Ω) + µ ∆tF ��∇θn u ��2 L2(Ω) ≤ µ 2∆tF ���∇θn u ��2 L2(Ω) + ���∇θn−1 u ��� 2 L2(Ω) � + 1 2 ���wn u ��2 L2(Ω) + ���∂θn��� 2 L2(Ω) � , which entails ���∂θn u ��� 2 L2(Ω) ≤ µ ∆tF ���∇θn−1 u ��� 2 L2(Ω) − µ ∆tF ��∇θn u ��2 L2(Ω) + ��wn u ��2 L2(Ω) , ∀n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , NT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (53) Summing over the time steps, we get n ∑ j=1 ���∂θj u ��� 2 L2(Ω) ≤ ��� n ∑ j=1 µ ∆tF ����∇θj−1 u ��� 2 L2(Ω) − ���∇θj u ��� 2 L2(Ω) � + ���wj u ��� 2 L2(Ω) ���, and we obtain n ∑ j=1 ���∂θn u ��� 2 L2(Ω) ≤ ��� µ ∆tF ����∇θ0 u ��� 2 L2(Ω) − ��∇θn u ��2 L2(Ω) ���� + n ∑ j=1 ���wj u ��� 2 L2(Ω) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' From the initial condition, θ0 u = 0, n ∑ j=1 ���∂θn u ��� 2 L2(Ω) ≤ µ ∆tF ��∇θn u ��2 L2(Ω) + n ∑ j=1 ���wj u ��� 2 L2(Ω) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (54) From (53) and by repeated application, we find for the first right-hand side term that ��∇θn u ��2 L2(Ω) ≤ ∆tF µ n ∑ j=1 ���wj u ��� 2 L2(Ω) , which gives for (54), multiplying by ∆tF to recover Tθ, n ∑ j=1 ∆tF ���∂θj u ��� 2 L2(Ω) ≤ 2 n ∑ j=1 ∆tF ���wj u ��� 2 L2(Ω) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (55) Now, going back to (51), we obtain ∆tF µ n ∑ j=1 ���∂uj h − ut(tj) ��� L2(Ω) ≤ C µ � n ∑ j=1 ∆tF ���wj u ��� 2 L2(Ω) � �� � Tw �1/2 ≤ C µ � n ∑ j=1 ∆tF ����wj 1,u ��� 2 L2(Ω) + ���wj 2,u ��� 2 L2(Ω) ��1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (56) It remains to estimate Tw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Let us first consider the construction for w1,u wj 1,u = (P1 h − I)∂u(tj) = 1 ∆tF (P1 h − I) � tj tj−1 ut ds = 1 ∆tF � tj tj−1(P1 h − I)ut ds , 14 since P1 h and the time integral commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Thus, from Cauchy-Schwarz inequality, ∆tF n ∑ j=1 ���wj 1,u ��� 2 L2(Ω) ≤ ∆tF n ∑ j=1 � Ω � 1 ∆t2 F � tj tj−1((P1 h − I)ut)2 ds ∆tF � ≤ n ∑ j=1 � tj tj−1 ���(P1 h − I)ut ��� 2 L2(Ω) ds , ≤ Ch4 n ∑ j=1 � tj tj−1∥ut∥2 H2(Ω) , by definition of P1 h, ≤ Ch4 � tn 0 ∥ut∥2 H2(Ω) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (57) To estimate the L2 norm of w2,u, we write wj 2,u = 1 ∆tF (u(tj) − u(tj−1)) − ut(tj) = − 1 ∆tF � tj tj−1(s − tj−1)utt(s) ds, such that we end up with ∆tF n ∑ j=1 ���wj 2,u ��� 2 L2(Ω) ≤ n ∑ j=1 ����� � tj tj−1(s − tj−1)utt(s) ds ����� 2 L2(Ω) ≤ ∆t2 F � tn 0 ∥utt∥2 L2(Ω) ds, (58) – We still have to find a bound for T3,n, defined in (49).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' For the estimates on wj 1, wj 1 = 1 ∆tF � tj tj−1(P1 h − I)Ψt ds , and thus, ∆tF n ∑ j=1 ���wj 1 ��� L2(Ω) ≤ Ch2 � tn 0 ∥Ψt∥H2(Ω) ds .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' For wj 2, we have ∆tFwj 2 = Ψ(tj) − Ψ(tj−1) − ∆tFΨt(tj) = − � tj tj−1(s − tj−1)Ψtt(s) ds , and therefore ∆tF n ∑ j=1 ���wj 2 ��� L2(Ω) ≤ n ∑ j=1 ����� � tj tj−1(s − tj−1)Ψtt(s) ds ����� L2(Ω) ≤ ∆tF � tn 0 ∥Ψtt∥L2(Ω) ds .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Altogether, T3,n ≤ Ch2 � tn 0 ∥Ψt∥H2(Ω) ds + ∆tF � tn 0 ∥Ψtt∥L2(Ω) ds , (59) and the proof ends by using (46), (49), (56), (57), (58), and (59).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' With the fully-discretized version (44), the following estimate holds with H1 norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Let Ω be a convex polyhedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Let A(µ) = µ Id, with µ ∈ R+∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Consider u ∈ H1(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' H2(Ω)) ∩ H2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' L2(Ω)) be the solution of (3) with u0 ∈ H2(Ω) and un h be the fully-discretized variational form (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Let Ψ and Ψn h be the corresponding sensitivities , respectively given by (23) and (44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Then ∀n = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , NT, ��∇Ψn h − ∇Ψ(t) �� L2(Ω) ≤ h � C ���Ψ0��� H2(Ω) + C(µ) � tn 0 ∥Ψt∥H2(Ω) ds + C(µ) � � tn 0 ∥ut∥2 H2(Ω) ds �1/2� + C(µ)∆tF � � tn 0 ∥Ψtt∥L2(Ω) ds + � � tn 0 ∥utt∥2 L2(Ω) ds �1/2�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 15 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' The proof combines the ideas of the two previous ones, since we seek the estimate in the H1 norm (as in the semi-discretized problem) but with the fully-discretized version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' In analogy with (26), the estimate on ρn is now given by ��∇ρn�� L2(Ω) ≤ Ch ��Ψ(tn) �� H2(Ω) ≤ Ch ����Ψ0��� H2(Ω) + � tn 0 ∥Ψt∥H2(Ω) ds � , ∀n = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , NT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (60) For θn, instead of choosing v = θn as in (47), we take v = ∂θn ���∂θn��� 2 L2(Ω) + µ(∇θn, ∇∂θn) = 1 µ(∂un h − ut(tn), ∂θn) − (wn, ∂θn), (61) and we obtain (as before with the semi-discretized version (40)) µ(∇θn, ∇∂θn) ≤ 1 2µ2 ���∂un h − ut(tn) ��� 2 L2(Ω) + 1 2 ��wn��2 L2(Ω) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' By definition of ∂ µ ��∇θn��2 L2(Ω) ≤ (√µ∇θn, √µ∇θn−1) + ∆tF 2µ2 ���∂un h − ut(tn) ��� 2 L2(Ω) + ∆tF 2 ��wn��2 L2(Ω) , which entails (by Young’s inequality) µ ��∇θn��2 L2(Ω) ≤ µ ���∇θn−1��� 2 L2(Ω) + ∆tF µ2 ���∂un h − ut(tn) ��� 2 L2(Ω) + ∆tF ��wn��2 L2(Ω) , and, by recursion (as in (49)) µ ��∇θn��2 L2(Ω) ≤ ∆tF µ2 n ∑ j=1 ���∂uj h − ut(tj) ��� 2 L2(Ω) � �� � T′ 2,n + ∆tF n ∑ j=1 ���wj��� 2 L2(Ω) � �� � T′ 3,n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (62) – To estimate T′ 2,n, we write T′ 2,n ≤ 2 µ2 � n ∑ j=1 ∆tF ���∂θj u ��� 2 L2(Ω) � �� � Tθ + n ∑ j=1 ∆tF ���wj u ��� 2 L2(Ω) � �� � Tw � , and thanks to the previous estimate on Tθ (55), we find that T′ 2,n ≤ 6 µ2 � n ∑ j=1 ∆tF ���wj u ��� 2 L2(Ω) � �� � Tw � , which, by (57) and (58), yields T′ 2,n ≤ C µ2 � h4 � tn 0 ∥ut∥2 H2(Ω) ds + ∆t2 F � tn 0 ∥utt∥2 L2(Ω) ds � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' – To find a bound for T′ 3,n, we simply use (57) and (58) again but with the sensitivity function Ψ instead of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Combining the estimates on T′ 2,n and T′ 3,n with (62), and (60) concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 16 With ∂Ψm H = Ψm H−Ψm−1 H ∆tG , on the coarse mesh TH with the Crank-Nicolson scheme, the fully-discretized system (11) yields � � � � � � � � � � � Find Ψm p,H ∈ VH for m ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , MT} such that (∂Ψm p,H, vH) + a( Ψm p,H+Ψm−1 p,H 2 , vH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' µ) = −( ∂A ∂µp (µ) ∇um H(µ)+∇um−1 H (µ) 2 , ∇vH), for vH ∈ VH, for m = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , MT}, (63) Ψ0 p,H(·) = P1 HΨ0 p(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' We have the following result in the L2 norm with the Crank-Nicolson scheme on the coarse mesh TH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Let Ω be a convex polyhedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Let A(µ) = µ Id, with µ ∈ R+∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Consider u ∈ H2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' H2(Ω)) ∩ H3(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' L2(Ω)) be the solution of (3) with u0 ∈ H2(Ω) and um H be the fully-discretized variational form (11) (on the coarse mesh TH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Let Ψ and Ψm H be the corresponding sensitivities , respectively given by (23) and (44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Then ∀m = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , MT, ���Ψm h − Ψ(�tm) ��� L2(Ω) ≤ CH2����Ψ0��� H2(Ω) + � �tm 0 ∥Ψt∥H2(Ω) ds + C(µ) � � �tm 0 ∥ut∥2 H2(Ω) ds �1/2� + C∆t2 G � � �tm 0 ∥Ψttt∥L2(Ω) ds + � � �tm 0 ∥∆utt∥2 L2(Ω) ds �1/2 + C(µ) �� � �tm 0 ∥uttt∥2 L2(Ω) ds]1/2 + � �tm 0 ∥∆Ψtt∥L2(Ω) ds �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' For ρm we have the same estimate as before (46) (but with the coarse size H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' We introduce the following notation � um H = 1 2(um H + um−1 H ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (64) Thanks to the Crank-Nicolson formulation on Ψm H (63) and um H (11) on the coarse mesh TH (and on the weak formulation on u (5) and by definition of P1 H (7)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (∂θm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' v) + µ(∇� θm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' ∇v) = (∂Ψm H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' v) − (∂P1 H(Ψ(�tm)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' v) + µ(∇� Ψm H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' ∇v) − µ 2 � (∇P1 HΨ(�tm),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' ∇v) + (∇P1 HΨ(�tm−1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' ∇v) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' = −(∇ � um H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' ∇v) − (∂P1 H(Ψ(�tm)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' v) − µ 2 � (∇Ψ(�tm),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' ∇v) + (∇Ψ(�tm−1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' ∇v) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' = −(∇ � um H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' ∇v) −(∂P1 H(Ψ(�tm)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' v) + (∂Ψ(�tm),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' v) � �� � −wm I −(∂Ψ(�tm),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' v) + (Ψt(�tm− 1 2 ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' v) � �� � −wm II − (Ψt(�tm− 1 2 ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' v) − µ 2 � (∇Ψ(�tm),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' ∇v) + (∇Ψ(�tm−1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' ∇v) � = −(∇ � um H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' ∇v) − wm I − wm II + (∇u(�tm− 1 2 ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' ∇v) + µ(∇Ψ(�tm− 1 2 ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' ∇v) − µ 2 � (∇Ψ(�tm),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' ∇v) + (∇Ψ(�tm−1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' ∇v) � = (∇u(�tm− 1 2 ) − � ∇um H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' ∇v) − (wm I + wm II + µwm III,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' v),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' where wm I ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' wm II and wm III are defined by wm I := (P1 H − I)∂Ψ(�tm),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' wm II := ∂Ψ(�tm) − Ψt(�tm− 1 2 ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' and wm III := ∆ψ(�tm− 1 2 ) − 1 2(∆Ψ(�tm) + ∆Ψ(�tm−1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (65) Thus, (47) with a Crank-Nicolson scheme and with v = � θm becomes (∂θm, � θm) + µ(∇� θm, ∇� θm) = 1 µ(∂um H − ut(�tm− 1 2 ), � θm) − (wm I + wm II + µwm III, � θm), = 1 µ(∂um H − ut(�tm− 1 2 ), � θm) − (wm T , � θm), (66) where wm T = wm I + wm II + µwm III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' By definition of ∂ (with the coarse time grid),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' and since the second term in (66) is always positive,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' we get (θm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' � θm) − (θm−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' � θm) ≤ ∆tG � 1 µ ���∂um H − ut(�tm− 1 2 ) ��� L2(Ω) + ��wm T �� L2(Ω) ����� θm ��� L2(Ω) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 17 and by definition of � θm (64),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' ��θm��2 L2(Ω) − ���θm−1��� 2 L2(Ω) ≤ ∆tG � 1 µ ���∂um h − ut(�tm− 1 2 ) ��� L2(Ω) + ��wm T �� L2(Ω) ����θm + θm−1��� L2(Ω) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' so that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' after cancellation of a common factor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' ��θm�� L2(Ω) − ���θm−1��� L2(Ω) ≤ ∆tG � 1 µ ���∂um H − ut(�tm− 1 2 ) ��� L2(Ω) + ��wm T �� L2(Ω) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' and by recursive application,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' it entails ��θm�� L2(Ω) ≤ ∆tG µ m ∑ j=1 ���∂uj H − ut(�tj− 1 2 ) ��� L2(Ω) � �� � T′′ 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='n + ∆tG m ∑ j=1 ���wj T ��� L2(Ω) � �� � T′′ 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (67) – To estimate T′′ 2,n, we use the same tricks as before (51).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' First,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' we can decompose T′′ 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='n in 2 contributions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' such that ∆tG µ m ∑ j=1 ���∂uj H − ut(�tj− 1 2 ) ��� L2(Ω) ≤ ∆tG µ m ∑ j=1 ���∂uj H − ∂Ph 1 u(�tj) ��� L2(Ω) � �� � ���∂θj u ��� L2(Ω) + ���∂Ph 1 u(�tj) − ut(�tj− 1 2 ) ��� L2(Ω) � �� � ���wj I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='u+wj II,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='u ��� L2(Ω) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' where we denote by wm I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' wm II,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='u the same terms respectively defined by wm I ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' wm II (65) but with u instead of Ψ wm I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='u := (P1 h − I)∂u(�tm),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' wm II,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='u := ∂u(�tm) − ut(�tm− 1 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (68) We now apply Cauchy-Schwarz inequality ∆tG µ m ∑ j=1 ���∂uj H − ut(�tj− 1 2 ) ��� L2(Ω) ≤ √�tm µ �� m ∑ j=1 ∆tG ���∂θj��� 2 L2(Ω) �1/2 + � m ∑ j=1 ∆tG ���wj I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='u + wj II,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='u ��� 2 L2(Ω) �1/2� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (69) ≤ √�tm µ �� m ∑ j=1 ∆tG ���∂θj��� 2 L2(Ω) �1/2 + � m ∑ j=1 ∆tG ���wj I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='u + wj II,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='u ��� 2 L2(Ω) + ���wj III,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='u ��� 2 L2(Ω) �1/2� To estimate the first term of (69),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' we use v = ∂θm u ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' and we now have (from the Crank-Nicolson scheme on u (11)) ���∂θm u ��� 2 L2(Ω) + µ(∇� θm u ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' ∇∂θm u ) = −(wm I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='u + wm II,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='u + µwm III,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' ∂θm u ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' where wm III,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='u := ∆u(�tm− 1 2 ) − 1 2(∆u(�tm) + ∆u(�tm−1)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (70) and θm u is the discrete version of (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' By definitions of ∂ and � θm u , it can be rewritten ���∂θm u ��� 2 L2(Ω) + µ 2∆tG ��∇θm u ��2 L2(Ω) − µ 2∆tF ���∇θm−1 u ��� 2 L2(Ω) = −(wm I,u + wm II,u + µwm III,u, ∂θm u ), and it leads to (using Young’s inequality) ���∂θm u ��� 2 L2(Ω) + µ ∆tG ��∇θm u ��2 L2(Ω) − µ ∆tG ���∇θm−1 u ��� 2 L2(Ω) ≤ ���wm I,u + wm II,u + µwm III,u ��� 2 L2(Ω) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (71) Now we find, as in (54) (by summing over all time steps in order to obtain a telescoping sum) m ∑ j=1 ���∂θj u ��� 2 L2(Ω) ≤ µ ∆tG ���∇θj u ��� 2 L2(Ω) + m ∑ j=1 ���wj I,u + wj II,u + wj III,u ��� 2 L2(Ω) , (72) 18 The term∥∇θm u ∥2 L2(Ω) can easily be bounded by repeated application using (71).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' We find that (since the first term of (71) is positive) µ ∆tG ��∇θm u ��2 L2(Ω) ≤ m ∑ j=1 ���wj I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='u + wj II,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='u + µwj III,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='u ��� 2 L2(Ω) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' and thus (72) gives m ∑ j=1 ���∂θj u ��� 2 L2(Ω) ≤ 2 m ∑ j=1 ���wj I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='u + wj II,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='u + µwj III,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='u ��� 2 L2(Ω) ≤ 4 m ∑ j=1 ���wj I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='u + wj II,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='u ��� 2 L2(Ω) + ���µwj III,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='u ��� 2 L2(Ω) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (73) therefore we obtain for (69) ∆tG µ m ∑ j=1 ���∂uj H − ut(�tj− 1 2 ) ��� L2(Ω) ≤ 3 � �tm ��∆tG µ2 m ∑ j=1 ���wj I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='u + wj II,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='u ��� 2 L2(Ω) + m ∑ j=1 ∆tG ���wj III,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='u ��� 2 L2(Ω) �1/2� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' which yields ∆tG µ m ∑ j=1 ���∂uj H − ut(�tj− 1 2 ) ��� L2(Ω) ≤ 3 � �tm �� 2 µ2 m ∑ j=1 ∆tG ���wj I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='u ��� 2 L2(Ω) + 2 µ2 m ∑ j=1 ∆tG ���wj II,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='u ��� 2 L2(Ω) + m ∑ j=1 ∆tG ���wj III,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='u ��� 2 L2(Ω) �1/2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (74) Now, we can estimate the right-hand side terms of (74) as done in [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' We first remark that wj I,u = wj 1,u (and the estimate is given by (57) but with the coarse spatial and time grids), so it remains to seek bounds for wj II,u and wj III,u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' For wj II,u, ∆tG m ∑ j=1 ���wj II,u ��� 2 L2(Ω) = 1 ∆tG m ∑ j=1 ���u(�tj) − u(�tj−1) − ∆tG ut(�tj− 1 2 ) ��� 2 L2(Ω) , = 1 4∆tG m ∑ j=1 ������ � �tj− 1 2 �tj−1 (s − �tj−1)2uttt(s) + � �tj �tj− 1 2 (s − �tj)2uttt(s) ds ������ 2 L2(Ω) ≤ C∆t3 G m ∑ j=1 ����� � �tj �tj−1 uttt(s) ds ����� 2 L2(Ω) , ≤ C∆t4 G m ∑ j=1 � �tj �tj−1∥uttt∥2 L2(Ω) ds, with Cauchy-Schwarz inequality, ≤ C∆t4 G � �tm �t0 ∥uttt∥2 L2(Ω) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (75) For wj III,u, ∆tG m ∑ j=1 ���wj III,u ��� 2 L2(Ω) = ∆tG m ∑ j=1 ����∆u(�tj− 1 2 ) − 1 2(∆u(�tj) + ∆u(�tj−1)) ���� 2 L2(Ω) , = ∆tG 4 m ∑ j=1 ������ � �tj− 1 2 �tj−1 (tj−1 − s)∆utt(s) ds + � �tj �tj− 1 2 (s − �tj)∆utt(s) ds ������ 2 L2(Ω) , ≤ C∆t3 G m ∑ j=1 ����� � �tj �tj−1 ∆utt ds ����� 2 L2(Ω) , ≤ C∆t4 G � �tm �t0 ∥∆utt∥2 L2(Ω) ds, by Cauchy-Schwarz inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (76) 19 Altogether, T′′ 2,n ≤ C � H2 µ � � �tm 0 ∥ut∥2 H2(Ω) ds �1/2 + ∆t2 G �� 1 µ � �tm 0 ∥uttt∥2 L2(Ω) �1/2 + � � �tm 0 ∥∆utt∥2 L2(Ω) �1/2 ds �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (77) – To estimate T′′ 3,n defined in (67), we remark that wj I = wj 1 (but with the coarse spatial and time grids), and for wj II, ∆tG ���wj II ��� L2(Ω) ≤ ���Ψ(�tj) − Ψ(�tj−1) − ∆tGΨt(�tj− 1 2 ) ��� L2(Ω) , = 1 2 ������ � �tj− 1 2 �tj−1 (s − �tj−1)2Ψttt(s) + � �tj �tj− 1 2 (s − �tj)2Ψttt(s) ds ������ L2(Ω) ≤ C∆t2 G � �tj �tj−1∥Ψttt∥L2(Ω) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (78) Finally, for wj III, ∆tG ���wj III ��� L2(Ω) = ∆tG ����Ψ(�tj− 1 2 ) − 1 2(Ψ(�tj) + Ψ(�tj−1)) ���� L2(Ω) ≤ C∆t2 G � �tj �tj−1∥∆Ψtt∥L2(Ω) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (79) Altogether, T′′ 3,n ≤ CH2 � �tm 0 ∥Ψt∥H2(Ω) ds + C∆t2 G � �tm 0 � ∥Ψttt∥L2(Ω) + µ∥∆Ψtt∥L2(Ω) � ds, (80) which concludes the proof (combining (67), (77) and (80)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' In analogy with the previous work on parabolic equations, we define � ΨH n = I2 n[Ψm H](µ), for n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , NT, (81) with I2 n defined by (12) as the quadratic interpolation in time of the coarse solution at time tn ∈ Im = [�tm−1,�tm] defined on [�tm−2,�tm] from the values Ψm−2 H , Ψm−1 H , and Ψm H, for all m = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , MT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' For tn ∈ I1 = [�t0,�t1], we use the same parabola defined by the values Ψ0 H, Ψ1 H, ψ2 H as the one used over [�t1,�t2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Note that, as before, we could have chosen another quadratic interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='10 (of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Under the assumptions of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='9, let um H be the fully-discretized solution (11) on the coarse mesh TH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Let Ψ and Ψm H be the corresponding sensitivities, respectively given by (23) and by (63).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Let � ΨH n be the quadratic interpolation of the coarse solution Ψm H given by (81).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Then, ∀n = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , NT, ���� Ψh n − Ψ(tn) ��� L2(Ω) ≤ CH2����Ψ0��� H2(Ω) + � tn 0 ∥Ψt∥H2(Ω) ds + C(µ) � � tn 0 ∥ut∥2 H2(Ω) ds �1/2� + C∆t2 G � � tn 0 ∥Ψttt∥L2(Ω) ds + � � tn 0 ∥∆utt∥2 L2(Ω) ds �1/2 + C(µ) �� � tn 0 ∥uttt∥2 L2(Ω) ds ]1/2 + � tn 0 ∥∆Ψtt∥L2(Ω) ds �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' In the next section, we proceed with the adjoint state formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='4 Sensitivity analysis: The adjoint problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' The adjoint method may be seen as an inverse method, where the goal is to retrieve the optimal parameter of an objective function F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' The objective function will have a different meaning whether the goal is to retrieve the parameters from several measures (for parameter identification) or if we want to optimize a function depending 20 on the variables (PDE-constrained optimization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' In the first case, F will have the following form (in its fully- discretized form) F(µ) = 1 2 NT ∑ n=1 ��un h(µ) − un��2 L2(Ω) � �� � ∥err(tn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' µ)∥ 2 L2(Ω) , (82) where un refer to the measures, which may be noisy (here for simplicity we consider the case of measures on the variables although it may be given by other outputs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' In the second setting, it will be written F(µ) = NT ∑ n=1 gnun h(µ), (83) with gn some suitable weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Note that by differentiating F with respect to the parameters µp, p = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , P, we can observe the influence on the objective function of the input parameters through the normalized sensitivity coefficients (also called elasticity of P) [4] Sk = ∂F ∂µk (µ) × µk F(µ), k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (84) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='1 The continuous setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Let us for instance consider the first case outlined above, given in the continuous version by F(µ) = 1 2 � T 0 ��err(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' µ) ��2 L2(Ω) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (85) To minimize F under the constraint that u is the solution of our model problem (3), we consider the following Lagrangian with (χ, ϕ) the Lagrangian multipliers L(u, χ, ϕ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' µ) = F(µ) + � T 0 (χ, (∇ · (A(µ)∇u) + f − ut)) ds + � T 0 (ϕ, u)L2(∂Ω) ds, (86) where – χ ∈ V is the multiplier associated to the constraint “u is a solution of (3)”, – ϕ ∈ R is the multiplier associated to the constraint of the Dirichlet boundary condition on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Since here we consider homogeneous condition, we just impose ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Differentiating L with respect to the parameter µp, for p = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , P, we obtain the following adjoint system in its variational form (see A for more details) � � � � � Find χ(t) ∈ V for t ∈ [0, T] such that (χt(t), v) = −( ∂err ∂u (t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' µ), v) + (A(µ)∇χ(t), ∇v), ∀v ∈ V, t < T, χ(·, T) = 0, in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (87) After solving (89) with the parameter µ, one can compute dF dµp by noticing that dF dµp = dL dµp = � T 0 � χ, ∇ · ( ∂A ∂µp (µ)∇u) � ds, from (124).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (88) Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' For a stable implementation, one may have to add a regularization term depending of the parameter to the cost function F(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 21 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='2 Discretized setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' In analogy with the direct method, we first discretize the system in space, and then we apply an Euler scheme with the fine grids and a Crank-Nicolson scheme with the coarse ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' The semi-discretized version on Th writes � � � � � � � Find χh(t) ∈ Vh for t ∈ [0, T] such that (χh,t(t), vh) − a(χh, vh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' µ) = −( ∂errh ∂uh (t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' µ), vh), ∀vh ∈ Vh, t < T, χh(·, T) = 0, in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (89) With the fully-discretized version, on the fine grids, the adjoint system becomes in its variational formulation � � � � � � � Find χn h ∈ Vh for n ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , NT} such that (∂χn h, vh) − a(χh, vh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' µ) = −(un h − un, vh), ∀n = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , NT − 1, χNT h (·) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (90) Note that to compute ∂errn h ∂uh , we need the fine solutions un h and the measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' As for the state variable (11), we also compute the adjoint on the coarse mesh with the Crank-Nicolson scheme, � � � � � � � � � Find χm H ∈ VH for m ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , MT} such that (∂χm H, vH) − a( 1 2(χm H + χm−1 H ), vH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' µ) = − 1 2 � (um H − um, vH) + (um−1 H − um−1, vH) � ), ∀vH ∈ VH, ∀m = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , MT − 1, (91) χMT H (·) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Finally, note that the problems (90) and (92) are well-posed, since they are solved backward in time (see [16] for precisions in the general setting of time-dependent PDEs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' The next section adapts the NIRB two-grid algorithm in the context of sensitivity analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 3 NIRB algorithms applied to sensitivity analysis 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='1 On the direct problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='1 NIRB algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Let u(µ) be the exact solution of problem (3) for a parameter µ ∈ G and Ψp(µ) its sensitivity with respect to the parameter µp, p = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' We consider P parameters of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' In this context, we use the following offline/online decomposition for the NIRB procedure: “Offline part” 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' For a set of training parameters (�µi)i=1,··· ,Np,train, we define Gp,train = ∪ i∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=',Np,train}�µi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Then, through a greedy algorithm 1, we adequately choose the parameters of the RB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' During this procedure, we compute fine fully-discretized solutions {Ψn p,h(�µi)}i∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='Nµ,p}, n={0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=',NT} (Nµ,p ≤ Np,train) with the HF solver, by solving either (44) or the following problem (where un h in (44) has been replaced by its NIRB approximation uN,n Hh or by its rectified version Rn u[uN,n Hh ] obtained from the algorithm of section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='2) � � � � � � � � � Find Ψn p,h ∈ Vh for n ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , NT} such that (∂Ψn p,h, vh) + a(Ψn p,h, vh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' �µ) = −( ∂A ∂µp (µ)∇uN,n Hh (µ), ∇vh) for n = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , NT}, (92) Ψ0 p,h(·) = P1 hΨ0 p(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' The term −( ∂A ∂µp (µ)∇uN,n Hh (µ), ∇vh) in (92) is replaced by −( ∂A ∂µp (µ)∇Rn u[uN,n Hh ](µ), ∇vh) in case of the rectification post-treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Note that un h can directly be used (as in (44)) since this step belongs to the offline part of the NIRB algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' However, if the number of parameters required for the initial RB is 22 lower than the number of parameters needed for the sensitivities RB or if one combine the sensitivities with an optimization algorithm, it may be convenient to employ (92) instead of (44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' In analogy to section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='2, a few time steps may be selected for each parameter of the RB, and thus, we obtain Np L2 orthogonal RB (time-independent) functions, denoted (ζh p,i)i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=',Np, and the reduced spaces X Np p,h := Span{ζh p,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , ζh p,Np} for p = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Then, for each p, we solve the eigenvalue problem (14) on X Np p,h: � � � � � Find ζh ∈ X Np p,h, and λ ∈ R such that: ∀v ∈ X Np p,h, � Ω ∇ζh · ∇v dx = λ � Ω ζh · v dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (93) For each parameter p ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , P}, we get an increasing sequence of eigenvalues λp i , and eigenfunc- tions (ζh p,i)i=1,··· ,Np, orthonormalized in L2(Ω) and orthogonalized in H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' As in the offline step 3 from section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='2, we enhance the NIRB approximation with a rectification post- processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Thus, we introduce the rectification matrices, denoted Rp,n Ψ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' They are associated to the sensitivities problem (44), defined for each p ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , P} and each fine time step n ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , NT}, and constructed from coarse snapshots, generated by solving (63) and whose parameters are the same as for the fine snapshots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Thus, for all n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , NT and all p = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , P, we compute the vectors Rp,n Ψ,i = ((Ap,n)TAp,n + δpINp)−1(Ap,n)TBp,n i , i = 1, · · · , Np, (94) where ∀i = 1, · · · , Np, and ∀�µk ∈ Gp, Ap,n k,i = � Ω � Ψp,H n(�µk) · ζh p,i dx, (95) Bp,n k,i = � Ω Ψn p,h(�µk) · ζh p,i dx, (96) and where INp refers to the identity matrix and δp is a regularization term (note that we used (81) for � Ψp,H n(�µk)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' In general, Np,train < Np and the parameter δp is required for the inversion of (Ap,n)TAp,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' “Online part” The online part of the algorithm is much faster than a double HF evaluation (to seek the sensitivity Ψn p,h, we also need the solution un h with a HF evaluation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Indeed, we first solve the problem (3) on the coarse mesh TH for a new parameter µ ∈ G at each time step m = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , MT using (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Then, for each p = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , P, we solve the coarse associated sensitivity problems (63) with the same parameter µ, at each time step m = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , MT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' We quadratically interpolate in time the coarse solution Ψm p,H on the fine time grid with (81).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Then, we linearly interpolate � Ψp,H n(µ) on the fine mesh in order to compute the L2-inner product with the basis functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' The approximation used in the two-grid method is For n = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , NT, Ψ Np,n p,Hh(µ) := Np ∑ i=1 (� Ψp,H n(µ), ζh p,i) ζh p,i, (97) and with the rectification post-treatment step, it becomes Rp,n Ψ [ΨN p,Hh](µ) := Np ∑ i,j=1 Rp,n ij (� Ψp,H n(µ), ζh p,j) ζh p,i, (98) where Rp,n Ψ is the rectification matrix at time tn, given by (94).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 23 In the next section, we propose an adaptation of this algorithm with a new post-treatment which reduces the online computational time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='2 New NIRB algorithm for the direct problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' The main drawback of the algorithm described in the previous section is that it requires 1 + P coarse systems in the online part (see the steps 4 and 5 in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' The online portion of the new algorithm described below only requires the resolution of two coarse problems, regardless the number of parameters of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' We refer to the following offline/online decomposition: “Offline part” 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' For a parameter training set Gtrain, we compute the RB functions of the initial problem, denoted (Φh i )i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=',N and generates XN h by the steps 1-2 of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='2 (see algorithm 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' As before, from the training sets Gp,train, we generate the reduced spaces X Np p,h, for p = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , P using steps 1 and 2 of section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='1, and at the end of this part, we obtain Np RB functions (time-independent), denoted (ζh p,i)i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=',Np for each p = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' We introduce GT defined by GT := Gtrain ∩ Gp,train, (99) and Nµ,T the number of parameters in GT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' We use the fact that the sensitivities are directly derived from the initial solutions, and we consider new rectification matrices, denoted �Rp,n and defined for each p ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , P} and each fine time step n ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , NT}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' In this new post-treatment, they are constructed from coarse snapshots of the initial solution, generated by solving (11) and whose parameters are the same as for the fine sensitivities, generated by solving (63).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Thus, for all n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , NT and all p = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , P, we compute the vectors �Rp,n i = ((An)TAn + δIN)−1(An)TBp,n i , i = 1, · · · , Np, (100) where this time ∀�µk ∈ GT, An k,i = � Ω � uH n(�µk) · Φh i dx, ∀i = 1, · · · , N, (101) Bp,n k,i = � Ω Ψn p,h(�µk) · ζh p,i dx, ∀i = 1, · · · , Np, (102) and where IN refers to the identity matrix and δ is a regularization term (required for the inversion of (An)TAn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Note that � uH n(�µk) is the quadratic interpolation given by (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' We highlight the fact that this step requires that GT ̸= ∅ (99).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' “Online step” 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' We solve the problem (3) on the coarse mesh TH for a new parameter µ ∈ G at each time step m = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , MT using (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' We quadratically interpolate in time the coarse solution um H on the fine time grid with (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Then, we linearly interpolate � uH n(µ) on the fine mesh in order to compute the L2-inner product with the basis functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' The new NIRB approximation is given by �Rp,n[ΨN p,Hh](µ) := Np ∑ i=1 N ∑ j=1 �Rp,n ij ( � uH n(µ), Φh j ) ζh p,i, (103) where �Rp,n is the rectification matrix at time tn, given by (100).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 24 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='2 On the adjoint formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' The adjoint formulation requires some modifications of the NIRB algorithm compared to section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Since in (88), for all n ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , NT}, the fine solution un h(µ) is required to obtain the sensitivities on F, it follows that here we have to compute two reductions: one for the initial solution u and one for the adjoint χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' As a matter of fact, in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='1), the RB generation for u was optional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' So let u(µ) be the exact solution of problem (3) for a parameter µ ∈ G and χ(µ) its adjoint given by (89).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' In this setting, we use the following offline/online decomposition for the NIRB procedure: “Offline part” 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' During the offline stage, we first construct the reduced space XN h and the RB function (Φh 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , Φh N) with the steps 1-2 of section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Then, we use steps 1-2 of section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='1, but instead of solving (44) on the sensitivities, we generate the reduced space XN1 1 by solving the adjoint problem on the fine mesh (90).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Thus, for a set of training parameters (�µi)i=1,··· ,N1,train, we define G1,train = ∪ i∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=',N1,train}�µi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Then, through a greedy procedure 1, we adequately choose the parameters of the RB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' During this proce- dure, we compute fine fully-discretized solutions {χn h(�µi)}i∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='Nµ,1}, n={0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=',NT} (Nµ,1 ≤ N1,train) with the HF solver, by solving either (90) or the following problem (where un h in (90) has been replaced by its NIRB approximation uN,n Hh or by its rectified version Rn u[uN,n Hh ] obtained from the algorithm of section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='2) � � � � � � � Find χn h ∈ Vh for n ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , NT} such that (∂χn h, vh) − a(χh, vh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' µ) = −(uN,n Hh − un, vh), ∀n = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , NT − 1, χNT h (·) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (104) The term −(uN,n Hh (µ) − un, vh) in (104) is replaced by −(Rn u[uN,n Hh ](µ) − un, vh) in case of the rectification post-treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' In practice, since in step 1 a RB for un h has already been generated, it is more convenient to employ (104) instead of (90).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' In analogy to section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='2, a few time steps may be selected for each parameter of the RB, and thus, we obtain N1 L2 orthogonal RB (time-independent) functions, denoted (ξh i )i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=',N1, and the reduced space XN1 h := Span{ξh 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , ξh N1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Then, we solve the eigenvalue problem (14) on XN1 h : � � � � � Find ξh ∈ XN1 h , and λ ∈ R such that: ∀v ∈ XN1 h , � Ω ∇ξh · ∇v dx = λ � Ω ξh · v dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (105) We get an increasing sequence of eigenvalues λi, and eigenfunctions (ξh i )i=1,··· ,N1, orthonormalized in L2(Ω) and orthogonalized in H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' As in the offline step 3 from section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='1, we enhance the NIRB approximation with a rectifica- tion post-processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Thus, we introduce a rectification matrix, denoted Rn χ for each fine time step n ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , NT}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' It is associated to the adjoint problem (90) and constructed from coarse snapshots, generated by solving (92) and whose parameters are the same as for the fine snapshots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Thus, for all n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , NT, we compute the vectors Rn χ,i = ((An)TAn + δIN1)−1(An)TBn i , i = 1, · · · , N1, (106) where ∀i = 1, · · · , N1, and ∀�µk ∈ Gp, An k,i = � Ω � χH n(�µk) · ξh i dx, (107) Bn k,i = � Ω χn h(�µk) · ξh i dx, (108) 25 and where IN1 refers to the identity matrix and δp is a regularization term required for the inversion of (An)TAn (note that we used (81) for � χH n(�µk)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' “Online part” 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' We first solve the problem (3) on the coarse mesh TH for a new parameter µ ∈ G at each time step m = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , MT using (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Then, we solve the coarse associated adjoint problem (92) with the same parameter µ, at each time step m = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , MT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' We quadratically interpolate in time the coarse solution χm H on the fine time grid with (81).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Then, we linearly interpolate � χH n(µ) on the fine mesh in order to compute the L2-inner product with the basis functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' The approximation used for the adjoint in the two-grid method is For n = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , NT, χN1,n Hh (µ) := N1 ∑ i=1 ( � χH n(µ), ξh i ) ξh i , (109) and with the rectification post-treatment step, it becomes Rn χ[χN1 Hh](µ) := N1 ∑ i,j=1 Rn χ,ij ( � χH n(µ), ξh j ) ξh i , (110) where Rn χ is the rectification matrix at time tn, given by (106).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Then, we use the steps 5 and 6 of section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='2 in order to obtain a NIRB approximation for u(µ) from the coarse solution um H given by step 4 of this online part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Finally, the sensitivities NIRB approximations of F are given by for p = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , P, [ ∂F ∂µp ]N1 Hh(µ) := tn ∑ j=1 ∆tF � χN1,j Hh , ∇ · ( ∂A ∂µp (µ)∇uN,j Hh) � , from (88), (111) and with the rectification post-treatment step, it becomes for p = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , P, Rχ[[ ∂F ∂µp ]N1 Hh](µ) := tn ∑ j=1 ∆tF � Rj χ[χN1,j Hh ](µ), ∇ · ( ∂A ∂µp (µ)∇Rj u[uN,j Hh](µ)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (112) The next section gives our main result on the NIRB two-grid method error estimate in the context of sensitivity analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 4 NIRB error estimate on the sensitivities Main result Our main result is the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (NIRB error estimate for the sensitivities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=') Let A(µ) = µ Id, with µ ∈ R+∗ , and let us consider the problem 3 with its exact solution u(x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' µ), and the full discretized solution un h(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' µ) to the problem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Let Ψ(x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' µ) and Ψn h(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' µ) respectively by the corresponding sensitivities , given by (23) and (44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Let (ζh i )i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=',N1 be the L2-orthonormalized and H1-orthogonalized RB generated with the greedy algorithm 1 through the NIRB algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Let us consider the NIRB approximation, For n = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , NT, ΨN,n Hh (µ) := N1 ∑ i=1 (� ΨH n(µ), ζh i ) ζh i , (113) where � ΨH n(µ) is given by (81).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Then, the following estimate holds ∀n = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , NT, ���Ψ(tn)(µ) − ΨN,n Hh (µ) ��� H1(Ω) ≤ ε(N) + C1(µ)h + C2(µ, N)H2 + C3(µ)∆tF + C4(µ, N)∆t2 G, (114) where C1, C2, C3 and C4 are constants independent of h and H, ∆tF and ∆tG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' The term ε depends on the Kolmogorov N-width and measures the error given by (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 26 If H is such as H2 ∼ h, ∆t2 G ∼ ∆tF, and ε(N) is small enough, with C2(µ, N) and C4(µ, N) not too large, it results in an error estimate in O(h + ∆tF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='1 then states that we recover optimal error estimates in L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' H1(Ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' We now go on with the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' The NIRB approximation at time step n = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , NT, for a new parameter µ ∈ G is defined by (97).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Thus, the triangle inequality gives ���Ψ(tn)(µ) − ΨN,n Hh (µ) ��� H1(Ω) ≤ ��Ψ(tn)(µ) − Ψn h(µ) �� H1(Ω) + ���Ψn h(µ) − ΨN,n hh (µ) ��� H1(Ω) + ���ΨN,n hh (µ) − ΨN,n Hh (µ) ��� H1(Ω) =: T1 + T2 + T3, (115) where ΨN1,n hh (µ) = N1 ∑ i=1 (Ψn h(µ), ζh i ) ζh i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' The first term T1 may be estimated using the inequality given by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='8, such that ��Ψ(tn)(µ) − Ψn h(µ) �� H1(Ω) ≤ C(µ) (h + ∆tF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (116) We then denote by S′ h = {Ψn h(µ, t), µ ∈ G, n = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' NT} the set of all the sensitivities .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' For our model problem, this manifold has a low complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' It means that for an accuracy ε = ε(N) related to the Kolmogorov N-width of the manifold S′ h, for any µ ∈ G, and any n ∈ 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , NT, T2 is bounded by ε which depends on the Kolmogorov N-width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' T2 = ������ Ψn h(µ) − N1 ∑ i=1 (Ψn h(µ), ζh i ) ζh i ������ H1(Ω) ≤ ε(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (117) Since (ζh i )i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=',N1 is a family of L2 and H1 orthogonalized RB functions (see [19] for only L2 orthonormalized RB functions) ���ΨN,n hh − ΨN,n Hh ��� 2 H1(Ω) = N1 ∑ i=1 |(Ψn h(µ) − � ΨH n(µ), ζh i )|2���ζh i ��� 2 H1(Ω) , (118) where � ΨH n(µ) is the quadratic interpolation of the coarse snapshots on time tn, ∀n = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , NT, defined by (81).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' From the RB orthonormalization in L2, the equation (105) yields ���ζh i ��� 2 H1 := ���∇ζh i ��� 2 L2(Ω) = λi ���ζh i ��� 2 L2(Ω) = λi ≤ max i=1,··· ,Nλi = λN, (119) such that the equation (118) leads to ���ΨN,n hh − ΨN,n Hh ��� 2 H1(Ω) ≤ CλN ���Ψn h(µ) − � ΨH n(µ) ��� 2 L2(Ω) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (120) Now by definition of � ΨH n(µ) and by corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='10 and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='7, for tn ∈ Im, ���Ψn h(µ) − � ΨH n(µ) ��� L2(Ω) ≤ C(µ)(H2 + ∆t2 G + h2 + ∆tF), (121) and we end up for equation (120) with ���ΨN,n hh − ΨN,n Hh ��� H1(Ω) ≤ C(µ) � λN(H2 + ∆t2 G + h2 + ∆tF), (122) where C(µ) does not depend on N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Combining these estimates (116), (117) and (122) concludes the proof and yields the estimate (114).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 27 Figure 1: H1 0 NIRB errors 5 Numerical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' In this section, we have applied the NIRB algorithms on several numerical tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' We have implemented both schemes (Euler and RK2) using FreeFem++ (version 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='9) [26] to compute the fine and coarse snapshots, and the solutions have been stored in VTK format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Then we have applied the plain NIRB and the NIRB rectified algorithms with python, in order to highlight the non-intrusive side of the two-grid method (as in [19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' After saving the NIRB approximations with Paraview module on Python, the errors have been computed with FreeFem++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='1 On the heat equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' We have solved (3) and (23) on the parameter set G = [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='5, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='5], with u0 solution of Poisson’s equation −∆u0 = f and Φ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' We have retrieved several snapshots on t = [0, 2] (note that the coarse time grid must belong to the interval of the fine one), and tried our algorithms on several size of meshes, always with ∆tF ≃ h and ∆tG ≃ H (both schemes are stables), and such that h = H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' We have taken 18 parameters in G for the RB construction such that µi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='5i, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , 19, i ̸= 2 and a reference solution to problem (92), with µ = 1 and its mesh and time step such that hre f ≃ ∆tF,re f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' In figure Figure 1, we present the errors of the FEM solutions and compare them to the one obtained with the NIRB algorithm with the rectification to observe the convergence rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' A Derivation of the adjoint for the heat equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' In this appendix, we recall the main steps to derive the adjoint of our model problem, in order to compute ( ∂F ∂µk )k=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=',P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' For a more general problem, we refer to [44] in case of FEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' 28 FEM H relative errors NIRB H relative error with H = V h 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='80- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='80- h h FEM coarse error NiRB+rectification 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='50 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='50- FEM fine error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='20 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='20 Error (log Error (log 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='05- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content='05 10-2 10-1 10-2 10-1 h (size of the fine mesh) h (size of the fine mesh)• We consider the Lagrangian formulation (86), denoted by L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Differentiating L with respect to the parameter µp, for p = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' , P, we obtain dL dµp (u, χ, ζ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' µ) = � T 0 � � Ω derr dµp (µ) dx + � χ, d[∇ · (A(µ)∇u) + f − ut] dµp �� ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (123) In our setting, the objective does not depend directly on the parameter µp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' The time and the parameter derivatives can commute ( d dt � du dµp � = d dµp � du dt � ), and since f is independent of µ, the term linked to f vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Therefore, using the chain rule, it may be rewritten dL dµp (u, χ, ζ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' µ) = � T 0 ��∂err ∂u , Ψp � + � χ, ∇ · ( ∂A ∂µp (µ)∇u) � + � χ, ∂[∇ · (A(µ)∇u)] ∂u Ψp � − � χ, Ψp,t � � �� � TIBP � ds, (124) where Ψp(t, x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' µ) := ∂u ∂µp (t, x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' As we saw before, a classical forward sensitivity computation would require P + 1 systems of PDEs to solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Here, we want to avoid calculating the sensitivities of the state variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' To do so, the strategy of the adjoint method is to factorize all the terms depending on Ψp, and to impose them to vanish by adequately choosing χ (which is arbitrary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' By IBP on TIBP, � T 0 � Ω χ · Ψp,t dx ds = � Ω � χ(T) · Ψp(T) − χ(0) · Ψp(0) � dx − � T 0 � Ω χt · Ψp dx ds , and choosing χ(T) = 0, and since in our example, u0 does not depend on µ, it yields dL dµp (u, χ, ζ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' µ) = � T 0 ��∂err ∂u , Ψp � + � χ, ∇ · ( ∂A ∂µp (µ)∇u) � + � χ, ∂[∇ · (A(µ)∇u)] ∂u Ψp � + � χt, Ψp �� ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Thus, we want the following term to vanish � T 0 � Ω �∂err ∂u · Ψp + χ∂[∇ · (A(µ)∇u)] ∂u Ψp � �� � TGF +χt · Ψp � dx ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (125) Now, applying Green’s formula twice, we have � T 0 � Ω TGF dx ds = � T 0 � Ω χ∇ · (A(µ)∇Ψp) dxds = � T 0 � − � Ω A(µ)∇χ · ∇Ψp dx + � ∂Ω A(µ)χ · ∇nΨp dσ � ds , = � T 0 � � Ω ∇ · (A(µ)∇χ) · Ψp dx − � ∂Ω A(µ)∇nχ · Ψp dσ + � ∂Ω A(µ)χ · ∇nΨp dσ � ds , with ∇n(·) the normal derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' Therefore, from the initial boundary conditions, since ∀t ≥ 0, ∀µ ∈ G, u(t) = 0 on ∂Ω, we also have Ψp(t) = 0 on ∂Ω and by imposing χ = 0 on ∂Ω, (125) becomes � T 0 � Ω ��∂err ∂u + ∇ · (A(µ)∇χ) + χt � Ψp � dxds = 0, and this equation leads us to the following adjoint state problem � � � � � � � � � � � Find χ ∈ V such that χt = − ∂err ∂u − ∇ · (A(µ)∇χ), in Ω × [0, T[, χ(x, T) = 0, in Ω, χ(x, t) = 0, on ∂Ω × [0, T[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQf2_mV/content/2301.00761v1.pdf'} +page_content=' (126) 29 Acknowledgment This work is supported by the SPP2311 program.' 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b/9dAzT4oBgHgl3EQf-_6e/content/tmp_files/2301.01942v1.pdf.txt @@ -0,0 +1,835 @@ +Compact and scalable polarimetric self- +coherent receiver using dielectric metasurface +GO SOMA,1,4 YOSHIRO NOMOTO,2 TOSHIMASA UMEZAWA,3 YUKI YOSHIDA,3 +YOSHIAKI NAKANO,1 AND TAKUO TANEMURA1,5 +1School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, Japan +2Central Research Laboratory, Hamamatsu Photonics K.K. 5000 Hirakuchi, Hamakita-ku, Hamamatsu +City, Shizuoka, Japan +3National Institute of Information and Communications Technologies (NICT), 4-2-1 Nukui-Kitamachi, +Koganei, Tokyo, Japan +4soma@hotaka.t.u-tokyo.ac.jp, 5tanemura@ee.t.u-tokyo.ac.jp +Abstract: The polarimetric self-coherent system using a direct-detection-based Stokes-vector +receiver (SVR) is a promising technology to meet both the cost and capacity requirements of +the short-reach optical interconnects. However, conventional SVRs require a number of optical +components to detect the state of polarization at high speed, resulting in substantially more +complicated receiver configurations compared with the current intensity-modulation-direct- +detection (IMDD) counterparts. Here, we demonstrate a simple and compact polarimetric self- +coherent receiver based on a thin dielectric metasurface and a photodetector array (PDA). With +a single 1.05-m-thick metasurface device fabricated on a compact silicon-on-quartz chip, we +implement functionalities of all the necessary passive components: a 1×3 splitter, three +polarization beam splitters with different polarization bases, and six focusing lenses. Combined +with a high-speed PDA, we demonstrate self-coherent transmission of 20-GBd 16-ary +quadrature amplitude modulation (16QAM) and 50-GBd quadrature phase-shift keying +(QPSK) signals over a 25-km single-mode fiber. Owing to the surface-normal configuration, it +can easily be scaled to receive spatially multiplexed channels from a multicore fiber or a fiber +bundle, enabling compact and low-cost receiver modules for the future highly parallelized self- +coherent systems. + +1. Introduction +Rapid spread of cloud computing, high-vision video streaming, and 5G mobile services has led +to a steady increase in information traffic in the datacenter interconnects and access networks +[1]. While intensity-modulation direct-detection (IMDD) formats such as 4-level pulse +amplitude modulation (PAM4) are employed in the current short-reach optical links, scaling +these IMDD-based transceivers beyond Tb/s is challenging due to the limited spectral +efficiency and severe signal distortion caused by the chromatic dispersion of fibers. On the +other hand, the digital coherent systems used in metro and long-haul networks can easily +expand the capacity by utilizing the full four-dimensional signal space of light and complete +compensation of linear impairments through digital signal processing (DSP). However, +substantially higher cost, complexity, and power consumption of coherent transceivers have +hindered their deployment in short-reach optical interconnects and access networks. +To address these issues, the self-coherent transmission scheme has emerged as a promising +approach that bridges the gap between the conventional IMDD and coherent systems [2-7]. In +this scheme, a continuous-wave (CW) tone is transmitted together with a high-capacity +coherent signal, which are mixed at a direct-detection-based receiver to recover the complex +optical field of the signal. Unlike the full coherent systems, this scheme eliminates the need for +a local oscillator (LO) laser at the receiver side as well as the stringent requirement of using +wavelength-tuned narrow-linewidth laser sources, suggesting that substantially low-cost broad- + +linewidth uncooled lasers can be used [4]. In addition, since the impacts of laser phase noise +and frequency offsets are mitigated, the computational cost of DSP can be reduced significantly +[7, 8]. The self-coherent systems thus enable low-cost, low-power-consumption, yet high- +capacity data transmission, required in the future datacenter interconnects and access networks. +Among several variations of implementing self-coherent systems, the polarimetric scheme +using a Stokes-vector receiver (SVR) [9-11] has an advantage in terms of simplicity. In this +scheme, the coherent signal is transmitted on a single polarization state, together with a CW +tone on the orthogonal polarization state. By retrieving the Stokes parameters 𝐒 = [𝑆1, 𝑆2, 𝑆3] +at the receiver side, the in-phase-and-quadrature (IQ) signal is demodulated through the DSP +after compensating for the effects of polarization rotation, chromatic dispersion, and other +signal distortions. To date, a number of high-speed polarimetric self-coherent transmission +experiments have been reported, where the SVRs were implemented using off-the-shelf +discrete components [3, 10-12]. Toward practical use, integrated waveguide-based SVRs were +also realized on Si [13, 14] and InP [15-18]. More recently, surface-normal SVRs were +demonstrated using nanophotonic circuits [19, 20] and liquid crystal gratings [21] with external +photodetectors (PDs). Compared with the conventional low-cost IMDD receivers, however, +these devices still suffer from a large fiber-to-chip coupling loss and/or need for external lenses +to focus light to PDs. +In this paper, we demonstrate high-speed polarimetric self-coherent signal detection using +a compact surface-normal SVR, composed of a metasurface-based polarization-sorting device +and a high-speed two-dimensional photodetector array (2D-PDA). A metasurface is a two- +dimensional array of subwavelength structures that can locally change the intensity, phase, and +polarization of input light [22]. Unlike the previous works on metasurface-based polarimeters +for imaging and sensing applications [23-27], our device enables efficient coupling of a self- +coherent optical signal from a single-mode fiber (SMF) and lens-less focusing to six high-speed +PDs. More specifically, by superimposing three types of meta-atom arrays, it implements the +functionalities of all the necessary passive components, namely a 1×3 splitter, three polarization +beam splitters (PBSs) with different polarization bases, and six lenses, inside a single ultrathin +device. Combined with an InP/InGaAs-based 2D-PDA chip, we demonstrate penalty-free +transmission of polarimetric self-coherent signals over a 25-km SMF in various formats such +as 20-GBd 16-ary quadrature amplitude modulation (16QAM) and 50-GBd quadrature phase- +shift keying (QPSK). Owing to the surface-normal configuration with the embedded focusing +functionality, highly efficient lens-free coupling to the 2D-PDA is achieved. The demonstrated +SVR, therefore, has a comparable complexity as a conventional low-cost IMDD receiver that +fits in a compact receiver optical subassembly (ROSA). Moreover, it can readily be extended +to receive spatially multiplexed channels from a multicore fiber (MCF) or a fiber bundle, which +are expected in the future >Tb/s highly parallelized optical interconnects [28-31]. + +2. Device concept +The schematic of the proposed surface-normal SVR is illustrated in Fig. 1(a). The light from +an SMF is incident to a thin metasurface-based polarization-sorting device, which is designed +to provide the same functionality as a conventional polarimeter shown in the inset. Namely, it +splits the light into three paths, resolves each of them to the orthogonal components in three +different polarization bases, and focuses them to six PDs integrated on a 2D-PDA chip. Unlike +previously demonstrated metasurface-based polarimeters [23-27], our proposed SVR +implements the 13 splitter and six metalenses as well to enable direct coupling from an SMF +to a high-speed 2D-PDA. As a result, the entire device can fit inside a compact ROSA module, +comparable to the current IMDD receivers. Moreover, owing to the surface-normal +configuration, this scheme can easily be scaled to receive multiple spatial channels without +increasing the number of components by simply replacing the input SMF to a MCF or a fiber +bundle and using a larger-scale PDA [32] as shown in Fig. 1(b). + +To enable three operations in parallel using a single metasurface layer, we adopt the spatial +multiplexing method [33, 34]; three independently designed meta-atom arrays are +superimposed as shown by MA1 (red), MA2 (blue), and MA3 (green) in Fig. 1(c). The phase +profile 𝜑(𝑥, 𝑦) of MA1 is designed to focus the x-polarized component of light to PDx and the +y-polarized component to PDy at the focal plane as shown in the inset. Similarly, MA2 and +MA3 function as PBSs with embedded metalenses for the ±45° polarization basis (a/b) and the +right/left-handed circular (RHC/LHC) polarization basis (r/l), respectively, and focus +respective components to PDa,b and PDr,l. The Stokes vector 𝐒 ≡ (𝑆1, 𝑆2, 𝑆3)𝑇 can then be +derived by taking the difference of the photocurrent signals as 𝑆1 = 𝐼x − 𝐼y, 𝑆2 = 𝐼a − 𝐼b, and +𝑆3 = 𝐼r − 𝐼l, where 𝐼p is the photocurrent at PDp. We should note that this scheme with three +balanced PDs without polarizers offers the maximum receiver sensitivity among various SVR +configurations [35] and is advantageous compared with the previous demonstrations that +employ a non-optimal polarization basis [19-21]. + + + +Fig. 1. Surface-normal SVR based on superimposed meta-atom arrays. (a) Schematic illustration +of the receiver module. A single metasurface device implements all the necessary passive optical +components of the equivalent circuit as shown in the inset. MS: metasurface. PDA: +photodetector array. IC: integrated circuit. HWP: half-wave plate. QWP: quarter-wave plate. +PBS: polarization beam splitter. (b) Scalable configuration to receive multiple input channels +from a MCF or fiber bundle. (c) Functionality and configuration of the designed metasurface. +The incident light from the SMF is split into three paths and focused to six PDs located at +different positions according to the input state of polarization. The superimposed meta-atom +arrays (MA1, 2, and 3) operate as PBSs and metalenses for x/y linear, ±45° linear, and RHC/LHC +polarization bases, respectively. + + + +ROSA +MS +2D-PDA +Atfocal plane +2D-PDA +PDr +PD, +S +PDp +PDx +PDa +PDy +Si +K +K +Quartz +MS +K +MA1 +PMA2 +MA3 +ROSA +MS +2D-PDA +ICF +r +ber +ndle +a3. Metasurface design and fabrication +As the dielectric metasurface, we employ 1050-nm-high elliptical Si nanoposts on a quartz +layer. The phase of the transmitted light and its polarization dependence can be controlled by +changing the lengths of two principal axes (𝐷𝑢, 𝐷𝑣) and the in-plane rotation angle θ of each +nanopost as defined in Fig. 2(a) [22]. Here, in each meta-atom array, MA1-3, we adopt the +triangular lattice with a sub-wavelength lattice constant of Λ = 700√3 nm, so that the non- +zero-order diffraction is prohibited. Then, three meta-atom arrays are superimposed by shifting +their positions by 𝑎 = 700 nm to form the overall metasurface, as shown in Fig. 1(c). +First, we set θ to 0 and simulate the transmission characteristics of uniform nanopost array +for the x- and y-polarized light at a wavelength of 1550 nm by the rigorous coupled-wave +analysis (RCWA) method [36]. From the simulated results, we first derive 𝑡𝑢(𝐷𝑢, 𝐷𝑣) and +𝑡𝑣(𝐷𝑢, 𝐷𝑣), which denote the complex transmittance for the x- and y-polarized light as a function +of 𝐷𝑢 and 𝐷𝑣. Then, we derive the required (𝐷𝑢, 𝐷𝑣) that provides a phase shift of (𝜑𝑢, 𝜑𝑣) for +each polarization component. The results are plotted in Fig. 2(b) (see Section S1 of Supplement 1 +for details). The amplitude of transmittance for each case is also shown in Fig. 2(c). We can confirm +that by setting the dimensions of the ellipse appropriately, arbitrary phase shifts for x- and y- +polarized components can be achieved with high transmittance. +By rotating the elliptical nanoposts by θ as shown in Fig. 2(a), such birefringence can be applied +to any linear polarization basis oriented at an arbitrary angle [37]. We should note that the phase +shifts and amplitudes of transmission are nearly insensitive to θ [22] and similar results as +shown in Fig. 2(b) and 2(c) are obtained for all θ. This is because the light is strongly confined +inside each Si nanopost, so that the optical coupling among neighboring meta-atoms has only +minor influence on the transmission. +We can also provide arbitrary phase shifts to orthogonal circular-polarization states by using the +geometric phase shift of meta-atoms [38]. First, we judiciously select 𝐷𝑢 and 𝐷𝑣 to satisfy 𝜑𝑣 = +𝜑𝑢 + 𝜋, so that each nanopost operates as a half-wave plate. In this case, input RHC and LHC states +are converted to LHC and RHC, respectively. In addition, their phases after transmission are written +as (𝜑𝑟, 𝜑𝑙) = (𝜑𝑢 + 2𝜃, 𝜑𝑢 − 2𝜃) (see Section S2 of Supplement 1 for the derivation). Therefore, +𝐷𝑢 and 𝐷𝑣 of each nanopost are selected to obtain desired 𝜑𝑢 (=(𝜑𝑟 + 𝜑𝑙)/2) while satisfying the +condition 𝜑𝑣 = 𝜑𝑢 + 𝜋. The angle 𝜃 is also determined to be (𝜑𝑟 − 𝜑𝑙)/4. +To realize the function of a metalens, each meta-atom array needs to impart a spatially +dependent phase profile given as [39] + +𝜑(𝑥, 𝑦) = − +2𝜋 +𝜆 (√(𝑥 − 𝑥0)2 + (𝑦 − 𝑦0)2 + 𝑓2 − 𝑓), +(1) +where (𝑥0, 𝑦0) is the in-plane position of the focal point, 𝑓 is the focal length, and 𝜆 is the +operating wavelength. In this work, we set 𝜆 = 1550 nm, 𝑓 = 10 mm, and the diameter of the +entire metasurface area to be 2 mm, corresponding to the numerical aperture (NA) of ~0.10. +The six focal points are arranged on a regular hexagon with a spacing of 60 µm, which are +matched to the positions of the high-speed 2D-PDA used in our self-coherent experiments. +Under these conditions, the phase profiles required for MA1, 2, and 3 are determined as shown +in Fig. 2(d). Note that a rather large (2 mm) metasurface is used in this work due to the limitation +in reducing the focal length 𝑓 in the current optical setup. In a fully packaged module as shown +in Fig. 1(a), we can readily shrink the entire area of the metasurface to a few tens of micrometers +by reducing 𝑓 and designing the geometrical parameters of each nanopost to satisfy the required +phase profiles given by Eq. (1). +The designed metasurface was fabricated using a silicon-on-quartz (SOQ) substrate with a +1050-nm-thick Si layer. The nanopost patterns were defined by electron-beam lithography with +ZEP520A resist. Then, the patterns were transferred to the Si layer by inductively-coupled- +plasma reactive-ion etching (ICP-RIE) using SF6, C4F8, and O2. An optical microscope image +and scanning electron microscopy (SEM) images of the fabricated metasurface are shown in +Fig. 2(e)-(g). + + + +Fig. 2. Metasurface design and fabrication. (a) Schematic of a periodic array of Si nano-posts +placed at the vertices of a triangular lattice with a lattice constant 𝑎 of 700 nm. The transmission +of x- and y-polarized light is simulated for various axes lengths (𝐷𝑢, 𝐷𝑣) of the elliptical posts. +(b) Required (𝐷𝑢, 𝐷𝑣) to obtain phase shifts (𝜑𝑢, 𝜑𝑣) for x- and y-polarized light. For ease of +fabrication, the ranges of 𝐷𝑢 and 𝐷𝑣 are limited from 100 nm to 650 nm. (c) The amplitude of +transmittance for each case in (b) as a function of (𝜑𝑢, 𝜑𝑣). (d) Required phase profiles for MA1, +2, and 3. (e) Optical microscope image and (f, g) SEM images of the fabricated device. In (g), +the image is false-colored to distinguish MA1, 2, and 3. + +4. Static characterization of the fabricated metasurface +We first characterized the fabricated metasurface by observing the intensity distribution at the +focal plane for various input states of polarization (SOPs). The experimental setup is shown in +Fig. 3(a). A CW light with a wavelength of 1550 nm was incident to the metasurface. The SOP +was modified by rotating a half-wave plate (HWP) and a quarter-wave plate (QWP). The image +at the focal plane was magnified at 50 times by a 4-f lens system and captured by an InGaAs +camera. From the detected intensity values at the six focal positions, the Stokes vector was +retrieved as described in Section 2. To enable quantitative measurement of the focused power, +we inserted a flip mirror and detected the total power by a bucket PD after spatially filtering +the focused beam at each target position using an iris. +Figure 3(b) shows the observed intensity distributions when the input Stokes vector is set +to (±1, 0, 0), (0, ±1, 0), and (0, 0, ±1). We can confirm that the incident light is focused to the +six well-defined points by transmitting through the metasurface. Moreover, its intensity +distribution changes with the SOP; x/y linear, 45 linear, and RHC/LHC components of light +are focused to the designed positions as expected. Figure 3(c) shows the retrieved Stokes + + + = 0 0 + = 700 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + 2 + 2 + + + + + + + + + + + + + + + + + + , + , + , + , + , + , + + + + + + +Raith +Mag-20.00KX +SE2 +EHT = 10.00KV +WD-17.$mmRaith +Mag +EHT = 10.00 KV +0=20.2mmvectors on the Poincaré sphere. The average error 〈|Δ𝐒|〉 is as small as 0.028. Figure 3(d) shows +the measured focusing efficiencies to the six positions. Subtracting the 4.8-dB intrinsic loss due +to the 13 splitter [see Fig. 1(a)], the excess loss is around 6.1 dB, whereas the crosstalk to the +orthogonal PD position is suppressed by 13-20 dB. While this excess loss is already comparable +to the coupling and propagation losses of the previously reported waveguide-based SVRs [13- +18], we expect further improvement by applying anti-reflection coating at the silica surface, +improving the fabrication processes to minimize the errors, and by adopting advanced +algorithms in designing the metasurface that take into account the nonzero interactions between +adjacent meta-atoms [40, 41]. + + + +Fig. 3. Experimental characterization of the fabricated metasurface. (a) Schematic of the optical +setup. The flip mirror is used to switch between capturing the intensity distribution at the focal +plane and measuring the power of each focused beam. TLS: tunable laser source. PC: +polarization controller. VOA: variable optical attenuator. FC: fiber collimator. Pol.: polarizer. +HWP: half-wave plate. QWP: quarter-wave plate. MS: metasurface. M: flip mirror. PD: +photodetector. (b) Measured intensity distributions at the focal plane for different input SOPs. +(c) Retrieved and input Stokes vectors on the Poincaré sphere. (d) Measured focusing efficiency +to each PD. The intrinsic loss due to splitting into three paths is shown by a green line. The input +polarization is labeled on the top of each bar. + +5. Self-coherent signal transmission experiment +We then performed the polarimetric self-coherent signal transmission experiment using the +fabricated metasurface. The experimental setup is shown in Fig. 4. We employed a 19-pixel +2D-PDA with InP/InGaAs-based p-i-n structure [42], from which six PDs were used as shown +in Fig. 4(b). Each PD had a diameter of 30 µm, the measured bandwidth above 10 GHz, and +the responsivity of 0.3 A/W. The 2D-PDA chip was packaged with the radio-frequency (RF) +coaxial connectors connected to each PD. The 2D-PDA was placed at the focal distance of 10 +mm from the metasurface as shown in Fig. 4(c). This distance was merely limited by the current +setup and should be reduced to a sub-millimeter scale in a practical fully packaged module, +which can be comparable to current IMDD receiver modules. +A CW light at a wavelength of 1550 nm was generated from a tunable laser source (TLS) +and split into two ports, which served as the signal and the pilot tone ports. At the signal port, +a LiNbO3 IQ modulator was used to generate a high-speed coherent optical signal. The Nyquist +filter was applied to the driving electrical signals from an arbitrary waveform generator (AWG). +The modulated optical signal was then combined with the pilot tone by a polarization beam +combiner (PBC). The optical power of the pilot tone was adjusted by a variable optical + + + + + + + + + + + + + + + + + + + + 2 + 3 + + + + + + + + + + + + + + + + + + + + +| +| +| +| +| +| +| +| +| +| +| +| + + +attenuator (VOA), so that their powers were nearly balanced. The self-coherent signal was then +transmitted over a 25-km SMF. At the receiver side, the optical signal-to-noise ratio (OSNR) +was controlled using another VOA, followed by an erbium-doped fiber amplifier (EDFA) and +an optical bandpass filter (OBPF). The electrical signals from the six PDs of the PDA were +amplified by differential RF amplifiers and then captured by a real-time oscilloscope (OSC). +At a baudrate beyond 20 GBd, we could not use the balanced PD (B-PD) configuration due to +the residual skew inside the PDA module. In these cases, we employed four single-ended PDs +(S-PDs), where the electrical signals from PDx, PDy, PDa, and PDl were independently captured +by a four-channel real-time oscilloscope, so that the skew could be calibrated during DSP. By +comparing the results using two configurations, the use of four S-PDs was validated (see +Section S3 of Supplement 1 for details). To equalize and reconstruct the original IQ signal, we +employed offline DSP with the 2×3 and 2×4 real-valued multi-input-multi-output (MIMO) +equalizers [43, 44] for three-B-PD and four-S-PD configurations, respectively. +Figures 5(a)-(c) show the BER curves and the constellations for 15-GBd 16QAM signals, +measured using the three-B-PD configuration. We can confirm that BERs well below the hard- +decision forward error correction (HD-FEC) threshold are obtained with a negligible penalty +even after 25-km transmission. Figures 5(d)-(f) show the results for 20-GBd 16QAM and 50- +GBd QPSK signals, measured by the four-S-PD configuration. Once again, BERs below the +HD-FEC threshold are obtained. Finally, Fig. 6 shows the measured BER curves and +constellation diagrams of 15-GBd 16QAM signal at 1540-nm and 1565-nm wavelengths, +demonstrating the wideband operation of our designed metasurface. While the baudrate in this +work was limited by the bandwidth of the 2D-PDA, beyond-100-GBd transmission should be +possible by using higher-speed surface-normal PDs with bandwidth exceeding 50 GHz [45, 46]. + + + +Fig. 4. Self-coherent transmission experiment using the fabricated metasurface and 2D-PDA. (a) +Experimental setup. AWG: arbitrary waveform generator. PBC: polarization beam combiner. +EDFA: erbium-doped fiber amplifier. OBPF: optical bandpass filter. Osc: oscilloscope. In the +insets, three-B-PD and four-S-PD configurations are depicted. (b) Optical microscope image of +the fabricated 19-pixel 2D-PDA. The six circled PDs were used in this experiment. (c) +Photograph of the receiver. + + +MS +2D-PDA +(a) +TLS +IQ mod. +PBC +AWG +FC +1:9 +coupler +EDFA OBPF +VOA +VOA +Real-time Osc +MS +2D-PDA +(c) +diff. Amp. +Real-time Osc +4 S-PDs +3 B-PDs +25 km +PDa +PDb +PDr +PDl +PDx +PDy +100 μ +(b) + + +Fig. 5. Experimental results of self-coherent signal transmission at a wavelength of 1550 nm. +(a)-(c) Measured BER curves and constellation diagrams of 15-GBd 16QAM signals before +(b2b) and after 25-km transmission using the three-B-PD configuration. (d)-(f) Measured BER +curves and constellation diagrams of 20-GBd 16QAM and 50-GBd QPSK signals after 25-km +transmission using the four-S-PD configuration. + + +Fig. 6. Experimental results of self-coherent signal transmission at wavelengths of (a) 1540 nm +and (b) 1565 nm. (a, b) Measured BER curves of 15-GBd 16QAM signals before (b2b) and after +25-km transmission using the three-B-PD configuration. The insets represent the retrieved +constellation diagrams. + +6. Conclusion +We have proposed and demonstrated a surface-normal SVR using a dielectric metasurface and +2D-PDA for the high-speed polarimetric self-coherent systems. Three independently designed +meta-atom arrays based on Si nanoposts were superimposed onto a single thin metasurface +layer to implement both the polarization-sorting and focusing functions simultaneously. Using +a compact metasurface chip fabricated on a SOQ substrate, we demonstrated 25-km +transmission of 20-GBd 16QAM and 50-GBd QPSK self-coherent signals. The operating +baudrate was merely limited by the 2D-PDA, so that higher-capacity transmission should be +possible by using a PDA with a broader bandwidth. Owing to the unique surface-normal +configuration with the embedded lens array functionality, a compact receiver module with +comparable size and complexity as the conventional IMDD receivers can be realized. Moreover, +it can easily be extended to receive spatially multiplexed channels by simply replacing the SMF +16 +18 +20 +22 +24 +26 +28 +10-5 +10-4 +10-3 +10-2 +10-1 +OSNR (dB) +BER +10-5 +10-4 +10-3 +10-2 +10-1 +BER +16 +18 +20 +22 +24 +26 +28 +OSNR (dB) +50-GBd QPSK +20-GBd 16QAM +15-GBd 16QAM +(a) +(d) +25km +b2b +25 km +25 km +(b) +(c) +(e) +(f) +7% HD-FEC +7% HD-FEC +25 km +b2b +25 km +b2b +22 +24 +26 +28 +30 +10-5 +10-4 +10-3 +10-2 +10-1 +OSNR (dB) +BER +14 +16 +18 +20 +22 +10-3 +10-2 +10-1 +OSNR (dB) +BER +(a) +(b) +1540 nm +1565 nm +25 km +b2b +25 km +b2b + +to a MCF and employing a larger-scale integrated PDA technology [32]. This work would, +therefore, pave the way toward realizing cost-effective receivers for the future >Tb/s spatially +multiplexed optical interconnects. +Funding. National Institute of Information and Communications Technology (NICT). +Acknowledgments. This work was obtained from the commissioned research 03601 by National Institute of +Information and Communications Technology (NICT), Japan. Portions of this work were presented at the Optical Fiber +Communications Conference (OFC) in 2022, M4J.5. A part of the device fabrication was conducted at the cleanroom +facilities of d.lab in the University of Tokyo, supported by MEXT Nanotechnology Platform, Japan. The authors also +thank all the technical staff at Advanced ICT device laboratory in NICT for supporting the PDA device fabrication. +G.S. acknowledges the financial support from Optics and Advanced Laser Science by Innovative Funds for Students +(OASIS) and World-leading Innovative Graduate Study Program - Quantum Science and Technology Fellowship +Program (WINGS-QSTEP). +References +1. +Cisco annual internet report (2018-2023), Cisco white paper. (2020). +2. +W. Shieh and H. Ji, “Advanced direct detection schemes,” in Proc. Opt. Fiber Commun. Conf. 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Lightwave Technol. 34, 3138–3147 (2016). + +Compact and scalable polarimetric self- +coherent receiver using dielectric +metasurface: Supplementary Information +GO SOMA,1,4 YOSHIRO NOMOTO,2 TOSHIMASA UMEZAWA,3 YUKI YOSHIDA,3 +YOSHIAKI NAKANO,1 AND TAKUO TANEMURA1,5 +1School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, Japan +2Central Research Laboratory, Hamamatsu Photonics K.K. 5000 Hirakuchi, Hamakita-ku, Hamamatsu +City, Shizuoka, Japan +3National Institute of Information and Communications Technologies (NICT), 4-2-1 Nukui-Kitamachi, +Koganei, Tokyo, Japan +4soma@hotaka.t.u-tokyo.ac.jp, 5tanemura@ee.t.u-tokyo.ac.jp + +S1. Derivation of 𝑫𝒖(𝝋𝒖, 𝝋𝒗) and 𝑫𝒗(𝝋𝒖, 𝝋𝒗) +From RCWA simulation, we obtain 𝑡𝑢(𝐷𝑢, 𝐷𝑣) and 𝑡𝑣(𝐷𝑢, 𝐷𝑣), which describe the complex +transmittance through an array of elliptical Si nanoposts with the principal axis lengths of 𝐷𝑢 +and 𝐷𝑣 for the linearly polarized light along 𝑢 and 𝑣 axes, respectively. The simulated intensity +|𝑡𝑢|2 and |𝑡𝑣|2 and the phase arg(𝑡𝑢) and arg(𝑡𝑣) are shown in Fig. S1(a) and (b). From these +results, we derive the required 𝐷𝑢(𝜑𝑢, 𝜑𝑣) and 𝐷𝑣(𝜑𝑢, 𝜑𝑣) to obtain desired phase shifts +(𝜑𝑢, 𝜑𝑣), by using the following the equation [1]: +(𝐷𝑢(𝜑𝑢, 𝜑𝑣), 𝐷𝑣(𝜑𝑢, 𝜑𝑣)) = arg min +(𝐷𝑢, 𝐷𝑣) +[|𝑡𝑢(𝐷𝑢, 𝐷𝑣) − 𝑒𝑖𝜑𝑢| +2 + |𝑡𝑣(𝐷𝑢, 𝐷𝑣) − 𝑒𝑖𝜑𝑣| +2]. + + +Fig. S1. Simulated intensity (a) and phase (b) of transmission coefficients as a function of 𝐷𝑢 +and 𝐷𝑣. +0 +0.7 +(μm) +0 +0.7 +(μm) +0 +1 +0 +0.7 +0 +0.7 +0 +0.7 +0 +0.7 +0 +0.7 +0 +0.7 +(μm) +(μm) +(μm) +(μm) +(μm) +(μm) +Transmittance +Phase (rad) +(a) +(b) + +S2. Derivation of phase shift by a meta-atom array for circularly polarized light +The Jones matrix, describing the transmittance through a lossless Si nanopost array can be +written as +𝐉 = 𝐑(𝜃) (𝑒𝑖𝜑𝑢 +0 +0 +𝑒𝑖𝜑𝑣) 𝐑(−𝜃) = (𝑒𝑖𝜑𝑢 cos2 𝜃 + 𝑒𝑖𝜑𝑣 sin2 𝜃 +(𝑒𝑖𝜑𝑢 − 𝑒𝑖𝜑𝑣) sin 𝜃 cos 𝜃 +(𝑒𝑖𝜑𝑢 − 𝑒𝑖𝜑𝑣) sin 𝜃 cos 𝜃 +𝑒𝑖𝜑𝑢 sin2 𝜃 + 𝑒𝑖𝜑𝑣 cos2 𝜃). +Here, 𝜑𝑢 and 𝜑𝑣 represent the phase shifts for the polarization components along the principal +axes of the elliptical nanoposts and 𝐑(𝜃) is a rotation matrix with a rotation angle of 𝜃. Here +we assume that the input lightwave is circularly-polarized and its Jones vector is written as +𝑬𝑟,𝑙 = 1/√2(1, ±𝑖)𝑇. Then, the output Jones vector is written as +𝐉𝑬𝑟,𝑙 = 𝑒𝑖𝜑𝑢 + 𝑒𝑖𝜑𝑣 +2 + 𝑬𝑟,𝑙 + 𝑒𝑖𝜑𝑢 − 𝑒𝑖𝜑𝑣 +2 + 𝑒±𝑖2𝜃𝑬𝑙,𝑟. +Therefore, when the meta-atom functions as a half-wave plate, i.e., 𝜑𝑣 = 𝜑𝑢 + 𝜋, the output +Jones vector becomes 𝑒𝑖(𝜑𝑢±2𝜃)𝑬𝑙,𝑟. In other words, the phase shifts given to the right-handed +and left-handed circularly-polarized waves are (𝜑𝑟, 𝜑𝑙) = (𝜑𝑢 + 2𝜃, 𝜑𝑢 − 2𝜃) , while the +output polarization handedness is reversed. + +S3. Comparison between three-B-PD and four-S-PD configurations +To compare the results obtained by three-B-PD and four-S-PD configurations, we performed +the self-coherent transmission experiment with 15-GBd 16QAM signals using the two +experimental setups shown in Fig. 4(a). The measured BER curves are shown in Fig. S2. Since +the BER was limited by the optical signal-to-noise ratio (OSNR), identical results were +obtained in two cases. + + +Fig. S2. Measured BER curves of 15-GBd 16QAM signals using the setup with three-B-PD and +four-S-PD configurations. + +References +1. +A. Arbabi, Y. Horie, M. Bagheri, and A. Faraon, “Dielectric metasurfaces for complete control of phase and +polarization with subwavelength spatial resolution and high transmission,” Nat. Nanotechnol. 10, 937–943 +(2015). + + + + + + + + + + + + + + + + diff --git a/9dAzT4oBgHgl3EQf-_6e/content/tmp_files/load_file.txt b/9dAzT4oBgHgl3EQf-_6e/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..890e1213785cf609b0fb1e53aa3a929ec3bc4585 --- /dev/null +++ b/9dAzT4oBgHgl3EQf-_6e/content/tmp_files/load_file.txt @@ -0,0 +1,787 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf,len=786 +page_content='Compact and scalable polarimetric self- coherent receiver using dielectric metasurface GO SOMA,1,4 YOSHIRO NOMOTO,2 TOSHIMASA UMEZAWA,3 YUKI YOSHIDA,3 YOSHIAKI NAKANO,1 AND TAKUO TANEMURA1,5 1School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, Japan 2Central Research Laboratory, Hamamatsu Photonics K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' 5000 Hirakuchi, Hamakita-ku, Hamamatsu City, Shizuoka, Japan 3National Institute of Information and Communications Technologies (NICT), 4-2-1 Nukui-Kitamachi, Koganei, Tokyo, Japan 4soma@hotaka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content='u-tokyo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content='jp, 5tanemura@ee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content='u-tokyo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content='jp Abstract: The polarimetric self-coherent system using a direct-detection-based Stokes-vector receiver (SVR) is a promising technology to meet both the cost and capacity requirements of the short-reach optical interconnects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' However, conventional SVRs require a number of optical components to detect the state of polarization at high speed, resulting in substantially more complicated receiver configurations compared with the current intensity-modulation-direct- detection (IMDD) counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Here, we demonstrate a simple and compact polarimetric self- coherent receiver based on a thin dielectric metasurface and a photodetector array (PDA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' With a single 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content='05-\uf06dm-thick metasurface device fabricated on a compact silicon-on-quartz chip, we implement functionalities of all the necessary passive components: a 1×3 splitter, three polarization beam splitters with different polarization bases, and six focusing lenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Combined with a high-speed PDA, we demonstrate self-coherent transmission of 20-GBd 16-ary quadrature amplitude modulation (16QAM) and 50-GBd quadrature phase-shift keying (QPSK) signals over a 25-km single-mode fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Owing to the surface-normal configuration, it can easily be scaled to receive spatially multiplexed channels from a multicore fiber or a fiber bundle, enabling compact and low-cost receiver modules for the future highly parallelized self- coherent systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Introduction Rapid spread of cloud computing, high-vision video streaming, and 5G mobile services has led to a steady increase in information traffic in the datacenter interconnects and access networks [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' While intensity-modulation direct-detection (IMDD) formats such as 4-level pulse amplitude modulation (PAM4) are employed in the current short-reach optical links, scaling these IMDD-based transceivers beyond Tb/s is challenging due to the limited spectral efficiency and severe signal distortion caused by the chromatic dispersion of fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' On the other hand, the digital coherent systems used in metro and long-haul networks can easily expand the capacity by utilizing the full four-dimensional signal space of light and complete compensation of linear impairments through digital signal processing (DSP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' However, substantially higher cost, complexity, and power consumption of coherent transceivers have hindered their deployment in short-reach optical interconnects and access networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' To address these issues, the self-coherent transmission scheme has emerged as a promising approach that bridges the gap between the conventional IMDD and coherent systems [2-7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' In this scheme, a continuous-wave (CW) tone is transmitted together with a high-capacity coherent signal, which are mixed at a direct-detection-based receiver to recover the complex optical field of the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Unlike the full coherent systems, this scheme eliminates the need for a local oscillator (LO) laser at the receiver side as well as the stringent requirement of using wavelength-tuned narrow-linewidth laser sources, suggesting that substantially low-cost broad- linewidth uncooled lasers can be used [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' In addition, since the impacts of laser phase noise and frequency offsets are mitigated, the computational cost of DSP can be reduced significantly [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' The self-coherent systems thus enable low-cost, low-power-consumption, yet high- capacity data transmission, required in the future datacenter interconnects and access networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Among several variations of implementing self-coherent systems, the polarimetric scheme using a Stokes-vector receiver (SVR) [9-11] has an advantage in terms of simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' In this scheme, the coherent signal is transmitted on a single polarization state, together with a CW tone on the orthogonal polarization state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' By retrieving the Stokes parameters 𝐒 = [𝑆1, 𝑆2, 𝑆3] at the receiver side, the in-phase-and-quadrature (IQ) signal is demodulated through the DSP after compensating for the effects of polarization rotation, chromatic dispersion, and other signal distortions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' To date, a number of high-speed polarimetric self-coherent transmission experiments have been reported, where the SVRs were implemented using off-the-shelf discrete components [3, 10-12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Toward practical use, integrated waveguide-based SVRs were also realized on Si [13, 14] and InP [15-18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' More recently, surface-normal SVRs were demonstrated using nanophotonic circuits [19, 20] and liquid crystal gratings [21] with external photodetectors (PDs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Compared with the conventional low-cost IMDD receivers, however, these devices still suffer from a large fiber-to-chip coupling loss and/or need for external lenses to focus light to PDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' In this paper, we demonstrate high-speed polarimetric self-coherent signal detection using a compact surface-normal SVR, composed of a metasurface-based polarization-sorting device and a high-speed two-dimensional photodetector array (2D-PDA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' A metasurface is a two- dimensional array of subwavelength structures that can locally change the intensity, phase, and polarization of input light [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Unlike the previous works on metasurface-based polarimeters for imaging and sensing applications [23-27], our device enables efficient coupling of a self- coherent optical signal from a single-mode fiber (SMF) and lens-less focusing to six high-speed PDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' More specifically, by superimposing three types of meta-atom arrays, it implements the functionalities of all the necessary passive components, namely a 1×3 splitter, three polarization beam splitters (PBSs) with different polarization bases, and six lenses, inside a single ultrathin device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Combined with an InP/InGaAs-based 2D-PDA chip, we demonstrate penalty-free transmission of polarimetric self-coherent signals over a 25-km SMF in various formats such as 20-GBd 16-ary quadrature amplitude modulation (16QAM) and 50-GBd quadrature phase- shift keying (QPSK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Owing to the surface-normal configuration with the embedded focusing functionality, highly efficient lens-free coupling to the 2D-PDA is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' The demonstrated SVR, therefore, has a comparable complexity as a conventional low-cost IMDD receiver that fits in a compact receiver optical subassembly (ROSA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Moreover, it can readily be extended to receive spatially multiplexed channels from a multicore fiber (MCF) or a fiber bundle, which are expected in the future >Tb/s highly parallelized optical interconnects [28-31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Device concept The schematic of the proposed surface-normal SVR is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' The light from an SMF is incident to a thin metasurface-based polarization-sorting device, which is designed to provide the same functionality as a conventional polarimeter shown in the inset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Namely, it splits the light into three paths, resolves each of them to the orthogonal components in three different polarization bases, and focuses them to six PDs integrated on a 2D-PDA chip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Unlike previously demonstrated metasurface-based polarimeters [23-27], our proposed SVR implements the 1\uf0b43 splitter and six metalenses as well to enable direct coupling from an SMF to a high-speed 2D-PDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' As a result, the entire device can fit inside a compact ROSA module, comparable to the current IMDD receivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Moreover, owing to the surface-normal configuration, this scheme can easily be scaled to receive multiple spatial channels without increasing the number of components by simply replacing the input SMF to a MCF or a fiber bundle and using a larger-scale PDA [32] as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' To enable three operations in parallel using a single metasurface layer, we adopt the spatial multiplexing method [33, 34];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' three independently designed meta-atom arrays are superimposed as shown by MA1 (red), MA2 (blue), and MA3 (green) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' 1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' The phase profile 𝜑(𝑥, 𝑦) of MA1 is designed to focus the x-polarized component of light to PDx and the y-polarized component to PDy at the focal plane as shown in the inset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Similarly, MA2 and MA3 function as PBSs with embedded metalenses for the ±45° polarization basis (a/b) and the right/left-handed circular (RHC/LHC) polarization basis (r/l), respectively, and focus respective components to PDa,b and PDr,l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' The Stokes vector 𝐒 ≡ (𝑆1, 𝑆2, 𝑆3)𝑇 can then be derived by taking the difference of the photocurrent signals as 𝑆1 = 𝐼x − 𝐼y, 𝑆2 = 𝐼a − 𝐼b, and 𝑆3 = 𝐼r − 𝐼l, where 𝐼p is the photocurrent at PDp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' We should note that this scheme with three balanced PDs without polarizers offers the maximum receiver sensitivity among various SVR configurations [35] and is advantageous compared with the previous demonstrations that employ a non-optimal polarization basis [19-21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Surface-normal SVR based on superimposed meta-atom arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' (a) Schematic illustration of the receiver module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' A single metasurface device implements all the necessary passive optical components of the equivalent circuit as shown in the inset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' MS: metasurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' PDA: photodetector array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' IC: integrated circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' HWP: half-wave plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' QWP: quarter-wave plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' PBS: polarization beam splitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' (b) Scalable configuration to receive multiple input channels from a MCF or fiber bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' (c) Functionality and configuration of the designed metasurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' The incident light from the SMF is split into three paths and focused to six PDs located at different positions according to the input state of polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' The superimposed meta-atom arrays (MA1, 2, and 3) operate as PBSs and metalenses for x/y linear, ±45° linear, and RHC/LHC polarization bases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' ROSA MS 2D-PDA Atfocal plane 2D-PDA PDr PD, S PDp PDx PDa PDy Si K K Quartz MS K MA1 PMA2 MA3 ROSA MS 2D-PDA ICF r ber ndle a3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Metasurface design and fabrication As the dielectric metasurface, we employ 1050-nm-high elliptical Si nanoposts on a quartz layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' The phase of the transmitted light and its polarization dependence can be controlled by changing the lengths of two principal axes (𝐷𝑢, 𝐷𝑣) and the in-plane rotation angle θ of each nanopost as defined in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' 2(a) [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Here, in each meta-atom array, MA1-3, we adopt the triangular lattice with a sub-wavelength lattice constant of Λ = 700√3 nm, so that the non- zero-order diffraction is prohibited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Then, three meta-atom arrays are superimposed by shifting their positions by 𝑎 = 700 nm to form the overall metasurface, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' 1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' First, we set θ to 0 and simulate the transmission characteristics of uniform nanopost array for the x- and y-polarized light at a wavelength of 1550 nm by the rigorous coupled-wave analysis (RCWA) method [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' From the simulated results, we first derive 𝑡𝑢(𝐷𝑢, 𝐷𝑣) and 𝑡𝑣(𝐷𝑢, 𝐷𝑣), which denote the complex transmittance for the x- and y-polarized light as a function of 𝐷𝑢 and 𝐷𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Then, we derive the required (𝐷𝑢, 𝐷𝑣) that provides a phase shift of (𝜑𝑢, 𝜑𝑣) for each polarization component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' The results are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' 2(b) (see Section S1 of Supplement 1 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' The amplitude of transmittance for each case is also shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' 2(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' We can confirm that by setting the dimensions of the ellipse appropriately, arbitrary phase shifts for x- and y- polarized components can be achieved with high transmittance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' By rotating the elliptical nanoposts by θ as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' 2(a), such birefringence can be applied to any linear polarization basis oriented at an arbitrary angle [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' We should note that the phase shifts and amplitudes of transmission are nearly insensitive to θ [22] and similar results as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' 2(b) and 2(c) are obtained for all θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' This is because the light is strongly confined inside each Si nanopost, so that the optical coupling among neighboring meta-atoms has only minor influence on the transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' We can also provide arbitrary phase shifts to orthogonal circular-polarization states by using the geometric phase shift of meta-atoms [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' First, we judiciously select 𝐷𝑢 and 𝐷𝑣 to satisfy 𝜑𝑣 = 𝜑𝑢 + 𝜋, so that each nanopost operates as a half-wave plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' In this case, input RHC and LHC states are converted to LHC and RHC, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' In addition, their phases after transmission are written as (𝜑𝑟, 𝜑𝑙) = (𝜑𝑢 + 2𝜃, 𝜑𝑢 − 2𝜃) (see Section S2 of Supplement 1 for the derivation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Therefore, 𝐷𝑢 and 𝐷𝑣 of each nanopost are selected to obtain desired 𝜑𝑢 (=(𝜑𝑟 + 𝜑𝑙)/2) while satisfying the condition 𝜑𝑣 = 𝜑𝑢 + 𝜋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' The angle 𝜃 is also determined to be (𝜑𝑟 − 𝜑𝑙)/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' To realize the function of a metalens, each meta-atom array needs to impart a spatially dependent phase profile given as [39] 𝜑(𝑥, 𝑦) = − 2𝜋 𝜆 (√(𝑥 − 𝑥0)2 + (𝑦 − 𝑦0)2 + 𝑓2 − 𝑓), (1) where (𝑥0, 𝑦0) is the in-plane position of the focal point, 𝑓 is the focal length, and 𝜆 is the operating wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' In this work, we set 𝜆 = 1550 nm, 𝑓 = 10 mm, and the diameter of the entire metasurface area to be 2 mm, corresponding to the numerical aperture (NA) of ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' The six focal points are arranged on a regular hexagon with a spacing of 60 µm, which are matched to the positions of the high-speed 2D-PDA used in our self-coherent experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Under these conditions, the phase profiles required for MA1, 2, and 3 are determined as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' 2(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Note that a rather large (2 mm) metasurface is used in this work due to the limitation in reducing the focal length 𝑓 in the current optical setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' In a fully packaged module as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' 1(a), we can readily shrink the entire area of the metasurface to a few tens of micrometers by reducing 𝑓 and designing the geometrical parameters of each nanopost to satisfy the required phase profiles given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' The designed metasurface was fabricated using a silicon-on-quartz (SOQ) substrate with a 1050-nm-thick Si layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' The nanopost patterns were defined by electron-beam lithography with ZEP520A resist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Then, the patterns were transferred to the Si layer by inductively-coupled- plasma reactive-ion etching (ICP-RIE) using SF6, C4F8, and O2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' An optical microscope image and scanning electron microscopy (SEM) images of the fabricated metasurface are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' 2(e)-(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Metasurface design and fabrication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' (a) Schematic of a periodic array of Si nano-posts placed at the vertices of a triangular lattice with a lattice constant 𝑎 of 700 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' The transmission of x- and y-polarized light is simulated for various axes lengths (𝐷𝑢, 𝐷𝑣) of the elliptical posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' (b) Required (𝐷𝑢, 𝐷𝑣) to obtain phase shifts (𝜑𝑢, 𝜑𝑣) for x- and y-polarized light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' For ease of fabrication, the ranges of 𝐷𝑢 and 𝐷𝑣 are limited from 100 nm to 650 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' (c) The amplitude of transmittance for each case in (b) as a function of (𝜑𝑢, 𝜑𝑣).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' (d) Required phase profiles for MA1, 2, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' (e) Optical microscope image and (f, g) SEM images of the fabricated device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' In (g), the image is false-colored to distinguish MA1, 2, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Static characterization of the fabricated metasurface We first characterized the fabricated metasurface by observing the intensity distribution at the focal plane for various input states of polarization (SOPs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' The experimental setup is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' A CW light with a wavelength of 1550 nm was incident to the metasurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' The SOP was modified by rotating a half-wave plate (HWP) and a quarter-wave plate (QWP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' The image at the focal plane was magnified at 50 times by a 4-f lens system and captured by an InGaAs camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' From the detected intensity values at the six focal positions, the Stokes vector was retrieved as described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' To enable quantitative measurement of the focused power, we inserted a flip mirror and detected the total power by a bucket PD after spatially filtering the focused beam at each target position using an iris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Figure 3(b) shows the observed intensity distributions when the input Stokes vector is set to (±1, 0, 0), (0, ±1, 0), and (0, 0, ±1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' We can confirm that the incident light is focused to the six well-defined points by transmitting through the metasurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Moreover, its intensity distribution changes with the SOP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' x/y linear, \uf0b145\uf0b0 linear, and RHC/LHC components of light are focused to the designed positions as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Figure 3(c) shows the retrieved Stokes = 0 0 = 700 2 2 , , , , , , Raith Mag-20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content='00KX SE2 EHT = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content='00KV WD-17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content='$mmRaith Mag EHT = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content='00 KV 0=20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content='2mmvectors on the Poincaré sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' The average error 〈|Δ𝐒|〉 is as small as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content='028.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Figure 3(d) shows the measured focusing efficiencies to the six positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Subtracting the 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content='8-dB intrinsic loss due to the 1\uf0b43 splitter [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' 1(a)], the excess loss is around 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content='1 dB, whereas the crosstalk to the orthogonal PD position is suppressed by 13-20 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' While this excess loss is already comparable to the coupling and propagation losses of the previously reported waveguide-based SVRs [13- 18], we expect further improvement by applying anti-reflection coating at the silica surface, improving the fabrication processes to minimize the errors, and by adopting advanced algorithms in designing the metasurface that take into account the nonzero interactions between adjacent meta-atoms [40, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Experimental characterization of the fabricated metasurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' (a) Schematic of the optical setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' The flip mirror is used to switch between capturing the intensity distribution at the focal plane and measuring the power of each focused beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' TLS: tunable laser source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' PC: polarization controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' VOA: variable optical attenuator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' FC: fiber collimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Pol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' : polarizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' HWP: half-wave plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' QWP: quarter-wave plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' MS: metasurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' M: flip mirror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' PD: photodetector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' (b) Measured intensity distributions at the focal plane for different input SOPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' (c) Retrieved and input Stokes vectors on the Poincaré sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' (d) Measured focusing efficiency to each PD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' The intrinsic loss due to splitting into three paths is shown by a green line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' The input polarization is labeled on the top of each bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Self-coherent signal transmission experiment We then performed the polarimetric self-coherent signal transmission experiment using the fabricated metasurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' The experimental setup is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' We employed a 19-pixel 2D-PDA with InP/InGaAs-based p-i-n structure [42], from which six PDs were used as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Each PD had a diameter of 30 µm, the measured bandwidth above 10 GHz, and the responsivity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content='3 A/W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' The 2D-PDA chip was packaged with the radio-frequency (RF) coaxial connectors connected to each PD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' The 2D-PDA was placed at the focal distance of 10 mm from the metasurface as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' 4(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' This distance was merely limited by the current setup and should be reduced to a sub-millimeter scale in a practical fully packaged module, which can be comparable to current IMDD receiver modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' A CW light at a wavelength of 1550 nm was generated from a tunable laser source (TLS) and split into two ports, which served as the signal and the pilot tone ports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' At the signal port, a LiNbO3 IQ modulator was used to generate a high-speed coherent optical signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' The Nyquist filter was applied to the driving electrical signals from an arbitrary waveform generator (AWG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' The modulated optical signal was then combined with the pilot tone by a polarization beam combiner (PBC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' The optical power of the pilot tone was adjusted by a variable optical 2 3 | | | | | | | | | | | | attenuator (VOA), so that their powers were nearly balanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' The self-coherent signal was then transmitted over a 25-km SMF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' At the receiver side, the optical signal-to-noise ratio (OSNR) was controlled using another VOA, followed by an erbium-doped fiber amplifier (EDFA) and an optical bandpass filter (OBPF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' The electrical signals from the six PDs of the PDA were amplified by differential RF amplifiers and then captured by a real-time oscilloscope (OSC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' At a baudrate beyond 20 GBd, we could not use the balanced PD (B-PD) configuration due to the residual skew inside the PDA module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' In these cases, we employed four single-ended PDs (S-PDs), where the electrical signals from PDx, PDy, PDa, and PDl were independently captured by a four-channel real-time oscilloscope, so that the skew could be calibrated during DSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' By comparing the results using two configurations, the use of four S-PDs was validated (see Section S3 of Supplement 1 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' To equalize and reconstruct the original IQ signal, we employed offline DSP with the 2×3 and 2×4 real-valued multi-input-multi-output (MIMO) equalizers [43, 44] for three-B-PD and four-S-PD configurations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Figures 5(a)-(c) show the BER curves and the constellations for 15-GBd 16QAM signals, measured using the three-B-PD configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' We can confirm that BERs well below the hard- decision forward error correction (HD-FEC) threshold are obtained with a negligible penalty even after 25-km transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Figures 5(d)-(f) show the results for 20-GBd 16QAM and 50- GBd QPSK signals, measured by the four-S-PD configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Once again, BERs below the HD-FEC threshold are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Finally, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' 6 shows the measured BER curves and constellation diagrams of 15-GBd 16QAM signal at 1540-nm and 1565-nm wavelengths, demonstrating the wideband operation of our designed metasurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' While the baudrate in this work was limited by the bandwidth of the 2D-PDA, beyond-100-GBd transmission should be possible by using higher-speed surface-normal PDs with bandwidth exceeding 50 GHz [45, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Self-coherent transmission experiment using the fabricated metasurface and 2D-PDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' (a) Experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' AWG: arbitrary waveform generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' PBC: polarization beam combiner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' EDFA: erbium-doped fiber amplifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' OBPF: optical bandpass filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Osc: oscilloscope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' In the insets, three-B-PD and four-S-PD configurations are depicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' (b) Optical microscope image of the fabricated 19-pixel 2D-PDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' The six circled PDs were used in this experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' (c) Photograph of the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' MS 2D PDA (a) TLS IQ mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' PBC AWG FC 1:9 coupler EDFA OBPF VOA VOA Real time Osc MS 2D PDA (c) diff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Amp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Real time Osc 4 S PDs 3 B PDs 25 km PDa PDb PDr PDl PDx PDy 100 μ (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Experimental results of self-coherent signal transmission at a wavelength of 1550 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' (a)-(c) Measured BER curves and constellation diagrams of 15-GBd 16QAM signals before (b2b) and after 25-km transmission using the three-B-PD configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' (d)-(f) Measured BER curves and constellation diagrams of 20-GBd 16QAM and 50-GBd QPSK signals after 25-km transmission using the four-S-PD configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Experimental results of self-coherent signal transmission at wavelengths of (a) 1540 nm and (b) 1565 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' (a, b) Measured BER curves of 15-GBd 16QAM signals before (b2b) and after 25-km transmission using the three-B-PD configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' The insets represent the retrieved constellation diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Conclusion We have proposed and demonstrated a surface-normal SVR using a dielectric metasurface and 2D-PDA for the high-speed polarimetric self-coherent systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Three independently designed meta-atom arrays based on Si nanoposts were superimposed onto a single thin metasurface layer to implement both the polarization-sorting and focusing functions simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Using a compact metasurface chip fabricated on a SOQ substrate, we demonstrated 25-km transmission of 20-GBd 16QAM and 50-GBd QPSK self-coherent signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' The operating baudrate was merely limited by the 2D-PDA, so that higher-capacity transmission should be possible by using a PDA with a broader bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Owing to the unique surface-normal configuration with the embedded lens array functionality, a compact receiver module with comparable size and complexity as the conventional IMDD receivers can be realized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Moreover,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' it can easily be extended to receive spatially multiplexed channels by simply replacing the SMF 16 18 20 22 24 26 28 10-5 10-4 10-3 10-2 10-1 OSNR (dB) BER 10-5 10-4 10-3 10-2 10-1 BER 16 18 20 22 24 26 28 OSNR (dB) 50-GBd QPSK 20-GBd 16QAM 15-GBd 16QAM (a) (d) 25km b2b 25 km 25 km (b) (c) (e) (f) 7% HD-FEC 7% HD-FEC 25 km b2b 25 km b2b 22 24 26 28 30 10-5 10-4 10-3 10-2 10-1 OSNR (dB) BER 14 16 18 20 22 10-3 10-2 10-1 OSNR (dB) BER (a) (b) 1540 nm 1565 nm 25 km b2b 25 km b2b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content='to a MCF and employing a larger-scale integrated PDA technology [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' This work would, therefore, pave the way toward realizing cost-effective receivers for the future >Tb/s spatially multiplexed optical interconnects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Funding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' National Institute of Information and Communications Technology (NICT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' This work was obtained from the commissioned research 03601 by National Institute of Information and Communications Technology (NICT), Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Portions of this work were presented at the Optical Fiber Communications Conference (OFC) in 2022, M4J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' A part of the device fabrication was conducted at the cleanroom facilities of d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content='lab in the University of Tokyo, supported by MEXT Nanotechnology Platform, Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' The authors also thank all the technical staff at Advanced ICT device laboratory in NICT for supporting the PDA device fabrication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' acknowledges the financial support from Optics and Advanced Laser Science by Innovative Funds for Students 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Akahane, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Yamamoto, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Kawanishi, “Bias-free operational UTC-PD above 110 GHz and its application to high baud rate fixed-fiber communication and W- band photonic wireless communication,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Lightwave Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' 34, 3138–3147 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Compact and scalable polarimetric self- coherent receiver using dielectric metasurface: Supplementary Information GO SOMA,1,4 YOSHIRO NOMOTO,2 TOSHIMASA UMEZAWA,3 YUKI YOSHIDA,3 YOSHIAKI NAKANO,1 AND TAKUO TANEMURA1,5 1School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, Japan 2Central Research Laboratory, Hamamatsu Photonics K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' 5000 Hirakuchi, Hamakita-ku, Hamamatsu City, Shizuoka, Japan 3National Institute of Information and Communications Technologies (NICT), 4-2-1 Nukui-Kitamachi, Koganei, Tokyo, Japan 4soma@hotaka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content='u-tokyo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content='jp, 5tanemura@ee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content='u-tokyo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content='jp S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Derivation of 𝑫𝒖(𝝋𝒖, 𝝋𝒗) and 𝑫𝒗(𝝋𝒖, 𝝋𝒗) From RCWA simulation, we obtain 𝑡𝑢(𝐷𝑢, 𝐷𝑣) and 𝑡𝑣(𝐷𝑢, 𝐷𝑣), which describe the complex transmittance through an array of elliptical Si nanoposts with the principal axis lengths of 𝐷𝑢 and 𝐷𝑣 for the linearly polarized light along 𝑢 and 𝑣 axes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' The simulated intensity |𝑡𝑢|2 and |𝑡𝑣|2 and the phase arg(𝑡𝑢) and arg(𝑡𝑣) are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' S1(a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' From these results, we derive the required 𝐷𝑢(𝜑𝑢, 𝜑𝑣) and 𝐷𝑣(𝜑𝑢, 𝜑𝑣) to obtain desired phase shifts (𝜑𝑢, 𝜑𝑣), by using the following the equation [1]: (𝐷𝑢(𝜑𝑢, 𝜑𝑣), 𝐷𝑣(𝜑𝑢, 𝜑𝑣)) = arg min (𝐷𝑢, 𝐷𝑣) [|𝑡𝑢(𝐷𝑢, 𝐷𝑣) − 𝑒𝑖𝜑𝑢| 2 + |𝑡𝑣(𝐷𝑢, 𝐷𝑣) − 𝑒𝑖𝜑𝑣| 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Simulated intensity (a) and phase (b) of transmission coefficients as a function of 𝐷𝑢 and 𝐷𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content='7 (μm) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content='7 (μm) 0 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content='7 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content='7 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content='7 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content='7 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content='7 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content='7 (μm) (μm) (μm) (μm) (μm) (μm) Transmittance Phase (rad) (a) (b) S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Derivation of phase shift by a meta-atom array for circularly polarized light The Jones matrix, describing the transmittance through a lossless Si nanopost array can be written as 𝐉 = 𝐑(𝜃) (𝑒𝑖𝜑𝑢 0 0 𝑒𝑖𝜑𝑣) 𝐑(−𝜃) = (𝑒𝑖𝜑𝑢 cos2 𝜃 + 𝑒𝑖𝜑𝑣 sin2 𝜃 (𝑒𝑖𝜑𝑢 − 𝑒𝑖𝜑𝑣) sin 𝜃 cos 𝜃 (𝑒𝑖𝜑𝑢 − 𝑒𝑖𝜑𝑣) sin 𝜃 cos 𝜃 𝑒𝑖𝜑𝑢 sin2 𝜃 + 𝑒𝑖𝜑𝑣 cos2 𝜃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Here, 𝜑𝑢 and 𝜑𝑣 represent the phase shifts for the polarization components along the principal axes of the elliptical nanoposts and 𝐑(𝜃) is a rotation matrix with a rotation angle of 𝜃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Here we assume that the input lightwave is circularly-polarized and its Jones vector is written as 𝑬𝑟,𝑙 = 1/√2(1, ±𝑖)𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Then, the output Jones vector is written as 𝐉𝑬𝑟,𝑙 = 𝑒𝑖𝜑𝑢 + 𝑒𝑖𝜑𝑣 2 𝑬𝑟,𝑙 + 𝑒𝑖𝜑𝑢 − 𝑒𝑖𝜑𝑣 2 𝑒±𝑖2𝜃𝑬𝑙,𝑟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Therefore, when the meta-atom functions as a half-wave plate, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=', 𝜑𝑣 = 𝜑𝑢 + 𝜋, the output Jones vector becomes 𝑒𝑖(𝜑𝑢±2𝜃)𝑬𝑙,𝑟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' In other words, the phase shifts given to the right-handed and left-handed circularly-polarized waves are (𝜑𝑟, 𝜑𝑙) = (𝜑𝑢 + 2𝜃, 𝜑𝑢 − 2𝜃) , while the output polarization handedness is reversed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Comparison between three-B-PD and four-S-PD configurations To compare the results obtained by three-B-PD and four-S-PD configurations, we performed the self-coherent transmission experiment with 15-GBd 16QAM signals using the two experimental setups shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' The measured BER curves are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Since the BER was limited by the optical signal-to-noise ratio (OSNR), identical results were obtained in two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Measured BER curves of 15-GBd 16QAM signals using the setup with three-B-PD and four-S-PD configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Arbabi, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Horie, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Bagheri, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Faraon, “Dielectric metasurfaces for complete control of phase and polarization with subwavelength spatial resolution and high transmission,” Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' Nanotechnol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} +page_content=' 10, 937–943 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAzT4oBgHgl3EQf-_6e/content/2301.01942v1.pdf'} diff --git a/BNE0T4oBgHgl3EQfxwK1/vector_store/index.faiss b/BNE0T4oBgHgl3EQfxwK1/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..cb4b295250d7f95f37ba8a8dbbe89c2cfa7c6d7e --- /dev/null +++ b/BNE0T4oBgHgl3EQfxwK1/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:e6172fa1d687fc2c9ebb7bf1f83835a269396e27c1b704c2110718470db90d99 +size 4915245 diff --git a/CtE2T4oBgHgl3EQf9Ant/content/tmp_files/2301.04225v1.pdf.txt b/CtE2T4oBgHgl3EQf9Ant/content/tmp_files/2301.04225v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..0c26c8cf6326da8501b3b56c7fa0d0a0237bc1d0 --- /dev/null +++ b/CtE2T4oBgHgl3EQf9Ant/content/tmp_files/2301.04225v1.pdf.txt @@ -0,0 +1,2539 @@ +1 +Inferring Gene Regulatory Neural Networks for +Bacterial Decision Making in Biofilms +Samitha Somathilaka, Student, IEEE, Daniel P. Martins, Member, IEEE, Xu Li, Yusong Li, +Sasitharan Balasubramaniam, Senior Member, IEEE, +Abstract—Bacterial cells are sensitive to a range of external sig- +nals used to learn the environment. These incoming external sig- +nals are then processed using a Gene Regulatory Network (GRN), +exhibiting similarities to modern computing algorithms. An in- +depth analysis of gene expression dynamics suggests an inherited +Gene Regulatory Neural Network (GRNN) behavior within the +GRN that enables the cellular decision-making based on received +signals from the environment and neighbor cells. In this study, +we extract a sub-network of Pseudomonas aeruginosa GRN that +is associated with one virulence factor: pyocyanin production as a +use case to investigate the GRNN behaviors. Further, using Graph +Neural Network (GNN) architecture, we model a single species +biofilm to reveal the role of GRNN dynamics on ecosystem-wide +decision-making. Varying environmental conditions, we prove +that the extracted GRNN computes input signals similar to +natural decision-making process of the cell. Identifying of neural +network behaviors in GRNs may lead to more accurate bacterial +cell activity predictive models for many applications, including +human health-related problems and agricultural applications. +Further, this model can produce data on causal relationships +throughout the network, enabling the possibility of designing +tailor-made infection-controlling mechanisms. More interestingly, +these GRNNs can perform computational tasks for bio-hybrid +computing systems. +Index Terms—Gene Regulatory Networks, Graph Neural Net- +work, Biofilm, Neural Network. +I. INTRODUCTION +B +ACTERIA are well-known for their capability to sense +external stimuli, for complex information computations +and for a wide range of responses [1]. The microbes can sense +numerous external signals, including a plethora of molecules, +temperatures, pH levels, and the presence of other microorgan- +isms [2]. The sensed signals then go through the Gene Regu- +latory Network (GRN), where a large number of parallel and +sequential molecular signals are collectively processed. The +GRN is identified as the main computational component of the +cell [3], which contains about 100 to more than 11000 genes +Samitha Somathilaka is with VistaMilk Research Centre, Walton Institute +for Information and Communication Systems Science, Waterford Institute +of Technology, Waterford, X91 P20H, Ireland and School of Computing, +University of Nebraska-Lincoln, 104 Schorr Center, 1100 T Street, Lincoln, +NE, 68588-0150, USA. E-mail: samitha.somathilaka@waltoninstitute.ie. +Daniel P. Martins are with VistaMilk Research Centre and the Wal- +ton Institute for Information and Communication Systems Science, Wa- +terford Institute of Technology, Waterford, X91 P20H, Ireland. E-mail: +daniel.martins@waltoninstitute.ie. +Xu Li and Yusong Li are with Department of Civil and Environmental +Engineering University of Nebraska-Lincoln 900 N. 16th Street Nebraska Hall +W181, Lincoln, NE 68588-0531 E-mail:xuli,yli7@unl.edu. +S. Balasubramaniam is with School of Computing, University of Nebraska- +Lincoln, 104 Schorr Center, 1100 T Street, Lincoln, NE, 68588-0150, USA. +E-mail:sasi@unl.edu +Full GNN +... +... +... +Extracted +NN +GRN +Biofilm +... +Fig. 1: Illustration of the Gene Regulatory Neural Networks +(GRNN) extraction and the implementation of the GNN to +model the biofilm. The diffusion of molecules from one cell +to another is modeled as a vector, where mq represents the +concentration of the qth molecular signal. +(the largest genome identified so far belongs to Sorangium +cellulosum strain So0157-2) [4]. Despite the absence of neural +components, the computational process through GRN allows +the bacteria to actuate through various mechanisms, such as +molecular production, motility, physiological state changes and +even sophisticated social behaviors. Understanding the natural +computing mechanism of cells can lead to progression of +key areas of machine learning in bioinformatics, including +prediction of biological processes, prevention of diseases and +personalized treatment [5]. +Bacterial cells are equipped with various regulatory sys- +tems, such as single/two/multi-component systems including +Quorum sensing (QS), to respond to environmental stimuli. +The receptors and transporters on cell membranes can react +and transport extracellular molecules, which subsequently in- +teract with respective genes. In turn, the GRN is triggered +to go through a complex non-linear computational process in +response to the input signals. In the literature, it has been +suggested that the computational process through the GRN of +a bacterial cell comprises a hidden neural network (NN)-like +architecture [6], [7]. This indicates that, even though bacterial +cells can be categorized as non-neural organisms, they perform +neural decision-making processes through the GRN. This re- +sults in recent attention towards Molecular Machine Learning +systems, where AI and ML are developed using molecular +systems [8]. In these systems, several neural components can +be identified in GRNs, in which genes may be regarded as +arXiv:2301.04225v1 [q-bio.MN] 10 Jan 2023 + +2 +C4- +RhlR +rhlI +3OC- +LasR +Phosphate +Fe(II) +BqsR +pqsABCDE +phz2 +phz1 +PQS- +PQSR +PhoB +pqsR +rhlR +lasR +LasI +pqsH +HHQ- +PQSR +? +3OC +C4 +PqsH +HHQ +PqsR +LasR +RhlR +PQS +AlgR +czcR +PhZ2 PhZ1 +3OC +C4 +PqsH +HHQ +PqsR LasR +RhlR +(a) +hn11 +hn12 +PhoB +hn13 +BqSR +hn14 +AlgR +hn21 +hn22 +hn23 +pqsABCDE +czcR +rhlI +pqsR lasR +lasI +rhlR +hn31 +phz2 +phz1 +hn24 +pqsH +(b) +GNN model +Inter-cellular diffusion +GRN +NN +... +... +... +... +(c) +Fig. 2: Extraction of a GRNN considering a specific sub-network of the GRN where a) is the two-component systems (TCSs) +and QS network that is associated with the pyocyanin production, b) is the derived GRNN that is equipped with hypothetical +nodes (hns) without affecting its computation process to form a symmetric network structure and c) is the conversion of real +biofilm to the suggested in-silico model. +computational units or neurons, transcription regulatory factors +as weights/biases and proteins/second messenger Molecular +Communications (MC) as neuron-to-neuron interactions. Ow- +ing to a large number of genes and the interactions in a GRN, +it is possible to infer sub-networks with NN behaviors that +we term Gene Regulatory Neural Networks (GRNN). The +non-linear computing of genes results from various factors +that expand through multi-omics layers, including proteomic, +transcriptomic and metabolomic data (further explained in +Section II-A). In contrast, the GRNN is a pure NN of genes +with summarized non-linearity stemmed from multi-omics +layers with weights/biases. +Identification of GRNNs can be used to model the decision- +making process of the cell precisely, especially considering +simultaneous multiple MC inputs or outputs. However, due +to the limited understanding and data availability, it is still +impossible to model the complete GRN with its NN-like +behaviors. Therefore, this study uses a GRNN of Pseudomonas +aeruginosa that is associated with PhoR-PhoB and BqsS- +BqsR two-component systems (TCSs) and three QS systems +related to pyocyanin production as a use case to explore the +NN-like behaviors. Although a single bacterium can do a +massive amount of computing, they prefer living in biofilms. +Hence, in order to understand the biofilm decision-making +mechanism, we extend this single-cell computational model +to an ecosystem level by designing an in-silico single species +biofilm with inter-cellular MC signaling as shown in Fig. 1. +The contributions of this study are as follows: +• Extracting a GRNN: Due to the complexity and insuf- +ficient understanding of the gene expression dynamics of +the full GRN, we only focus on a sub-network associated +with pyocyanin production (shown in Fig. 2a) to inves- +tigate the NN-like computational behavior of the GRN. +Further, the genes of extracted sub-network are arranged +following a NN structure that comprises input, hidden +and output layers, as shown in Fig. 2b. +• Modeling a biofilm as a GNN: The GRNN only repre- +sents the single-cell activities. To model the biofilm-wide +decision-making process, we use a Graph Neural Network +(GNN). First, we create a graph network of the bacterial +cell and convert it to a GNN by embedding each node +with the extracted GRNN as the update function. Second, +the diffusion-based MCs between bacterial cells in the +biofilm are encoded as the message-passing protocol of +the GNN, as shown in Fig. 2c. +• Exploring the behaviors of the GRNN and intra- +cellular MC dynamics to predict cell decisions: The +output of the GRNN is evaluated by comparing it with +the transcriptomic and pyocyanin production data from +the literature. Finally, an edge-level analysis of the GRNN +is conducted to explore the causal relationships between +gene expression and pyocyanin production. +This paper is organized as follows: Section II explains +the background of bacterial decision-making in two levels: +cellular-level in Section II-A and population-level in Section +II-B, while the background on the P. aeruginosa is introduced +in Section II-C. Section III is dedicated to explaining the model +design of cellular and population levels. The results related +to model validation and the intergenic intra-cellular signaling +pattern analysis are presented in Section IV and the study is +concluded in Section V. +II. BACKGROUND +As the model expands through single cellular and biofilm- +wide decision-making layers, this section provides the back- +ground of how a bacterium uses the GRN to make decisions +and how bacterial cells make decisions in biofilms. Moreover, +we briefly discuss the cellular activities of the Pseudomonas +aeruginosa as it is the use case species of this study. +A. Decision-Making Process of an Individual Cell +Prokaryotic cells are capable of sensing the environment +through multiple mechanisms, including TCSs that have been +widely studied and it is one of the focal points of this +study. The concentrations of molecular-input signals from +the extracellular environment influence the bacterial activities +at the cellular and ecosystem levels [9]. Apart from the +extracellular signals of nutrients, it is evident that the QS +input signals have a diverse set of regulative mechanisms in +biofilm-wide characteristics, including size and shape [10]. +These input signals undergo a computational process through +the GRN, exhibiting a complex decision-making mechanism. +Past studies have explored and suggested this underpinning +computational mechanism in a plethora of directions, such + +3 +Prom +Op +Enh +GeneA +... +Sil +GeneB +... +Fig. 3: Illustration of gene expression regulators that are +considered the weight influencers of the edges of GRNN. +Here, the α(σ), α(∼σ), α(T F ), α(Rep), α(eT F ) and α(sT F ) +are relative concentrations of sigma factors, anti-sigma factors, +transcription factors (TFs), repressors, enhancer-binding TFs +and silencer-binding TFs respectively. Moreover, β(P rom), +β(Op), β(Enh), and β(Sil) are the binding affinities of the pro- +moter, operator, enhancer and silencers regions respectively. +as using differential equations [11] and probabilistic Boolean +networks [12] and logic circuit [13]. All of these models +mainly infer that the bacterial cells can make decisions not +just based on the single input-output combinations, but they +can integrate several incoming signals non-linearly to produce +outputs. +The studies that focus on differences in gene expression +levels suggest that a hidden weight behavior controls the +impact of one gene on another [6]. This weight behavior +emerges through several elements, such as the number of +transcription factors that induce the expression, the affinity +of the transcription factor binding site, and machinery such +as thermoregulators and enhancers/silencers [14], [15]. Fig. +3 depicts a set of factors influencing the weight between +genes. The weight of an edge between two genes has a +higher dynamicity as it is combinedly determined by several +of these factors. Based on environmental conditions, the GRN +of the bacterial cell adapts various weights to increase the +survivability and repress unnecessary cellular functions to +preserve energy. An example of such regulation is shown in +Fig. 4 where a P. aeruginosa cell uses a thermoregulator to +regulate the QS behaviors. Fig. 4a has a set of relative weights +based on cellar activities in an environment at 37 ◦C, while +Fig. 4b represents weights at 30 ◦C. The weights between the +hn21 and rhlR are different in two conditions, and these cellar +activities are further explained in [14]. +B. Biofilm Decision-Making +Even though an individual cell is capable of sensing, +computing, and actuating, the majority of bacterial cells live +in biofilms, where the survivability is significantly increased +compared to their planktonic state. Biofilm formation can +cause biofouling and corrosion in water supply and industrial +systems [16]. However, biofilms formation can be desirable +in many situations, for example, bioreactors in wastewater +treatment [17], bioremediation of contaminated groundwater +[18], [19], where biofilms serve as platforms for biogeochem- +ical reactions. A massive number of factors can influence +biofilm formation, including substratum surface geometrical +characteristics, diversity of species constituting the biofilm, +hydrodynamic conditions, nutrient availability, and especially +communication patterns [20] where the TCS and QS play +rhlI +BqsR +pqsABCDE +phz2 +phz1 +PhoB +hn11 +pqsR +rhlR +lasR +LasI +pqsH +hn31 +hn12 +hn13 +hn14 +hn23 +hn24 +hn21 +hn22 +AlgR +czcR +1 +-1 +0 +(a) +rhlI +BqsR +pqsABCDE +phz2 +phz1 +PhoB +hn11 +pqsR +rhlR +lasR +LasI +pqsH +hn31 +hn12 +hn13 +hn14 +hn23 +hn24 +hn21 +hn22 +AlgR +czcR +1 +-1 +0 +(b) +Fig. 4: Two GRNN setups with different weights associated +with two environmental conditions. a) is the relative weight +setup of P. aeruginosa cell in 37 ◦C and b) is in 30 ◦C. +significant roles. A TCS comprises a histidine kinase that +is the sensor for specific stimulus and a cognate response +regulator that initiates expressions of a set of genes [21]. +Hence, in each stage, essential functions can be traced back +to their gene expression upon a response to the input signals +detected by bacterial cells. For instance, in the first stage +of biofilm formation, the attachment of bacteria to a surface +is associated with sensing a suitable surface and altering +the activities of the flagella. In the next stage, rhamnolipids +production is associated with ferric iron Fe3+ availability in +the environment, mostly sensed through BqsS-BqsR TCSs. +Further, Fe3+ was identified as a regulator of pqsA, pqsR, and +pqsE gene expressions that are associated with the production +of two critical components for the formation of microcolonies: +eDNA and EPS [22]. Similarly, in the final stage, the dis- +persion process can also be traced back to a specific set +of gene regulations, including bdlA an rbdA [23], [24]. An +understanding of the underlying decision-making process of +bacteria may enable us to control their cellular activities. +C. Pseudomonas Aeruginosa +The main reason for selecting P. aeruginosa in this work +lies in its alarming role in human health. For example, this +species is the main cause of death in cystic fibrosis patients +[25]. P. aeruginosa is a gram-negative opportunistic pathogen +with a range of virulence factors, including pyocyanin and +cytotoxin secretion [26]. These secreted molecules can lead to +complications such as respiratory tract ciliary dysfunction and +induce proinflammatory and oxidative effects damaging the +host cells [27]. The biofilms are being formed on more than +90% endotracheal tubes implanted in patients who are getting +assisted ventilation, causing upper respiratory tract infections +[28]. In addition, another important reason for targeting P. +aeruginosa is the data availability for the GRN structure [29], +pathways [30], genome [31], transcriptome [32] and data from +mutagenesis studies [33], [34]. Compared to the complexity of +the GRN, the amount of data and information available on the + +4 +C4- +RhlR +3OC- +LasR +PQS- +PQSR +HHQ- +PQSR +3OC +C4 +PqsH +HHQ +PqsR +LasR +RhlR +PQS +Fig. 5: Illustrations of intra-cellular metabolite interaction. +gene-to-gene interactions and expression patterns is insuffi- +cient to develop an accurate full in-silico model. Therefore, +we chose a set of specific genes that are associated with QS, +TCS, and pyocyanin production. +III. SYSTEM DESIGN +This section explains the system design in two main phases, +extracting a NN-like architecture from the GRN targeting the +set of genes and creating a model of the biofilm ecosystem. +A. Extracting Natural Neural Network from GRN +We first fetch the structure of the GRN graph from Reg- +ulomePA [29] database that contains only the existence of +interactions and their types (positive or negative). As the next +step, using information from the past studies [35]–[38], we +identified the genes involved in the Las, Rhl and PQS QS +systems, PhoR-PhoB and BqsS-BqsR TCSs, and pyocyanin +production to derive the sub-network of GRN as shown in +Fig 2a. We further explored the expression dynamics using +transcriptomic data [39], [40] where we observed the non- +linearity in computations that are difficult to capture with +existing approaches such as logic circuits, etc. [6], making +the NN approach more suitable. However, a NN model with a +black box that is trained on a large amount of transcriptomic +data records to do computations similar to the GRN has a +number of limitations, especially in understanding the core +of the computational process [41]. Our model does not use a +conventional NN model; instead, we extract a NN from the in- +teraction patterns of the GRN, which we consider a pre-trained +GRNN. In this sub-network, we observed that the lengths of +expression pathways are not equal. For example, the path from +PhoR-PhoB to the phz2 gene has two hops, but the path from +the BqsS-BqsR system to the rhlR gene only has one hop. The +extracted network has the structure of a random NN. Hence, +we transform this GRNN to Gene Regulatory Feedforward +Neural Network by introducing hypothetical nodes (hns) that +do not affect the behaviors of the GRNN as shown in Fig 2b. +In this transformation, we decide the number of hidden layers +based on the maximum number of hops in gene expression +pathways. In our network, the maximum number of hops is +two, which determines the number of hidden layers as one, +and then the number of hops of all the pathways is leveled +by introducing hns. If a hn is introduced between a source +and target genes, the edge weights from the source node to +the hn and from hn to the target node are made “1” so that +the hn does not have an influence on the regulation of genes. +Moreover, if a gene does not induce another in the network, +the weight of the edge between that pair is made “0”. +Here, we summarize multiple factors of interaction into +a weight that determines the transcriptional regulation of +a particular gene. This regulation process occurs when the +gene products get bound to the promoter region of another, +influencing the transcriptional machinery. Hence, we observe +this regulation process of a target gene as a multi-layered +model that relies on the products of a set of source genes, the +interaction between gene products, and the diffusion dynamics +within the cell. Creating a framework to infer an absolute +weight value using all the above factors is a highly complex +task. In order to infer weight, one method is to train a NN +model with the same structure as the GRN using a series of +transcriptomic data. However, this approach also has numerous +challenges, such as the lack of a sufficient amount of data in +similar environments. +Therefore, we estimate a set of relative weights based on +genomic, transcriptomic, and proteomic explanations of each +interaction from the literature. The weights were further fine- +tuned using the transcriptomics data. A relative weight value +of an edge can be considered a summarizing of multi-layer +transcriptional-translation to represent the impact of the source +gene on a target gene. +In this computational process, we identify another layer of +interactions that occur within the cell. The produced molecules +by the considered TCs network go through a set of metabolic +interactions that are crucial for the functionality of the cell. +Since our primary goal is to explore the NN behaviors of +GRN, we model these inter-cellular chemical reactions as a +separate process, keeping the gene-to-gene interactions and +metabolic interactions in two different layers. To model the +complete pyocyanin production functionality of the cell, we +use the inter-cellular molecular interactions shown in Fig 5. +Here, RhlR is a transcriptional regulator of P. aeruginosa that +forms a complex by getting attached to its cognate inducer +C4-HSL and then binds to the promoter regions of relevant +genes [42]. Similarly, LasR transcriptional regulator protein +and 3-oxo-C12-HSL (3OC), and PqsR with PQS and HHQ +form complexes and get involved in the regulation of a range +of genes [43], [44]. Further, C10H10O6 in the environment are +converted by the P. aeruginosa cells in multiple steps using +the products of the GRNN we consider. First, C10H10O6 +is converted into phenazine-1-carboxylic using the enzymes +of pqsABCDEFG genes. Later, phenazine-1-carboxylic was +converted into 5-Methylphenazine-1-carboxylate, and finally, +5-Methylphenazine-1-carboxylate into Pyocyanin by PhzM +and PhzS, respectively [45]. +Molecular accumulation within a bacterial cell can be +considered its memory module where certain intra-cellular +interactions occurs. Therefore, we define an internal memory +matrix IM as, +IM(t) = +im1 +im2 +... +imJ +� +� +� +� +� +� +� +� +� +� +� +� +B1 +C(t) +(1,im1) +C(t) +(1,im2) +... +C(t) +(1,imJ) +B2 +C(t) +(2,im1) +C(t) +(2,im2) +... +C(t) +(2,imJ) +... +... +... +... +... +BP +C(t) +(P,im1) +C(t) +(P,im2) +... +C(t) +(P,imJ) +, +(1) +where the concentration of the internal molecule imj is + +5 +C(t) +(i,imj). +GRNN process molecular signals from the environment and +other cells. Hence, we used the approach of GNN as a scalable +mechanism to model the MCs and biofilm wide decision- +making process. The extreme computational power demand of +modeling the diffusion-based MCs of each cell is also avoided +by using this approach. +B. Graph Neural Network Modeling of Biofilm +First, the biofilm is created as a graph network of bacterial +cells where each node is a representation of a cell, and an edge +between two nodes is a MC channel. We convert the graph +network into a Graph Neural Network (GNN) in three steps: 1) +embedding the extracted GRNN of pyocyanin production into +each node as the update function, 2) encoding the diffusion- +based cell-to-cell MC channels as the message passing scheme, +and 3) creating an aggregation function at the reception of +molecular messages by a node as shown in Fig. 6. Next, we +define feature vectors of each node of the GNN to represent +the gene expression profile of the individual cell at a given +time. Subsequently, considering L is the number of genes in +the GRNN, P is the number of bacterial cells in the biofilm +and b(t) +(i,gl) is the expression of gene gl by the bacteria Bi, +we derive the following matrix FV(t) that represents all the +feature vectors of the GNN at time t. +FV(t) = +g1 +g2 +... +gL +� +� +� +� +� +� +� +� +� +� +� +� +B1 +b(t) +(1,g1) +b(t) +(1,g2) +... +b(t) +(1,gL) +B2 +b(t) +(2,g1) +b(t) +(2,g2) +... +b(t) +(2,gL) +... +... +... +... +... +BP +b(t) +(P,g1) +b(t) +(P,g2) +... +b(t) +(P,gL) +(2) +The computational output of the GRNN of each node results +in the secretion of a set of molecules that are considered +messages in our GNN model as illustrated in the Fig. 7. +When the number of molecular species considered in the +network is Q and output mq molecular message from bacterial +cell Bi at TS t is msg(t) +(i,mq), we derive the matrix +MSG(t) = +m1 +m2 +... +mQ +� +� +� +� +� +� +� +� +� +� +� +� +B1 +msg(t) +(1,m1) +msg(t) +(1,m2) +... +msg(t) +(1,mQ) +B2 +msg(t) +(2,m1) +msg(t) +(2,m2) +... +msg(t) +(2,mQ) +... +... +... +... +... +BP +msg(t) +(P,m1) +msg(t) +(P,m2) +... +msg(t) +(P,mQ) +. +(3) +Further, we use a static diffusion coefficients vector +D = {Dm1, Dm2, ..., DmQ}, +(4) +where Dmq is diffusion coefficient of molecular species mq. +Gene expression profile of bacterial cell b1 at time step t +(a) +B2 +B5 +B6 +B1 +B3 +B7 +B2 +B5 +B6 +B1 +B3 +B7 +B2 +B5 +B6 +B1 +B3 +B7 +B2 +B5 +B6 +B1 +B3 +B7 +t=0 +t=1 +t=2 +t=T +... +(b) +Fig. 6: Illustration of the GNN components where a) is a +snapshot of the bacterial network that has the gene expression +profile as the feature vector. Further, this gene expression +pattern of a cell is encoded to a message of secreted molecules +where MC plays a crucial role. Moreover, b) shows the +temporal behavior of the GNN, that the output of one graph +snapshot influences the next. +... +... +Fig. 7: The process of one GRNN outputs reaching another +GRNN as molecular messages. +We define another matrix ED that contains the euclidean +distances between bacterial cells in the biofilm as follows +ED = +B1 +B2 +... +BP +� +� +� +� +� +� +� +� +B1 +d(1,1) +d(1,2) +... +d(1,P ) +B2 +d(2,1) +d(2,2) +... +d(2,P ) +... +... +... +... +... +BP +d(P,1) +d(P,2) +... +d(P,P ) +(5) +where di,j is the euclidean distance between the ith and jth +cells. + +6 +TABLE I: Parameters utilised in the system development +Parameter +Value +Description +No. of cells +2000 +The number of cells is limited due to the memory availability of the server. +No. of genes +13 +The network only consists of the gene that are directly associated with QS, PhoR-PhoB and BqsS-BqsR +TCSs, and pyocyanin production. +No. internal memory molecules +16 +The set of molecules that involved in QS, PhoR-PhoB and BqsS-BqsR TCSs,and pyocyanin production. +No. messenger molecules +4 +The number of molecules that were exchanged between cells in the sub network. +Dimensions of the environment +20x20x20µm +The dimensions were fixed considering the average sizes of P. aeruginosa biofilms and computational +demand of the model. +Duration +150 TSs +The number of TSs can be modified to explore the cellular and ecosystem level activities. For this +experiment we fixed a TS to represent 30mins. +No. iterations per setup +10 +Considering the stochasticity ranging from the gene expression to ecosystem-wide communications, the +experiments were iterated 10 times. +The feature vector of ith bacterial cell at the TS t + 1 is +then modeled as, +FV(t+1) +i += GRNNi(MSG(t) +i ++ S(t) +i ) +(6) +where MSG(t) +i +is the message generated by the same cell +in the previous TS. The GRNNi is the extracted GRNN +that is the update function in the GNN learning process +and S(t) +i += R(t) +i ++ K(t) +i , is the aggregate function. In the +aggregation component, the R(t) +i +is the incoming signals from +peer bacterial cells and K(t+1) +(i:mq) is the external molecule input +vector at the location of Bi and the TS t that is expressed as +K(t+1) +i += +� +K(t+1) +i:m1 , K(t+1) +i:m2 , ..., K(t+1) +i:mQ +� +. +(7) +In order to compute R(t+1) +i +, we use a matrix Yi; +Yi = +↔ +1 [Q×1] ×EDi, where +↔ +1 [Q×1] is an all-ones matrix of +dimension Q × 1. The ˆg matrix is then defined as follows, +ˆg(D⊺, Y, t) = +� +���� +g(Dm1, d(i,1), t) +g(Dm1, d(i,2), t) +... +g(Dm1, d(i,P ), t) +g(Dm2, d(i,1), t) +g(Dm2, d(i,2), t) +... +g(Dm2, d(i,P ), t) +... +... +... +... +g(DmQ, d(i,1), t) +g(DmQ, d(i,2), t) +... +g(DmQ, d(i,P ), t) +� +���� . +(8) +In the above matrix, g(Dml, d(i,j), t) is the Green’s function +of the diffusion equation as shown below, +ˆG(Dml, d(i,j), t) = +1 +(4πDmlt) +3 +2 exp +� +− +d2 +(i,j) +4Dmlt +� +. +(9) +Further, the incoming signal vector R(t+1) +i +is denoted as +below, +R(t+1) +i += diag +� +ˆg(D⊺, Y, t) × MSG(t)� +. +(10) +Further, we equip our model with a 3-D environment +to compensate for the noise element and external molecule +inputs. Environment-layer is designed as a 3-D grid of voxels +that can store precise information on external nutrients (simi- +larly to our previous model in [46]). The diffusion of nutrient +molecules through the medium is modeled as a random-walk +process. This layer allows us to enrich the model with the +dynamics of nutrient accessibility of bacterial cells due to +diffusion variations between the medium and the Extracellular +Polymeric Substance (EPS). +The bacterial cells in the ecosystem also perform their +own computing tasks individually, resulting in a massively +parallel processing framework. Hence, we use the python-cuda +platform to make our model closer to the parallel processing +architecture of the biofilm, where we dedicate a GPU block for +each bacterial cell and the threads of each block for the matrix +multiplication of the GRNN computation associated with the +particular cell. Additionally, due to the massive number of +iterative components in the model, the computational power +demand faces significant challenges with serial programming +making parallelization the best match for the model. +IV. SIMULATIONS +In this section, we first explain the simulation setup and +then discuss the results of gene expression and molecular +production dynamics to prove the accuracy of the extracted +GRNN, emphasizing that it works similarly to the real GRN. +Later, we use computing through the GRNN to explain certain +activities of the biofilm. +A. Simulation Setup +As our interest is to investigate the NN-like computational +process, we do not model the formation process of the biofilm, +but we only remodel a completely formed biofilm and disre- +garding the maturation and dispersion stages. In this model, we +consider the biofilm as a static 3-D structure of bacterial cells. +Hence, we first place bacterial cells randomly in the model in a +paraboloid shape using the equation, z < x2 +5 + y2 +5 +20 where x, +y and z are the components of 3-D Cartesian coordinates. This +paraboloid shape is chosen to make the spacial arrangement of +the cells close to real biofilm while keeping the cell placement +process mathematically simple. Within this 3-D biofilm region, +we model the diffusivity according to DB/Daq = 0.4, which +is the mean relative diffusion [47] where DB and Daq are +the average molecular diffusion coefficients of the biofilm +and pure water, respectively. Further, to start the simulation +at a stage where the biofilm is fully formed and the MC is +already taking place, we filled the internal memory vector +of each cell with the average molecular level at the initial +TS. Each bacterial cell will use the initial signals from the +internal memory and use its GRNN to process and update the +feature vector for the next TS. Table I presents the parameter +descriptions and values used for the simulation. As shown in +Table I, the model runs for 150 TSs, generating data on a +range of functions for the system. For instance, this model can +produce data on feature vector of each cell, MC between cells, + +7 +Cell count % +0 +20 +Phos. Concentration % +40 +60 +0 +20 +40 +60 +80 +100 +80 +60 +40 +20 +0 +Time steps +(a) +Cell count % +0 +20 +Phos. Concentration % +40 +60 +0 +20 +40 +60 +80 +100 +80 +60 +40 +20 +0 +Time steps +(b) +Fig. 8: The nutrient accessibility variations of cells is ex- +pressed in two different environment conditions: a) low phos- +phate and b) high phosphate concentrations. +LP_WD +HP_WD +0 +25 50 75 100 125 150 +100 +80 +60 +40 +20 +0 +TS +Rel. pyocyanin acc. +(a) +0 +25 50 75 100125150 +100 +80 +60 +40 +20 +0 +TS +Rel. pyocyanin acc. +LP_lasR +HP_lasR +(b) +LP_phoB +HP_phoB +0 +25 50 75 100 125 150 +100 +80 +60 +40 +20 +0 +TS +Rel. pyocyanin acc. +(c) +LP_lasRphoB +HP_lasRphoB +0 +25 50 75 100 125 150 +100 +80 +60 +40 +20 +0 +TS +Rel. pyocyanin acc. +(d) +Fig. 9: Relative Pyocyanin accumulation of four different +biofilms of a) WD, b) lasR∆, c) phob∆ and d) lasR∆phob∆ +in both low and high phosphate levels. +molecular consumption by cells, secretion to the environment, +and nutrient accessibility of cells for each TS. +In order to prove that our GRNN computes similarly to +the natural bacterial cell and collective behaviors of the cells +are the same as the natural biofilm, we conduct a series of +experiments. We explore the GRNN computation and biofilm +activities under High Phosphate (HP) and Low Phosphate (LP) +levels using eight experimental setups as follows, 1) wild- +type bacteria (WD) in LP, 2) lasR mutant (lasR∆) in LP, 3) +phoB mutant (phoB∆) in LP, 4) lasR & PhoB double mutant +(LasR∆PhoB∆) in LP, 5) WD in HP, 6) lasR∆ in HP, 7) +PhoB∆ in HP and LasR∆PhoB∆ in HP. While the WD uses +the full GRNN, lasR∆ is created by making the weight of +the link between hn22 and lasR as “0”. Further, the GRNN of +phoB∆ is created by making the weights of links from PhoB +to hn23 and PhoB to pqsABCDE also “0”. +B. Model Validation +First, we show the nutrient accessibility variation in the +biofilm through Fig 8. The cells in the biofilm core have +less accessibility while the cells closer to the periphery have +more access to nutrients due to variations in diffusion between +100 +80 +60 +40 +20 +0 +WD +lasR +phoB +lasRphoB +Model +Wet-lab +HP to LP Pyocyanin +production ratio (%) +Fig. 10: Evaluation of the model accuracy by comparing HP +to LP pyocyanin production ratio with wet-lab data from [48]. +the environment and the EPS. Fig 8a shows that when a low +phosphate concentration is introduced to the environment, the +direct access to the nutrient by the cells is limited. After the +TS = 10, around 60% of cells have access to 20% of the +nutrient concentration. Further, Fig 8b shows that the increased +nutrient introduction to the environment positively reflects on +the accessibility. This accessibility plays a role mainly in the +deviation of gene expression patterns resulting in phenotypic +differentiations that is further analyzed in Section IV-C. +Comparing the predictions of molecular production through +GRNN computing with the wet-lab experimental data from +the literature, we are able to prove that the components of +the GRN work similarly to a NN. Fig. 9 shows the pyocyanin +accumulation variations of the environment in the eight setups +mentioned earlier as results of decision-making of the GRNN. +Production of pyocyanin of the WD P. aeruginosa biofilms +is high in LP, compared to the HP environments as shown +in Fig: 9a. Further, the same pattern can be observed in the +lasR∆ biofilms, but with a significantly increased pyocyanin +production in LP as shown in Fig. 9b. The phob∆ and +LasR∆phob∆ biofilms produce a reduced level of pyocyanin +compared to WD and LasR∆ that are shown in Fig. 9c and +Fig. 9d respectively. We then present a comparison between +GRNN prediction and wet-lab experimental data [48] as ratios +of HP to LP in Fig. 10. The differences between pyocyanin +production through GRNN in HP and LP condition for all the +four setups in Fig. 10 are fairly close to the wet-lab data. In +the WD setup, the difference between the GRNN model and +wet-lab data only has around 5% difference, while deviations +around 10% can be observed in lasR∆ and phoB∆. The most +significant deviation around 20% of pyocyanin production +difference is visible in lasR∆phoB∆ that is caused by the lack +of interaction from other gene expression pathways, as we only +extracted a sub-network portion of the GRN. Therefore, these +results prove that the extracted GRNN behaves similarly to +the GRN dynamics. +We further prove that the GRNN computing process per- +forms similarly to the GRN by comparing the gene expression +behaviors of the model with the wet-lab data [48] as shown in +Fig. 11. First, we show the expression dynamics of genes lasI, +pqsA and rhlR of WD in LP in Fig. 11a, Fig. 11b and Fig. 11c +respectively. All the figures depict that gene expression levels +are higher in LP compared to HP until around TS = 100. +Beyond that point, relative gene expression levels are close to +zero as the the environment run out of nutrients. Moreover, the +differences in gene expression levels predicted by the GRNN + +12 +10 +8 +6 +4 +2 +0 +120 +100 +80 +0 +10 +60 +20 +30 +40 +40 +50 +20 +60 +70 +0100 +Relative Pyocyanin Accumulation +LP WD +HP WD +80 +60 +40 +20 +0 +0 +25 +50 +75 +100 +125 +150 +Timesteps(Hrs)100 +Relative Pyocyanin Accumulation +80 +60 +40 +20 +LP lasR△ +HP lasR△ +0 +0 +25 +50 +75 +100 +125 +150 +Timesteps(Hrs)100 +LP PhoB△ +HP PhoB△ +80 +60 +40 +20 +0 +0 +25 +50 +75 +100 +125 +150 +Timesteps(Hrs)100 +LP LASR△ PHOB△ +HP LASRA PHOBA +80 +60 +40 +20 +0 +0 +25 +50 +75 +100 +125 +150 +Timesteps(Hrs)100 +Model +80 +Real +60 +40 +20 +0 +WD +lasRA +PHOBAlasRA PHOBA8 +7 +6 +5 +4 +3 +2 +1 +0 +120 +100 +80 +0 +10 +60 +20 +30 +40 +40 +50 +20 +60 +70 +08 +0 +25 +50 +75 +100 +125 +150 +8 +7 +6 +5 +4 +3 +2 +Relative lasI expression +POA1_LP +POA1_HP +TS +(a) +0 +25 +50 +75 +100 +125 +150 +POA1_LP +POA1_HP +TS +12 +10 +8 +6 +4 +2 +0 +Relative pqsA expression +(b) +0 +25 +50 +75 +100 +125 +150 +6 +5 +4 +3 +2 +1 +0 +Relative rhlR expression +POA1_LP +POA1_HP +TS +(c) +100 +80 +60 +40 +20 +0 +Model +Wet-lab +lasI +pqsA +rhlR +HP to LP gene +expression ratio (%) +(d) +Fig. 11: Expression levels of three different genes to that were used to prove the accuracy of the GRNN: a) lasI, b) pqsA, c) +rhlR expression levels in LP and HP and d) comparison between GRNN computing results and wet-lab data. +rhlI +BqsR +pqsABCDE +phz2 phz1 +PhoB +hn11 +pqsR +rhlR +lasR +lasI +pqsH +hn31 +hn12 +hn13 +hn14 +hn23 +hn24 +hn21 +hn22 +AlgR +czcR +TS +Genes +pqsABCDE +czcR +rhlI +pqsR +lasR +lasI +lasR +pqsH +phz2 +phz1 +0 3 6 9 12151821242730333639424548 +(a) +rhlI +BqsR +pqsABCDE +phz2 phz1 +PhoB +hn11 +pqsR +rhlR +lasR +lasI +pqsH +hn31 +hn12 +hn13 +hn14 +hn23 +hn24 +hn21 +hn22 +AlgR +czcR +TS +0 3 6 9 12151821242730333639424548 +(b) +rhlI +BqsR +pqsABCDE +phz2 phz1 +PhoB +hn11 +pqsR +rhlR +lasR +lasI +pqsH +hn31 +hn12 +hn13 +hn14 +hn23 +hn24 +hn21 +hn22 +AlgR +czcR +TS +0 3 6 9 12151821242730333639424548 +(c) +Fig. 12: Gene expression and associated information flow variations in GRNNs of a) WD b) lasR∆ and c) phoB∆ in LP. +rhlI +BqsR +pqsABCDE +phz2 phz1 +PhoB +hn11 +pqsR +rhlR +lasR +lasI +pqsH +hn31 +hn12 +hn13 +hn14 +hn23 +hn24 +hn21 +hn22 +AlgR +czcR +TS +Genes +pqsABCDE +czcR +rhlI +pqsR +lasR +lasI +lasR +pqsH +phz2 +phz1 +0 3 6 9 12151821242730333639424548 +(a) +TS +rhlI +BqsR +pqsABCDE +phz2 phz1 +PhoB +hn11 +pqsR +rhlR +lasR +lasI +pqsH +hn31 +hn12 +hn13 +hn14 +hn23 +hn24 +hn21 +hn22 +AlgR +czcR +0 3 6 9 12151821242730333639424548 +(b) +rhlI +BqsR +pqsABCDE +phz2 phz1 +PhoB +hn11 +pqsR +rhlR +lasR +lasI +pqsH +hn31 +hn12 +hn13 +hn14 +hn23 +hn24 +hn21 +AlgR +czcR +hn22 +TS +0 3 6 9 12151821242730333639424548 +(c) +Fig. 13: Gene expression and associated information flow variations in GRNNs of a) WD b) lasR∆ and c) phoB∆ in HP. +computing for LP and HP are also compared with the wet- +lab data in Fig. 11d. In this comparison, it is evident that +the predicted gene expression differences of all three genes +are close to the wet-lab data with only around 10% variation. +The performance similarities between the GRNN and real cell +activities once again prove that the GRN has underpinning +NN-like behaviors. +C. Analysis of GRNN Computing +Fig. 12 and Fig. 13 are used to show the diverse information +flow of the GRNN that cause the variations in pyocyanin + +Lasl +8 +POA1 LP +Relative Gene expression +¥7 +POA1 HP +2 +0 +25 +50 +75 +100 +125 +150 +Timesteps(Hrs)PqsA +12 +POA1 LP +Relative Gene expression +POA1 HP +10 +8 +6 +4 +2 +0 +0 +25 +50 +75 +100 +125 +150 +Timesteps(Hrs)RhiR +6 +POA1 LP +Relative Gene expression +¥5 +POA1 HP +¥32 +1 +0 +0 +25 +50 +75 +100 +125 +150 +Timesteps(Hrs)100 +Model +80 +Wet-lab +60 +40 +20 +0 +WD +lasRA +PHOBApqSABCDE +10 +CZCR +rhll +8 +pqsR +lasR- +6 +Lasl +rhIR - +4 +pqsH +phz2 +2 +phz1 +0 +912151821242730333639424548 +TSPqSABCDE +10 +CZCR +hll - +8 +pqsR - +lasR +6 +Lasl - +rhIR - +4 +pqsH +phz2 - +2 +phz1 +0 +J +9 12151821242730333639424548 +TSpqSABCDE +5 +CZCR +rhll : +pqsR +lasR - +3 +Lasl - +mIR - +2 +pqsH - +phz2 +phzl +0 +9 12151821242730333639424548 +TS8 +PqSABCDE +CZCR +rhli +6 +pqsR - +lasR- +Lasl - +4 +rhIR - +pqsH - +2 +phz2 +phz1 +0 +9 12151821242730333639424548 +TSPqSABCDE +CZCR +6 +rhll - +5 +pqsR - +lasR- +4 +Lasl- +3 +mIR - +pqsH - +2 +phz2- +1 +phz1 +0 +9 12151821242730333639424548 +TSPqSABCDE +CZCR +6 +rhll - +5 +pqsR - +lasR- +4 +Lasl- +3 +mIR - +pqsH - +2 +phz2- +1 +phz1 +0 +9 12151821242730333639424548 +TS9 +5 +4 +3 +2 +1 +0 +5 +4 +3 +2 +1 +0 +5 +4 +3 +2 +1 +0 +X +rhlI +BqsR +pqsABCDE +phz2 phz1 +PhoB +hn11 +pqsR +rhlR +lasR +lasI +pqsH +hn31 +hn12 +hn13 +hn14 +hn23 +hn24 +hn21 +hn22 +AlgR +czcR +rhlI +BqsR +pqsABCDE +phz2 phz1 +PhoB +hn11 +pqsR +rhlR +lasR +lasI +pqsH +hn31 +hn12 +hn13 +hn14 +hn23 +hn24 +hn21 +hn22 +AlgR +czcR +rhlI +BqsR +pqsABCDE +phz2 phz1 +PhoB +hn11 +pqsR +rhlR +lasR +lasI +pqsH +hn31 +hn12 +hn13 +hn14 +hn23 +hn24 +hn21 +AlgR +czcR +hn22 +rhlI +BqsR +pqsABCDE +phz2 phz1 +PhoB +hn11 +pqsR +rhlR +lasR +lasI +pqsH +hn31 +hn12 +hn13 +hn14 +hn23 +hn24 +hn21 +AlgR +czcR +hn22 +20.0 +20.0 +15.0 +10.0 +5.0 +0.0 +15.0 +10.0 +5.0 +0.0 +20.0 +15.0 +10.0 +5.0 +0.0 +Y +Z +TS +pqsABCDE +czcR +rhlI +pqsR +lasR +lasI +lasR +phz2 +phz1 +0 3 6 9 12 1518 21 24 2730 33 3639 42 45 48 +pqsH +TS +pqsABCDE +czcR +rhlI +pqsR +lasR +lasI +lasR +phz2 +phz1 +0 3 6 9 12151821 24 273033 3639 42 4548 +pqsH +TS +pqsABCDE +czcR +rhlI +pqsR +lasR +lasI +lasR +phz2 +phz1 +0 3 6 9 12 1518 21 24 2730 33 3639 42 45 48 +pqsH +TS +pqsABCDE +czcR +rhlI +pqsR +lasR +lasI +lasR +phz2 +phz1 +0 3 6 9 12 1518 21 24 2730 33 3639 42 45 48 +pqsH +5 +4 +3 +2 +1 +0 +(a) +(b) +(c) +(d) +Fig. 14: Illustration of GRNN information flow variations concerning the particular positions of cells within the biofilm. We +selected four cells at a) [10, 10, 0] – close to the attached surface, b) [10, 10, 5]- close to the periphery, c) [7, 10, 13] – at +the center and d) [3, 15, 0] – close to the attached surface and the periphery of the biofilm. +production in LP and HP conditions, respectively. Here, we +use gene expression profiles extracted from one bacterial cell +located at (7, 9, 2)µm in the Cartesian coordinates that is in +the middle region of the biofilm with limited access to the +nutrients. First, the gene expression variations of WD, lasR∆, +and phob∆ bacterial cells in LP (Fig. 12) and HP (Fig. 13) +are shown for TS < 50. Next, the information flow through +the GRNN is illustrated above each expression profile at time +TS = 20, where the variations will be discussed. In Fig. 12a, +impact of the inputs 3OC-LasR and phosphate cause higher +expression levels of the nodes hn12 and phoB in the input +layer that cascade the nodes phZ1, phZ2, pqsR, lasR, 3OC, +rhlR and PqsH in the output layer at TS = 20. Fig. 12b has +significantly higher pqsA operon expression levels compared +to HP conditions (Fig. 13b), reflecting higher pyocyanin pro- +duction that can be seen in Fig. 9b. Nevertheless, the reduced +gene expression levels, except pqsA operon, of lasR∆ biofilm +in both LP (Fig. 12b) and HP (Fig. 13b) conditions compared +to the other setups emphasize that the inputs via inter-cellular +MC significantly alter GRNN computing outputs. In contrast, +only a smaller gene expression difference can be observed +between the two setups of phob∆ in LP (Fig. 12c and phob∆ +in HP (Fig. 13c) resulting in minimized pyocyanin production +differences as shown earlier in Fig. 9c. +The GRNN model supports the understanding of the gene +expression variations due to factors such as nutrient acces- +sibility, where in our case is a single species biofilm. Fig. +14 depicts the variability in the gene expression levels for +four different locations of the biofilm at TS = 3. Fig. 14a +and Fig. 14b are the gene expression profiles and the signal +flow through GRNN pairs of two cells located close to the +attached surface and the center of the biofilm. The phosphate +accessibility for these two locations is limited. Hence, edges +from phob have a higher information flow compared to the +other two cells near the periphery of the biofilm, which can be +observed in Fig. 14c and Fig. 14d. The microbes in the center +(Fig. 14a) and the bottom (Fig. 14b) mainly have access to the +inter-cellular MCs, while the other two bacteria have direct +access to the extracellular phosphate. +This GRNN produced data can further be used to understand +the spatial and temporal dynamics of phenotypic clustering +of gene expressions which is important in predicting and +diagnosis of diseases [49]. Fig. 15 shows the phenotypic +variation of WD biofilm in LP. Fig. 15a shows the number of +cluster variations over the first 30 TSs when the significant +phenotypic changes of the biofilm is evident. At around +TS = 9 and TS = 10, the bacterial cells have the most diverse +expression patterns due to the highest extracellular nutrient +penetration (can be seen in Fig. 8a) to the biofilm and inter- +cellular communications. Here we use four TSs (TS = 5 - Fig. +15b, TS = 15 - Fig. 15c, TS = 23 - Fig. 15d and TS = 30 - +Fig. 15e) to analyze this phenotypic differentiation. Each pair +of Uniform Manifold Approximation and Projection (UMAP) +plot and diagram of cell locations of each cluster explain how +nutrient accessibility contribute to the phenotypic clustering. +Although at TS = 5 (Fig. 15) the average number of clusters +is over four, there are only two major clusters that can be +observed with higher proportions, as shown in the pie chart. +Among the two major clusters (blue and green) of Fig. 15b, +the bacteria in the blue cluster can mostly be found in the + +8 +PqSABCDE +CZCR +rhli +6 +pqsR - +lasR- +Lasl - +4 +rhIR - +pqsH - +2 +phz2 +phz1 +0 +912151821242730333639424548 +TS8 +pqsABCDE +CZCR +rhll - +6 +pqsR - +lasR - +Lasl - +4 +rhIR - +pqsH - +2 +phz2- +phzl +-0 +0 +6 +912151821242730333639424548 +TSpqsABCDE +5 +CZCR +rhll - +4 +pqsR +lasR - +3 +Lasl - +mIR - +2 +pqsH - +phz2 +phzl +0 +9 12151821242730333639424548 +TS17.5 +15.0 +12.5 +10.0 +7.5 +5.0 +2.5 +0.0 +20.0 17.5 15.0 12.5 10.0 +7.5 +5.0 +2.5 +X Label +YLabel +0.0PqSABCDE +5 +CZCR +rhll : +pqsR +lasR - +3 +Lasl - +mIR - +2 +pqsH - +phz2 +phzl +0 +9 12151821242730333639424548 +TS10 +5 +15 +TS = 5 +TS = 15 +TS = 25 +TS = 30 +30 +20 +10 +0 +-10 +-20 +-30 +-20 +-10 +0 +10 +20 +30 +30 +20 +10 +0 +-10 +-20 +-30 +-20 +-10 +0 +10 +20 +30 +30 +20 +10 +0 +-10 +-20 +-30 +-20 +-10 +0 +10 +20 +30 +30 +20 +10 +0 +-10 +-20 +-30 +-20 +-10 +0 +10 +20 +30 +20 +15 +10 +5 +0 +20 +15 +10 +5 +0 +0 +5 +10 +15 +20 +X +Y +Z +20 +15 +10 +5 +0 +20 +15 +10 +5 +0 +0 +5 +10 +15 +20 +X +Y +Z +20 +15 +10 +5 +0 +20 +15 +10 +5 +0 +0 +5 +10 +15 +20 +X +Y +Z +20 +15 +10 +5 +0 +20 +15 +10 +5 +0 +0 +5 +10 +15 +20 +X +Y +Z +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 +26 +27 +28 +29 +30 +20 +16 +12 +8 +4 +0 +No. of cluster +(b) +(c) +(d) +(e) +(a) +UMAP2 +UMAP2 +UMAP2 +UMAP2 +UMAP1 +UMAP1 +UMAP1 +UMAP 1 +TS +Fig. 15: Illustration of GRNN-driven phenotypic cluster formation behaviors. a) shows the number of clusters (with their +proportions via pie charts) for TS < 30, b), c), d) and e) are pairs of the UMAP clustering based on gene expressions of cells +and their locations in the biofilm at TS = 5, TS = 15, TS = 23 and TS = 30, respectively. +center of the biofilm, while the green cluster cells are close +to the periphery. Fig. 15c and Fig. 15d have more clusters +as the nutrient accessibility among cells is high. In contrast, +due to the lack of nutrients in the biofilm, a limited number +of clusters can be seen in the biofilm after around TS = 30, +which can be observed Fig. 15e. +V. CONCLUSION +The past literature has captured the non-linear signal com- +puting mechanisms of Bacterial GRNs, suggesting under- +pinning NN behaviors. This study extracts a GRNN with +summarized multi-omics gene expression regulation mecha- +nisms as weights that can further analyze gene expression +dynamics, design predictive models, or even conduct in-vivo +computational tasks. We used P. aeruginosa single species +biofilm as a use case and extracted relevant gene expression +data from databases such as RegulomePA and transcriptomic +data from databases including GEO. Due to the complexity +of the GRN and expression dynamics, we only considered a +smaller sub-network of the GRN as a GRNN that is associated +with QS, iron and phosphate inputs, and pyocyanin produc- +tion. Considering this GRNN, we modeled the computation +process that drives cellular decision-making mechanism. As +bacteria live in ecosystems in general where intra-cellular +communication play a significant role in cellular activities, +an in-silico biofilm is modeled using GNN to further analyze +the biofilm-wide decision-making. A comparison between +the GRNN generated data and the transcriptomic data from +the literature exhibits that the GRN behaves similarly to a +NN. Hence, this model can explore the causal relationships +between gene regulation and cellular activities, predict the +future behaviors of the biofilm as well as conduct bio-hybrid +computing tasks. 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S. Stewart, “Diffusion in biofilms,” Journal of Bacteriology, vol. 185, +no. 5, pp. 1485–1491, 2003. +[48] X. Meng, S. D. Ahator, and L.-H. Zhang, “Molecular mechanisms of +phosphate stress activation of pseudomonas aeruginosa quorum sensing +systems,” MSphere, vol. 5, no. 2, pp. e00119–20, 2020. +[49] P. Dutta, S. Saha, and S. Gulati, “Graph-based hub gene selection +technique using protein interaction information: Application to sample +classification,” IEEE journal of biomedical and health informatics, +vol. 23, no. 6, pp. 2670–2676, 2019. + diff --git a/CtE2T4oBgHgl3EQf9Ant/content/tmp_files/load_file.txt b/CtE2T4oBgHgl3EQf9Ant/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4b4b234fd1b66c56a25cb9bba27a336301e3beee --- /dev/null +++ b/CtE2T4oBgHgl3EQf9Ant/content/tmp_files/load_file.txt @@ -0,0 +1,1876 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf,len=1875 +page_content='1 Inferring Gene Regulatory Neural Networks for Bacterial Decision Making in Biofilms Samitha Somathilaka, Student, IEEE, Daniel P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Martins, Member, IEEE, Xu Li, Yusong Li, Sasitharan Balasubramaniam, Senior Member, IEEE, Abstract—Bacterial cells are sensitive to a range of external sig- nals used to learn the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' These incoming external sig- nals are then processed using a Gene Regulatory Network (GRN), exhibiting similarities to modern computing algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' An in- depth analysis of gene expression dynamics suggests an inherited Gene Regulatory Neural Network (GRNN) behavior within the GRN that enables the cellular decision-making based on received signals from the environment and neighbor cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' In this study, we extract a sub-network of Pseudomonas aeruginosa GRN that is associated with one virulence factor: pyocyanin production as a use case to investigate the GRNN behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Further, using Graph Neural Network (GNN) architecture, we model a single species biofilm to reveal the role of GRNN dynamics on ecosystem-wide decision-making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Varying environmental conditions, we prove that the extracted GRNN computes input signals similar to natural decision-making process of the cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Identifying of neural network behaviors in GRNs may lead to more accurate bacterial cell activity predictive models for many applications, including human health-related problems and agricultural applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Further, this model can produce data on causal relationships throughout the network, enabling the possibility of designing tailor-made infection-controlling mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' More interestingly, these GRNNs can perform computational tasks for bio-hybrid computing systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Index Terms—Gene Regulatory Networks, Graph Neural Net- work, Biofilm, Neural Network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' INTRODUCTION B ACTERIA are well-known for their capability to sense external stimuli, for complex information computations and for a wide range of responses [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' The microbes can sense numerous external signals, including a plethora of molecules, temperatures, pH levels, and the presence of other microorgan- isms [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' The sensed signals then go through the Gene Regu- latory Network (GRN), where a large number of parallel and sequential molecular signals are collectively processed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' The GRN is identified as the main computational component of the cell [3], which contains about 100 to more than 11000 genes Samitha Somathilaka is with VistaMilk Research Centre, Walton Institute for Information and Communication Systems Science, Waterford Institute of Technology, Waterford, X91 P20H, Ireland and School of Computing, University of Nebraska-Lincoln, 104 Schorr Center, 1100 T Street, Lincoln, NE, 68588-0150, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' E-mail: samitha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='somathilaka@waltoninstitute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='ie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Daniel P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Martins are with VistaMilk Research Centre and the Wal- ton Institute for Information and Communication Systems Science, Wa- terford Institute of Technology, Waterford, X91 P20H, Ireland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' E-mail: daniel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='martins@waltoninstitute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='ie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Xu Li and Yusong Li are with Department of Civil and Environmental Engineering University of Nebraska-Lincoln 900 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' 16th Street Nebraska Hall W181, Lincoln, NE 68588-0531 E-mail:xuli,yli7@unl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Balasubramaniam is with School of Computing, University of Nebraska- Lincoln, 104 Schorr Center, 1100 T Street, Lincoln, NE, 68588-0150, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' E-mail:sasi@unl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='edu Full GNN .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Extracted NN GRN Biofilm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' 1: Illustration of the Gene Regulatory Neural Networks (GRNN) extraction and the implementation of the GNN to model the biofilm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' The diffusion of molecules from one cell to another is modeled as a vector, where mq represents the concentration of the qth molecular signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' (the largest genome identified so far belongs to Sorangium cellulosum strain So0157-2) [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Despite the absence of neural components, the computational process through GRN allows the bacteria to actuate through various mechanisms, such as molecular production, motility, physiological state changes and even sophisticated social behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Understanding the natural computing mechanism of cells can lead to progression of key areas of machine learning in bioinformatics, including prediction of biological processes, prevention of diseases and personalized treatment [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Bacterial cells are equipped with various regulatory sys- tems, such as single/two/multi-component systems including Quorum sensing (QS), to respond to environmental stimuli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' The receptors and transporters on cell membranes can react and transport extracellular molecules, which subsequently in- teract with respective genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' In turn, the GRN is triggered to go through a complex non-linear computational process in response to the input signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' In the literature, it has been suggested that the computational process through the GRN of a bacterial cell comprises a hidden neural network (NN)-like architecture [6], [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' This indicates that, even though bacterial cells can be categorized as non-neural organisms, they perform neural decision-making processes through the GRN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' This re- sults in recent attention towards Molecular Machine Learning systems, where AI and ML are developed using molecular systems [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' In these systems, several neural components can be identified in GRNs, in which genes may be regarded as arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='04225v1 [q-bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='MN] 10 Jan 2023 2 C4- RhlR rhlI 3OC- LasR Phosphate Fe(II) BqsR pqsABCDE phz2 phz1 PQS- PQSR PhoB pqsR rhlR lasR LasI pqsH HHQ- PQSR ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' 3OC C4 PqsH HHQ PqsR LasR RhlR PQS AlgR czcR PhZ2 PhZ1 3OC C4 PqsH HHQ PqsR LasR RhlR (a) hn11 hn12 PhoB hn13 BqSR hn14 AlgR hn21 hn22 hn23 pqsABCDE czcR rhlI pqsR lasR lasI rhlR hn31 phz2 phz1 hn24 pqsH (b) GNN model Inter-cellular diffusion GRN NN .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' (c) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' 2: Extraction of a GRNN considering a specific sub-network of the GRN where a) is the two-component systems (TCSs) and QS network that is associated with the pyocyanin production, b) is the derived GRNN that is equipped with hypothetical nodes (hns) without affecting its computation process to form a symmetric network structure and c) is the conversion of real biofilm to the suggested in-silico model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' computational units or neurons, transcription regulatory factors as weights/biases and proteins/second messenger Molecular Communications (MC) as neuron-to-neuron interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Ow- ing to a large number of genes and the interactions in a GRN, it is possible to infer sub-networks with NN behaviors that we term Gene Regulatory Neural Networks (GRNN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' The non-linear computing of genes results from various factors that expand through multi-omics layers, including proteomic, transcriptomic and metabolomic data (further explained in Section II-A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' In contrast, the GRNN is a pure NN of genes with summarized non-linearity stemmed from multi-omics layers with weights/biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Identification of GRNNs can be used to model the decision- making process of the cell precisely, especially considering simultaneous multiple MC inputs or outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' However, due to the limited understanding and data availability, it is still impossible to model the complete GRN with its NN-like behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Therefore, this study uses a GRNN of Pseudomonas aeruginosa that is associated with PhoR-PhoB and BqsS- BqsR two-component systems (TCSs) and three QS systems related to pyocyanin production as a use case to explore the NN-like behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Although a single bacterium can do a massive amount of computing, they prefer living in biofilms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Hence, in order to understand the biofilm decision-making mechanism, we extend this single-cell computational model to an ecosystem level by designing an in-silico single species biofilm with inter-cellular MC signaling as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' The contributions of this study are as follows: Extracting a GRNN: Due to the complexity and insuf- ficient understanding of the gene expression dynamics of the full GRN, we only focus on a sub-network associated with pyocyanin production (shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' 2a) to inves- tigate the NN-like computational behavior of the GRN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Further, the genes of extracted sub-network are arranged following a NN structure that comprises input, hidden and output layers, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Modeling a biofilm as a GNN: The GRNN only repre- sents the single-cell activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' To model the biofilm-wide decision-making process, we use a Graph Neural Network (GNN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' First, we create a graph network of the bacterial cell and convert it to a GNN by embedding each node with the extracted GRNN as the update function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Second, the diffusion-based MCs between bacterial cells in the biofilm are encoded as the message-passing protocol of the GNN, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' 2c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Exploring the behaviors of the GRNN and intra- cellular MC dynamics to predict cell decisions: The output of the GRNN is evaluated by comparing it with the transcriptomic and pyocyanin production data from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Finally, an edge-level analysis of the GRNN is conducted to explore the causal relationships between gene expression and pyocyanin production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' This paper is organized as follows: Section II explains the background of bacterial decision-making in two levels: cellular-level in Section II-A and population-level in Section II-B, while the background on the P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' aeruginosa is introduced in Section II-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Section III is dedicated to explaining the model design of cellular and population levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' The results related to model validation and the intergenic intra-cellular signaling pattern analysis are presented in Section IV and the study is concluded in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' BACKGROUND As the model expands through single cellular and biofilm- wide decision-making layers, this section provides the back- ground of how a bacterium uses the GRN to make decisions and how bacterial cells make decisions in biofilms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Moreover, we briefly discuss the cellular activities of the Pseudomonas aeruginosa as it is the use case species of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Decision-Making Process of an Individual Cell Prokaryotic cells are capable of sensing the environment through multiple mechanisms, including TCSs that have been widely studied and it is one of the focal points of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' The concentrations of molecular-input signals from the extracellular environment influence the bacterial activities at the cellular and ecosystem levels [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Apart from the extracellular signals of nutrients, it is evident that the QS input signals have a diverse set of regulative mechanisms in biofilm-wide characteristics, including size and shape [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' These input signals undergo a computational process through the GRN, exhibiting a complex decision-making mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Past studies have explored and suggested this underpinning computational mechanism in a plethora of directions, such 3 Prom Op Enh GeneA .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Sil GeneB .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' 3: Illustration of gene expression regulators that are considered the weight influencers of the edges of GRNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Here, the α(σ), α(∼σ), α(T F ), α(Rep), α(eT F ) and α(sT F ) are relative concentrations of sigma factors, anti-sigma factors, transcription factors (TFs), repressors, enhancer-binding TFs and silencer-binding TFs respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Moreover, β(P rom), β(Op), β(Enh), and β(Sil) are the binding affinities of the pro- moter, operator, enhancer and silencers regions respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' as using differential equations [11] and probabilistic Boolean networks [12] and logic circuit [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' All of these models mainly infer that the bacterial cells can make decisions not just based on the single input-output combinations, but they can integrate several incoming signals non-linearly to produce outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' The studies that focus on differences in gene expression levels suggest that a hidden weight behavior controls the impact of one gene on another [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' This weight behavior emerges through several elements, such as the number of transcription factors that induce the expression, the affinity of the transcription factor binding site, and machinery such as thermoregulators and enhancers/silencers [14], [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' 3 depicts a set of factors influencing the weight between genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' The weight of an edge between two genes has a higher dynamicity as it is combinedly determined by several of these factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Based on environmental conditions, the GRN of the bacterial cell adapts various weights to increase the survivability and repress unnecessary cellular functions to preserve energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' An example of such regulation is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' 4 where a P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' aeruginosa cell uses a thermoregulator to regulate the QS behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' 4a has a set of relative weights based on cellar activities in an environment at 37 ◦C, while Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' 4b represents weights at 30 ◦C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' The weights between the hn21 and rhlR are different in two conditions, and these cellar activities are further explained in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Biofilm Decision-Making Even though an individual cell is capable of sensing, computing, and actuating, the majority of bacterial cells live in biofilms, where the survivability is significantly increased compared to their planktonic state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Biofilm formation can cause biofouling and corrosion in water supply and industrial systems [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' However, biofilms formation can be desirable in many situations, for example, bioreactors in wastewater treatment [17], bioremediation of contaminated groundwater [18], [19], where biofilms serve as platforms for biogeochem- ical reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' A massive number of factors can influence biofilm formation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' including substratum surface geometrical characteristics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' diversity of species constituting the biofilm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' hydrodynamic conditions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' nutrient availability,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' and especially communication patterns [20] where the TCS and QS play rhlI BqsR pqsABCDE phz2 phz1 PhoB hn11 pqsR rhlR lasR LasI pqsH hn31 hn12 hn13 hn14 hn23 hn24 hn21 hn22 AlgR czcR 1 1 0 (a) rhlI BqsR pqsABCDE phz2 phz1 PhoB hn11 pqsR rhlR lasR LasI pqsH hn31 hn12 hn13 hn14 hn23 hn24 hn21 hn22 AlgR czcR 1 1 0 (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' 4: Two GRNN setups with different weights associated with two environmental conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' a) is the relative weight setup of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' aeruginosa cell in 37 ◦C and b) is in 30 ◦C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' significant roles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' A TCS comprises a histidine kinase that is the sensor for specific stimulus and a cognate response regulator that initiates expressions of a set of genes [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Hence, in each stage, essential functions can be traced back to their gene expression upon a response to the input signals detected by bacterial cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' For instance, in the first stage of biofilm formation, the attachment of bacteria to a surface is associated with sensing a suitable surface and altering the activities of the flagella.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' In the next stage, rhamnolipids production is associated with ferric iron Fe3+ availability in the environment, mostly sensed through BqsS-BqsR TCSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Further, Fe3+ was identified as a regulator of pqsA, pqsR, and pqsE gene expressions that are associated with the production of two critical components for the formation of microcolonies: eDNA and EPS [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Similarly, in the final stage, the dis- persion process can also be traced back to a specific set of gene regulations, including bdlA an rbdA [23], [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' An understanding of the underlying decision-making process of bacteria may enable us to control their cellular activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Pseudomonas Aeruginosa The main reason for selecting P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' aeruginosa in this work lies in its alarming role in human health.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' For example, this species is the main cause of death in cystic fibrosis patients [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' aeruginosa is a gram-negative opportunistic pathogen with a range of virulence factors, including pyocyanin and cytotoxin secretion [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' These secreted molecules can lead to complications such as respiratory tract ciliary dysfunction and induce proinflammatory and oxidative effects damaging the host cells [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' The biofilms are being formed on more than 90% endotracheal tubes implanted in patients who are getting assisted ventilation, causing upper respiratory tract infections [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' In addition, another important reason for targeting P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' aeruginosa is the data availability for the GRN structure [29], pathways [30], genome [31], transcriptome [32] and data from mutagenesis studies [33], [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Compared to the complexity of the GRN, the amount of data and information available on the 4 C4- RhlR 3OC- LasR PQS- PQSR HHQ- PQSR 3OC C4 PqsH HHQ PqsR LasR RhlR PQS Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' 5: Illustrations of intra-cellular metabolite interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' gene-to-gene interactions and expression patterns is insuffi- cient to develop an accurate full in-silico model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Therefore, we chose a set of specific genes that are associated with QS, TCS, and pyocyanin production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' SYSTEM DESIGN This section explains the system design in two main phases, extracting a NN-like architecture from the GRN targeting the set of genes and creating a model of the biofilm ecosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Extracting Natural Neural Network from GRN We first fetch the structure of the GRN graph from Reg- ulomePA [29] database that contains only the existence of interactions and their types (positive or negative).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' As the next step, using information from the past studies [35]–[38], we identified the genes involved in the Las, Rhl and PQS QS systems, PhoR-PhoB and BqsS-BqsR TCSs, and pyocyanin production to derive the sub-network of GRN as shown in Fig 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' We further explored the expression dynamics using transcriptomic data [39], [40] where we observed the non- linearity in computations that are difficult to capture with existing approaches such as logic circuits, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' [6], making the NN approach more suitable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' However, a NN model with a black box that is trained on a large amount of transcriptomic data records to do computations similar to the GRN has a number of limitations, especially in understanding the core of the computational process [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Our model does not use a conventional NN model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' instead, we extract a NN from the in- teraction patterns of the GRN, which we consider a pre-trained GRNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' In this sub-network, we observed that the lengths of expression pathways are not equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' For example, the path from PhoR-PhoB to the phz2 gene has two hops, but the path from the BqsS-BqsR system to the rhlR gene only has one hop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' The extracted network has the structure of a random NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Hence, we transform this GRNN to Gene Regulatory Feedforward Neural Network by introducing hypothetical nodes (hns) that do not affect the behaviors of the GRNN as shown in Fig 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' In this transformation, we decide the number of hidden layers based on the maximum number of hops in gene expression pathways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' In our network, the maximum number of hops is two, which determines the number of hidden layers as one, and then the number of hops of all the pathways is leveled by introducing hns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' If a hn is introduced between a source and target genes, the edge weights from the source node to the hn and from hn to the target node are made “1” so that the hn does not have an influence on the regulation of genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Moreover, if a gene does not induce another in the network, the weight of the edge between that pair is made “0”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Here, we summarize multiple factors of interaction into a weight that determines the transcriptional regulation of a particular gene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' This regulation process occurs when the gene products get bound to the promoter region of another, influencing the transcriptional machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Hence, we observe this regulation process of a target gene as a multi-layered model that relies on the products of a set of source genes, the interaction between gene products, and the diffusion dynamics within the cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Creating a framework to infer an absolute weight value using all the above factors is a highly complex task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' In order to infer weight, one method is to train a NN model with the same structure as the GRN using a series of transcriptomic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' However, this approach also has numerous challenges, such as the lack of a sufficient amount of data in similar environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Therefore, we estimate a set of relative weights based on genomic, transcriptomic, and proteomic explanations of each interaction from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' The weights were further fine- tuned using the transcriptomics data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' A relative weight value of an edge can be considered a summarizing of multi-layer transcriptional-translation to represent the impact of the source gene on a target gene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' In this computational process, we identify another layer of interactions that occur within the cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' The produced molecules by the considered TCs network go through a set of metabolic interactions that are crucial for the functionality of the cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Since our primary goal is to explore the NN behaviors of GRN, we model these inter-cellular chemical reactions as a separate process, keeping the gene-to-gene interactions and metabolic interactions in two different layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' To model the complete pyocyanin production functionality of the cell, we use the inter-cellular molecular interactions shown in Fig 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Here, RhlR is a transcriptional regulator of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' aeruginosa that forms a complex by getting attached to its cognate inducer C4-HSL and then binds to the promoter regions of relevant genes [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Similarly, LasR transcriptional regulator protein and 3-oxo-C12-HSL (3OC), and PqsR with PQS and HHQ form complexes and get involved in the regulation of a range of genes [43], [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Further, C10H10O6 in the environment are converted by the P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' aeruginosa cells in multiple steps using the products of the GRNN we consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' First, C10H10O6 is converted into phenazine-1-carboxylic using the enzymes of pqsABCDEFG genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Later, phenazine-1-carboxylic was converted into 5-Methylphenazine-1-carboxylate, and finally, 5-Methylphenazine-1-carboxylate into Pyocyanin by PhzM and PhzS, respectively [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Molecular accumulation within a bacterial cell can be considered its memory module where certain intra-cellular interactions occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Therefore, we define an internal memory matrix IM as, IM(t) = im1 im2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' imJ � � � � � � � � � � � � B1 C(t) (1,im1) C(t) (1,im2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' C(t) (1,imJ) B2 C(t) (2,im1) C(t) (2,im2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' C(t) (2,imJ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' BP C(t) (P,im1) C(t) (P,im2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' C(t) (P,imJ) , (1) where the concentration of the internal molecule imj is 5 C(t) (i,imj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' GRNN process molecular signals from the environment and other cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Hence, we used the approach of GNN as a scalable mechanism to model the MCs and biofilm wide decision- making process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' The extreme computational power demand of modeling the diffusion-based MCs of each cell is also avoided by using this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Graph Neural Network Modeling of Biofilm First, the biofilm is created as a graph network of bacterial cells where each node is a representation of a cell, and an edge between two nodes is a MC channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' We convert the graph network into a Graph Neural Network (GNN) in three steps: 1) embedding the extracted GRNN of pyocyanin production into each node as the update function, 2) encoding the diffusion- based cell-to-cell MC channels as the message passing scheme, and 3) creating an aggregation function at the reception of molecular messages by a node as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Next, we define feature vectors of each node of the GNN to represent the gene expression profile of the individual cell at a given time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Subsequently, considering L is the number of genes in the GRNN, P is the number of bacterial cells in the biofilm and b(t) (i,gl) is the expression of gene gl by the bacteria Bi, we derive the following matrix FV(t) that represents all the feature vectors of the GNN at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' FV(t) = g1 g2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' gL � � � � � � � � � � � � B1 b(t) (1,g1) b(t) (1,g2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' b(t) (1,gL) B2 b(t) (2,g1) b(t) (2,g2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' b(t) (2,gL) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' BP b(t) (P,g1) b(t) (P,g2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' b(t) (P,gL) (2) The computational output of the GRNN of each node results in the secretion of a set of molecules that are considered messages in our GNN model as illustrated in the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' When the number of molecular species considered in the network is Q and output mq molecular message from bacterial cell Bi at TS t is msg(t) (i,mq), we derive the matrix MSG(t) = m1 m2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' mQ � � � � � � � � � � � � B1 msg(t) (1,m1) msg(t) (1,m2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' msg(t) (1,mQ) B2 msg(t) (2,m1) msg(t) (2,m2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' msg(t) (2,mQ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' BP msg(t) (P,m1) msg(t) (P,m2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' msg(t) (P,mQ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' (3) Further, we use a static diffusion coefficients vector D = {Dm1, Dm2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=', DmQ}, (4) where Dmq is diffusion coefficient of molecular species mq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Gene expression profile of bacterial cell b1 at time step t (a) B2 B5 B6 B1 B3 B7 B2 B5 B6 B1 B3 B7 B2 B5 B6 B1 B3 B7 B2 B5 B6 B1 B3 B7 t=0 t=1 t=2 t=T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' 6: Illustration of the GNN components where a) is a snapshot of the bacterial network that has the gene expression profile as the feature vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Further, this gene expression pattern of a cell is encoded to a message of secreted molecules where MC plays a crucial role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Moreover, b) shows the temporal behavior of the GNN, that the output of one graph snapshot influences the next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' 7: The process of one GRNN outputs reaching another GRNN as molecular messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' We define another matrix ED that contains the euclidean distances between bacterial cells in the biofilm as follows ED = B1 B2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' BP � � � � � � � � B1 d(1,1) d(1,2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' d(1,P ) B2 d(2,1) d(2,2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' d(2,P ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' BP d(P,1) d(P,2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' d(P,P ) (5) where di,j is the euclidean distance between the ith and jth cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' 6 TABLE I: Parameters utilised in the system development Parameter Value Description No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' of cells 2000 The number of cells is limited due to the memory availability of the server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' of genes 13 The network only consists of the gene that are directly associated with QS, PhoR-PhoB and BqsS-BqsR TCSs, and pyocyanin production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' internal memory molecules 16 The set of molecules that involved in QS, PhoR-PhoB and BqsS-BqsR TCSs,and pyocyanin production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' messenger molecules 4 The number of molecules that were exchanged between cells in the sub network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Dimensions of the environment 20x20x20µm The dimensions were fixed considering the average sizes of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' aeruginosa biofilms and computational demand of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Duration 150 TSs The number of TSs can be modified to explore the cellular and ecosystem level activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' For this experiment we fixed a TS to represent 30mins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' iterations per setup 10 Considering the stochasticity ranging from the gene expression to ecosystem-wide communications, the experiments were iterated 10 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' The feature vector of ith bacterial cell at the TS t + 1 is then modeled as, FV(t+1) i = GRNNi(MSG(t) i + S(t) i ) (6) where MSG(t) i is the message generated by the same cell in the previous TS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' The GRNNi is the extracted GRNN that is the update function in the GNN learning process and S(t) i = R(t) i + K(t) i , is the aggregate function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' In the aggregation component, the R(t) i is the incoming signals from peer bacterial cells and K(t+1) (i:mq) is the external molecule input vector at the location of Bi and the TS t that is expressed as K(t+1) i = � K(t+1) i:m1 , K(t+1) i:m2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=', K(t+1) i:mQ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' (7) In order to compute R(t+1) i , we use a matrix Yi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Yi = ↔ 1 [Q×1] ×EDi, where ↔ 1 [Q×1] is an all-ones matrix of dimension Q × 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' The ˆg matrix is then defined as follows, ˆg(D⊺, Y, t) = � ���� g(Dm1, d(i,1), t) g(Dm1, d(i,2), t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' g(Dm1, d(i,P ), t) g(Dm2, d(i,1), t) g(Dm2, d(i,2), t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' g(Dm2, d(i,P ), t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' g(DmQ, d(i,1), t) g(DmQ, d(i,2), t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' g(DmQ, d(i,P ), t) � ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' (8) In the above matrix, g(Dml, d(i,j), t) is the Green’s function of the diffusion equation as shown below, ˆG(Dml, d(i,j), t) = 1 (4πDmlt) 3 2 exp � − d2 (i,j) 4Dmlt � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' (9) Further, the incoming signal vector R(t+1) i is denoted as below, R(t+1) i = diag � ˆg(D⊺, Y, t) × MSG(t)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' (10) Further, we equip our model with a 3-D environment to compensate for the noise element and external molecule inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Environment-layer is designed as a 3-D grid of voxels that can store precise information on external nutrients (simi- larly to our previous model in [46]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' The diffusion of nutrient molecules through the medium is modeled as a random-walk process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' This layer allows us to enrich the model with the dynamics of nutrient accessibility of bacterial cells due to diffusion variations between the medium and the Extracellular Polymeric Substance (EPS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' The bacterial cells in the ecosystem also perform their own computing tasks individually, resulting in a massively parallel processing framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Hence, we use the python-cuda platform to make our model closer to the parallel processing architecture of the biofilm, where we dedicate a GPU block for each bacterial cell and the threads of each block for the matrix multiplication of the GRNN computation associated with the particular cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Additionally, due to the massive number of iterative components in the model, the computational power demand faces significant challenges with serial programming making parallelization the best match for the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' SIMULATIONS In this section, we first explain the simulation setup and then discuss the results of gene expression and molecular production dynamics to prove the accuracy of the extracted GRNN, emphasizing that it works similarly to the real GRN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Later, we use computing through the GRNN to explain certain activities of the biofilm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Simulation Setup As our interest is to investigate the NN-like computational process, we do not model the formation process of the biofilm, but we only remodel a completely formed biofilm and disre- garding the maturation and dispersion stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' In this model, we consider the biofilm as a static 3-D structure of bacterial cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Hence, we first place bacterial cells randomly in the model in a paraboloid shape using the equation, z < x2 5 + y2 5 +20 where x, y and z are the components of 3-D Cartesian coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' This paraboloid shape is chosen to make the spacial arrangement of the cells close to real biofilm while keeping the cell placement process mathematically simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Within this 3-D biofilm region, we model the diffusivity according to DB/Daq = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='4, which is the mean relative diffusion [47] where DB and Daq are the average molecular diffusion coefficients of the biofilm and pure water, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Further, to start the simulation at a stage where the biofilm is fully formed and the MC is already taking place, we filled the internal memory vector of each cell with the average molecular level at the initial TS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Each bacterial cell will use the initial signals from the internal memory and use its GRNN to process and update the feature vector for the next TS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Table I presents the parameter descriptions and values used for the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' As shown in Table I, the model runs for 150 TSs, generating data on a range of functions for the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' For instance, this model can produce data on feature vector of each cell, MC between cells, 7 Cell count % 0 20 Phos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Concentration % 40 60 0 20 40 60 80 100 80 60 40 20 0 Time steps (a) Cell count % 0 20 Phos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Concentration % 40 60 0 20 40 60 80 100 80 60 40 20 0 Time steps (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' 8: The nutrient accessibility variations of cells is ex- pressed in two different environment conditions: a) low phos- phate and b) high phosphate concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' LP_WD HP_WD 0 25 50 75 100 125 150 100 80 60 40 20 0 TS Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' pyocyanin acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' (a) 0 25 50 75 100125150 100 80 60 40 20 0 TS Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' pyocyanin acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' LP_lasR HP_lasR (b) LP_phoB HP_phoB 0 25 50 75 100 125 150 100 80 60 40 20 0 TS Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' pyocyanin acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' (c) LP_lasRphoB HP_lasRphoB 0 25 50 75 100 125 150 100 80 60 40 20 0 TS Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' pyocyanin acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' (d) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' 9: Relative Pyocyanin accumulation of four different biofilms of a) WD, b) lasR∆, c) phob∆ and d) lasR∆phob∆ in both low and high phosphate levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' molecular consumption by cells, secretion to the environment, and nutrient accessibility of cells for each TS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' In order to prove that our GRNN computes similarly to the natural bacterial cell and collective behaviors of the cells are the same as the natural biofilm, we conduct a series of experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' We explore the GRNN computation and biofilm activities under High Phosphate (HP) and Low Phosphate (LP) levels using eight experimental setups as follows, 1) wild- type bacteria (WD) in LP, 2) lasR mutant (lasR∆) in LP, 3) phoB mutant (phoB∆) in LP, 4) lasR & PhoB double mutant (LasR∆PhoB∆) in LP, 5) WD in HP, 6) lasR∆ in HP, 7) PhoB∆ in HP and LasR∆PhoB∆ in HP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' While the WD uses the full GRNN, lasR∆ is created by making the weight of the link between hn22 and lasR as “0”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Further, the GRNN of phoB∆ is created by making the weights of links from PhoB to hn23 and PhoB to pqsABCDE also “0”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Model Validation First, we show the nutrient accessibility variation in the biofilm through Fig 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' The cells in the biofilm core have less accessibility while the cells closer to the periphery have more access to nutrients due to variations in diffusion between 100 80 60 40 20 0 WD lasR phoB lasRphoB Model Wet-lab HP to LP Pyocyanin production ratio (%) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' 10: Evaluation of the model accuracy by comparing HP to LP pyocyanin production ratio with wet-lab data from [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' the environment and the EPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Fig 8a shows that when a low phosphate concentration is introduced to the environment, the direct access to the nutrient by the cells is limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' After the TS = 10, around 60% of cells have access to 20% of the nutrient concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Further, Fig 8b shows that the increased nutrient introduction to the environment positively reflects on the accessibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' This accessibility plays a role mainly in the deviation of gene expression patterns resulting in phenotypic differentiations that is further analyzed in Section IV-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Comparing the predictions of molecular production through GRNN computing with the wet-lab experimental data from the literature, we are able to prove that the components of the GRN work similarly to a NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' 9 shows the pyocyanin accumulation variations of the environment in the eight setups mentioned earlier as results of decision-making of the GRNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Production of pyocyanin of the WD P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' aeruginosa biofilms is high in LP, compared to the HP environments as shown in Fig: 9a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Further, the same pattern can be observed in the lasR∆ biofilms, but with a significantly increased pyocyanin production in LP as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' 9b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' The phob∆ and LasR∆phob∆ biofilms produce a reduced level of pyocyanin compared to WD and LasR∆ that are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' 9c and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' 9d respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' We then present a comparison between GRNN prediction and wet-lab experimental data [48] as ratios of HP to LP in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' The differences between pyocyanin production through GRNN in HP and LP condition for all the four setups in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' 10 are fairly close to the wet-lab data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' In the WD setup, the difference between the GRNN model and wet-lab data only has around 5% difference, while deviations around 10% can be observed in lasR∆ and phoB∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' The most significant deviation around 20% of pyocyanin production difference is visible in lasR∆phoB∆ that is caused by the lack of interaction from other gene expression pathways, as we only extracted a sub-network portion of the GRN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Therefore, these results prove that the extracted GRNN behaves similarly to the GRN dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' We further prove that the GRNN computing process per- forms similarly to the GRN by comparing the gene expression behaviors of the model with the wet-lab data [48] as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' First, we show the expression dynamics of genes lasI, pqsA and rhlR of WD in LP in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' 11a, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' 11b and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' 11c respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' All the figures depict that gene expression levels are higher in LP compared to HP until around TS = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Beyond that point, relative gene expression levels are close to zero as the the environment run out of nutrients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Moreover,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='differences in gene expression levels predicted by the GRNN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='120 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='0100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='Relative Pyocyanin Accumulation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='LP WD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='HP WD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='125 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='Timesteps(Hrs)100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='Relative Pyocyanin Accumulation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='LP lasR△ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='HP lasR△ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='125 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='Timesteps(Hrs)100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='LP PhoB△ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='HP PhoB△ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='125 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='Timesteps(Hrs)100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='LP LASR△ PHOB△ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='HP LASRA PHOBA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='0 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='Timesteps(Hrs)100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='Model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='Real ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='WD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='lasRA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='PHOBAlasRA PHOBA8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='5 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='125 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='Relative lasI expression ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='125 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='POA1_LP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='POA1_HP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='TS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='Relative pqsA expression ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='(b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='125 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='Relative rhlR expression ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='POA1_LP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='POA1_HP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='TS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='(c) ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' 11: Expression levels of three different genes to that were used to prove the accuracy of the GRNN: a) lasI, b) pqsA, c) rhlR expression levels in LP and HP and d) comparison between GRNN computing results and wet-lab data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='rhlI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='BqsR ' metadata={'source': 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variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' The performance similarities between the GRNN and real cell activities once again prove that the GRN has underpinning NN-like behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Analysis of GRNN Computing Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' 12 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' 13 are used to show the diverse information ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='0 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='0 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='0 Y Z TS pqsABCDE czcR rhlI pqsR lasR lasI lasR phz2 phz1 0 3 6 9 12 1518 21 24 2730 33 3639 42 45 48 pqsH TS pqsABCDE czcR rhlI pqsR lasR lasI lasR phz2 phz1 0 3 6 9 12151821 24 273033 3639 42 4548 pqsH TS pqsABCDE czcR rhlI pqsR lasR lasI lasR phz2 phz1 0 3 6 9 12 1518 21 24 2730 33 3639 42 45 48 pqsH TS pqsABCDE czcR rhlI pqsR lasR lasI lasR phz2 phz1 0 3 6 9 12 1518 21 24 2730 33 3639 42 45 48 pqsH 5 4 3 2 1 0 (a) (b) (c) (d) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' 14: Illustration of GRNN information flow variations concerning the particular positions of cells within the biofilm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' We selected four cells at a) [10, 10, 0] – close to the attached surface, b) [10, 10, 5]- close to the periphery, c) [7, 10, 13] – at the center and d) [3, 15, 0] – close to the attached surface and the periphery of the biofilm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' production in LP and HP conditions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Here, we use gene expression profiles extracted from one bacterial cell located at (7, 9, 2)µm in the Cartesian coordinates that is in the middle region of the biofilm with limited access to the nutrients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' First, the gene expression variations of WD, lasR∆, and phob∆ bacterial cells in LP (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' 12) and HP (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' 13) are shown for TS < 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Next, the information flow through the GRNN is illustrated above each expression profile at time TS = 20, where the variations will be discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' 12a, impact of the inputs 3OC-LasR and phosphate cause higher expression levels of the nodes hn12 and phoB in the input layer that cascade the nodes phZ1, phZ2, pqsR, lasR, 3OC, rhlR and PqsH in the output layer at TS = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' 12b has significantly higher pqsA operon expression levels compared to HP conditions (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' 13b), reflecting higher pyocyanin pro- duction that can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' 9b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Nevertheless, the reduced gene expression levels, except pqsA operon, of lasR∆ biofilm in both LP (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' 12b) and HP (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' 13b) conditions compared to the other setups emphasize that the inputs via inter-cellular MC significantly alter GRNN computing outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' In contrast, only a smaller gene expression difference can be observed between the two setups of phob∆ in LP (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' 12c and phob∆ in HP (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' 13c) resulting in minimized pyocyanin production differences as shown earlier in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' 9c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' The GRNN model supports the understanding of the gene expression variations due to factors such as nutrient acces- sibility, where in our case is a single species biofilm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' 14 depicts the variability in the gene expression levels for four different locations of the biofilm at TS = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' 14a and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' 14b are the gene expression profiles and the signal flow through GRNN pairs of two cells located close to the attached surface and the center of the biofilm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' The phosphate accessibility for these two locations is limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Hence, edges from phob have a higher information flow compared to the other two cells near the periphery of the biofilm, which can be observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' 14c and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' 14d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' The microbes in the center (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' 14a) and the bottom (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' 14b) mainly have access to the inter-cellular MCs, while the other two bacteria have direct access to the extracellular phosphate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' This GRNN produced data can further be used to understand the spatial and temporal dynamics of phenotypic clustering of gene expressions which is important in predicting and diagnosis of diseases [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' 15 shows the phenotypic variation of WD biofilm in LP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' 15a shows the number of cluster variations over the first 30 TSs when the significant phenotypic changes of the biofilm is evident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' At around TS = 9 and TS = 10, the bacterial cells have the most diverse expression patterns due to the highest extracellular nutrient penetration (can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' 8a) to the biofilm and inter- cellular communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Here we use four TSs (TS = 5 - Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' 15b, TS = 15 - Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' 15c, TS = 23 - Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' 15d and TS = 30 - Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' 15e) to analyze this phenotypic differentiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Each pair of Uniform Manifold Approximation and Projection (UMAP) plot and diagram of cell locations of each cluster explain how nutrient accessibility contribute to the phenotypic clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Although at TS = 5 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' 15) the average number of clusters is over four, there are only two major clusters that can be observed with higher proportions, as shown in the pie chart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Among the two major clusters (blue and green) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' 15b, the bacteria in the blue cluster can mostly be found in the 8 PqSABCDE CZCR rhli 6 pqsR - lasR- Lasl - 4 rhIR - pqsH - 2 phz2 phz1 0 912151821242730333639424548 TS8 pqsABCDE CZCR rhll - 6 pqsR - lasR - Lasl - 4 rhIR - pqsH - 2 phz2- phzl 0 0 6 912151821242730333639424548 TSpqsABCDE 5 CZCR rhll - 4 pqsR lasR - 3 Lasl - mIR - 2 pqsH - phz2 phzl 0 9 12151821242730333639424548 TS17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='5 15.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content='No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' of cluster (b) (c) (d) (e) (a) UMAP2 UMAP2 UMAP2 UMAP2 UMAP1 UMAP1 UMAP1 UMAP 1 TS Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' 15: Illustration of GRNN-driven phenotypic cluster formation behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' a) shows the number of clusters (with their proportions via pie charts) for TS < 30, b), c), d) and e) are pairs of the UMAP clustering based on gene expressions of cells and their locations in the biofilm at TS = 5, TS = 15, TS = 23 and TS = 30, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' center of the biofilm, while the green cluster cells are close to the periphery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' 15c and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' 15d have more clusters as the nutrient accessibility among cells is high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' In contrast, due to the lack of nutrients in the biofilm, a limited number of clusters can be seen in the biofilm after around TS = 30, which can be observed Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' 15e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' CONCLUSION The past literature has captured the non-linear signal com- puting mechanisms of Bacterial GRNs, suggesting under- pinning NN behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' This study extracts a GRNN with summarized multi-omics gene expression regulation mecha- nisms as weights that can further analyze gene expression dynamics, design predictive models, or even conduct in-vivo computational tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' We used P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' aeruginosa single species biofilm as a use case and extracted relevant gene expression data from databases such as RegulomePA and transcriptomic data from databases including GEO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Due to the complexity of the GRN and expression dynamics, we only considered a smaller sub-network of the GRN as a GRNN that is associated with QS, iron and phosphate inputs, and pyocyanin produc- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Considering this GRNN, we modeled the computation process that drives cellular decision-making mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' As bacteria live in ecosystems in general where intra-cellular communication play a significant role in cellular activities, an in-silico biofilm is modeled using GNN to further analyze the biofilm-wide decision-making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' A comparison between the GRNN generated data and the transcriptomic data from the literature exhibits that the GRN behaves similarly to a NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Hence, this model can explore the causal relationships between gene regulation and cellular activities, predict the future behaviors of the biofilm as well as conduct bio-hybrid computing tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Further, in the GRNN extraction phase, we were able to identify the possibility of modeling more network structures with various number of input nodes, hidden layers, and output nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' In addition, GRN components including auto regulated genes and bidirectional intergenic interactions hints the possibility of extracting more sophisticated types of GRNNs such as Recurrent NN and Residual NN in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' The idea of extracting sub-networks as NNs can lead to more intriguing intra-cellular distributed computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Further, this model can be extended to multi-species ecosystems for more advanced predictive models as well as distributed computing architectures combining various NNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' REFERENCES [1] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Alm, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf'} +page_content=' Huang, and A.' metadata={'source': 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221268 diff --git a/E9E0T4oBgHgl3EQfhAEU/content/tmp_files/2301.02424v1.pdf.txt b/E9E0T4oBgHgl3EQfhAEU/content/tmp_files/2301.02424v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e49f2adeeb87f52b6342c146952f1068b775364e --- /dev/null +++ b/E9E0T4oBgHgl3EQfhAEU/content/tmp_files/2301.02424v1.pdf.txt @@ -0,0 +1,1455 @@ +THIS WORK HAS BEEN SUBMITTED TO THE IEEE FOR POSSIBLE PUBLICATION. +1 +Conformal Loss-Controlling Prediction +Di Wang, Ping Wang, Zhong Ji, Xiaojun Yang, Hongyue Li +Abstract—Conformal prediction is a learning framework con- +trolling prediction coverage of prediction sets, which can be +built on any learning algorithm for point prediction. This work +proposes a learning framework named conformal loss-controlling +prediction, which extends conformal prediction to the situation +where the value of a loss function needs to be controlled. +Different from existing works about risk-controlling prediction +sets and conformal risk control with the purpose of controlling +the expected values of loss functions, the proposed approach +in this paper focuses on the loss for any test object, which is +an extension of conformal prediction from miscoverage loss to +some general loss. The controlling guarantee is proved under the +assumption of exchangeability of data in finite-sample cases and +the framework is tested empirically for classification with a class- +varying loss and statistical postprocessing of numerical weather +forecasting applications, which are introduced as point-wise +classification and point-wise regression problems. All theoretical +analysis and experimental results confirm the effectiveness of our +loss-controlling approach. +Index Terms—Conformal prediction, Loss-controlling predic- +tion, Finite-sample guarantee, Weather forecasting. +I. INTRODUCTION +Prediction sets convey uncertainty or confidence information +for users, which is more preferred than prediction points, +especially for sensitive applications such as medicine, finance +and weather forecasting [1] [2] [3]. Conformal prediction +(CP) is a learning framework providing prediction sets for +test labels, which guarantees the finite-sample coverage at a +user-preset level under the assumption of exchangeability of +data samples [4]. This property of validity has been proved +both theoretically and empirically in many works and applied +to many areas [5] [6]. Besides, many researches extend CP +to more general cases, such as conformal prediction for +multi-label learning [7] [8], functional data [9] [10], few-shot +learning [11] and distribution shift [12] [13]. +However, existing CP methods mainly make promise about +the coverage of prediction sets, which limits its use to other +broad applications concerning controlling general losses. To +tackle this issue, two works have been proposed recently. +[14] employs DeepSets to estimate the expected value or +This work has been submitted to the IEEE for possible publication. +Copyright may be transferred without notice, after which this version may +no longer be accessible. +This work was supported by the National Natural Science Foundation of +China under Grant 62106169. (Corresponding author: Hongyue Li) +Di Wang and Zhong Ji are with School of Electrical and Information Engi- +neering, Tianjin University, Tianjin 300072, China, and also with Tianjin Key +Laboratory of Brain-inspired Intelligence Technology, School of Electrical and +Information Engineering, Tianjin University, Tianjin 300072, China. (email: +wangdi2015@tju.edu.cn; jizhong@tju.edu.cn;). +Ping Wang and Hongyue Li are with School of Electrical and In- +formation Engineering, Tianjin University, Tianjin 300072, China. (email: +wangps@tju.edu.cn; lihongyue@tju.edu.cn). +Xiaojun Yang is with Tianjin Meteorological Observatory, Tianjin 300074, +China. (email: boluo0127@yeah.net) +the cumulative distribution function of the number of false +positives, and then use calibration data to control the number +of false positives of prediction sets. Conformal risk control +(CRC) [15] extends CP to prediction tasks of controlling +the expected value of a general loss based on finding the +optimal parameter for nested prediction sets. The spirit is to +employ calibration data to obtain the information of the upper +bound of the expected value of the loss function at hand and +control the expected value for the test object, whose main idea +was originally proposed from their pioneer work named risk- +controlling prediction sets [16]. +This paper extends CP to the situation where the value of a +general loss function needs to be controlled, which has been +not considered in the literature to our best knowledge. This +situation is reasonable and practical since one may only care +about the loss value for a specific test object, just like the +coverage guarantee made by CP. Our approach is similar to +CRC with the main difference being that we focus on finding +the optimal parameter for nested prediction sets to control the +loss. Therefore, we also concentrate on inductive conformal +prediction [17] or split conformal prediction [18] process like +CRC. +Recall that inductive conformal prediction makes the cov- +erage guarantee as follows, +P +� +Yn+1 ∈ C(n) +1−δ(Xn+1) +� +≥ 1 − δ, +where δ is the significance level preset by users, C(n) +1−δ +is the set predictor made by CP based on n calibration +data {(Xi, Yi)}n +i=1, (Xn+1, Yn+1) is the test feature-response +pair, and the randomness is from both {(Xi, Yi)}n +i=1 and +(Xn+1, Yn+1). By comparison, conformal loss-controlling +prediction (CLCP), the learning framework proposed in this +paper, provides the controlling guarantee as follows, +P +� +L +� +Yn+1, Cλ∗(Xn+1) +� +≤ α +� +≥ 1 − δ, +where L is a loss function satisfying some monotonic con- +ditions as in [15], α is the preset level of loss, Cλ(·) is the +prediction set usually constructed by an underlying algorithm +and a parameter λ. The optimal λ∗ is obtained based on α, +δ and calibration data. The controlling guarantee needs two +levels α and δ to be chosen by users, which is similar with +that in [16], i.e., CLCP guarantees that the prediction loss +is not greater than α with high probability 1 − δ when δ is +small such as 0.1. If L is replaced by false positive for multi- +label classification, the controlling guarantee above is also the +(α, δ)-FP validity defined in Definition 4.2 in [14]. +We prove the controlling guarantee for distribution-free and +finite-sample settings with the assumption of exchangeability +arXiv:2301.02424v1 [cs.LG] 6 Jan 2023 + +THIS WORK HAS BEEN SUBMITTED TO THE IEEE FOR POSSIBLE PUBLICATION. +2 +of data samples. The main idea is that we find the λ∗ to make +the 1 − δ quantile of the loss values on calibration data not +greater than α, which is inspired by CRC focusing on making +the mean of the loss values not greater than α. Furthermore, +we test our approach in classification with a class-varying loss +introduced in [16], and postprocessing of numerical weather +forecasts, which we consider as point-wise classification and +point-wise regression problems. The experimental results em- +pirically confirm the theoretical guarantee we prove in this +paper. +In summary, the main contributions of this paper are: +• A learning framework named conformal loss-controlling +prediction (CLCP) is proposed for controlling the pre- +diction loss for the test object. The approach is simple +to implement and can be built on any machine learning +algorithm for point prediction. +• The controlling guarantee is proved mathematically for +finite-sample cases with the exchangeability assumption, +without any further assumption for data distribution. +• The controlling guarantee is empirically verified by clas- +sification with a class-varying loss and weather forecast- +ing problems, which confirms the effectiveness of CLCP. +The rest of this paper is organized as follows. Section II +reviews inductive conformal prediction and conformal risk +control. Section III introduces conformal loss-controlling pre- +diction and its theoretical guarantee. Section IV conducts +experiments to test the proposed method and the conclusions +are drawn in Section V. +II. INDUCTIVE CONFORMAL PREDICTION AND +CONFORMAL RISK CONTROL +This section reviews inductive conformal prediction and +conformal risk control. Throughout this paper, {(Xi, Yi)}n+1 +i=1 +denotes n + 1 data drawn exchangeably from PXY +on +X × Y, where {(Xi, Yi)}n +i=1 is the calibration dataset and +(Xn+1, Yn+1) is the test object-response pair. We use lower- +case letter (xi, yi) to represent the realization of (Xi, Yi). +The set-valued function and loss function considered in +this paper are the same as those in [15] and [16], which we +formally introduce as follows. Let Cλ : X → Y′ be a set- +valued function with a parameter λ ∈ R, where Y′ represents +some space of sets and R is the set of real numbers. Taking +single-label classification for example, Y′ can be the power +set of Y. For binary image segmentation, Y′ can be equal to +Y as the space of all possible results of image segmentation, +where the sets here stand for all of the pixels of positive class +for the image. +We also introduce the nesting property for prediction sets +and losses as in [16] as follows. For each realization of input +object x, we assume that Cλ(x) satisfies the following nesting +property: +λ1 < λ2 =⇒ Cλ1(x) ⊆ Cλ2(x). +(1) +Let L : Y × Y′ → R be a loss function respecting the nesting +property for each realization of response y: +C1 ⊆ C2 ⊆ Y′ =⇒ L(y, C2) ≤ L(y, C1) ≤ B, +(2) +where B is the upper bound of the loss function. +A. Inductive Conformal Prediction +Inductive conformal prediction (ICP) is a computationally +efficient version of the original conformal prediction approach. +It starts with any measurable function named nonconformity +measure A : X × Y → R and obtain n nonconformity scores +as +Ai = A(Xi, Yi), +for i = 1, · · · , n. Then, with the exchangeable assumption and +a preset δ ∈ (0, 1), one can conclude that +P +� +A(Xn+1, Yn+1) ≤ Q(n) +1−δ +� +≥ 1 − δ, +where Q(n) +1−δ is the 1 − δ quantile of {Ai}n +i=1 ∪ {∞} [12]. +Therefore, the prediction set made by ICP is +C(n) +1−δ(Xn+1) = {y : A(Xn+1, y) ≤ Q(n) +1−δ}, +which satisfies +P +� +Yn+1 ∈ C(n) +1−δ(Xn+1) +� +≥ 1 − δ. +The nonconformity measure A is often defined based on +a point prediction model ˆf learned from some other training +samples, each of which is also drawn from PXY . +Here is an example of constructing prediction sets with CP. +For a classification problem with K classes, one can first train +a classifier ˆf : X → [0, 1]K with the ith output being the +estimation of the probability of the ith class, and calculate the +nonconformity scores as +A(x, y) = 1 − ˆfk(x), +where ˆfk is the kth output of ˆf(x), if y stands for the kth +class. Therefore, the corresponding prediction set for an input +object x is +C(n) +1−δ(x) = {k : ˆfk(x) ≥ 1 − Q(n) +1−δ}, +which indicates that k ∈ C(n) +1−δ(x) if the estimated probability +of kth class is not less than 1 − Q(n) +1−δ. +B. Conformal Risk Control +Different from conformal prediction, CRC starts with a set- +valued function with the nesting property, whose approach is +inspired by nested conformal prediction [19] and was first +proposed in the researches about risk-controlling prediction +sets. +Assume one has a way of constructing a set-valued function +Cλ with the nesting property of formula (1). Given a loss +function L with the nesting property of formula (2), the +purpose of CRC is to find λ∗ such that +E +� +L(Yn+1, Cλ∗(Xn+1)) +� +≤ α, +(3) +i.e., the expected loss or the risk is not greater than α. +To do so, CRC first calculates Li as +Li(λ) = L(Yi, Cλ(Xi)), +(4) + +THIS WORK HAS BEEN SUBMITTED TO THE IEEE FOR POSSIBLE PUBLICATION. +3 +with the fact that Li(λ) is a monotone decreasing function of +λ based on the nesting properties. Then, CRC searches for λ∗ +using the following equation, +λ∗ = inf +� +λ : +n +n + 1 +ˆRn(λ) + +B +n + 1 ≤ α +� +, +where ˆRn(λ) = (L1(λ) + · · · + Ln(λ))/n is an estimation of +the risk on calibration data and B is introduced to make the +estimation not overconfident. Also, to make λ∗ well defined, +one should assume that ˆRn(λ) is right continuous. +These two steps of CRC are too simple that one may +surprise about its theoretical conclusion that with the assump- +tion of exchangeability of data samples, the prediction set +Cλ∗(Xn+1) obtained by CRC satisfies formula (3), which has +been also proved empirically in [15]. CRC extends CP from +controlling the expected value of miscoverage loss to some +general loss, which can be applied to the cases where Y is +beyond real numbers or vectors, such as images, fields and +even graphs. +After tackling the theoretical issue, the problem for CRC +is how to construct Cλ. Here, we also give an example of a +classification problem with K classes. In fact, with the same +notations of the example in Section II-A, CRC can construct +the prediction set as +Cλ(x) = {k : ˆfk(x) ≥ 1 − λ}. +Therefore, as long as L satisfies formula (2), such as L is the +indicator of miscoverage, CRC guarantees to control the risk +as formula (3). +III. CONFORMAL LOSS-CONTROLLING PREDICTION AND +ITS THEORETICAL ANALYSIS +This section introduces the approach of CLCP and its +theoretical analysis. CLCP also has two steps like CRC, and +the main difference between them is that CLCP focuses on +whether the estimation of the 1 − δ quantile of the losses is +not greater than α while CRC concentrates on whether the +mean of the losses not greater than α. The controlling of the +1−δ quantile of the losses makes CLCP be able to control the +value of a general loss by employing the probability inequation +derived from the exchangeability assumption, which is also +employed by ICP if the loss is seen as the nonconformity +score. +Suppose one has a way of constructing a set-valued function +Cλ with the nesting property of formula (1), which can be the +same as that used in CRC. Here, we assume that the parameter +λ is selected from a discrete set Λ, such as from 0 to 1 with a +step size 0.01, which avoids us from the assumption of right +continuous for the loss function in theoretical analysis, and +is also reasonable since we actually search for λ∗ with some +step size in practice [15] [16]. Besides, the latest paper about +risk-controlling prediction also makes this discrete assumption +for general cases [20]. After determining Cλ(·) and Λ, CLCP +first calculates Li(λ) on calibration data as formula (4). Then, +for any preset α ∈ R and δ ∈ (0, 1), CLCP searches for λ∗ +such that +λ∗ = min +� +λ ∈ Λ : Q(n) +1−δ(λ) ≤ α +� +, +(5) +with Q(n) +1−δ(λ) being the 1 − δ quantile of {Li(λ)}n +i=1 ∪ {B}. +The approach of CLCP is summarised in Algorithm 1, which +is easy to implement. +Algorithm 1 Conformal Loss-Controlling Prediction +Input: +Calibration dataset {(xi, yi)}n +i=1, test input object xn+1, +the set predictor Cλ satisfies formula (1), the loss function +L satisfies formula (2), preset α ∈ R and δ ∈ (0, 1). +Output: +Predictive set for yn+1. +1: Based on formula (4), calculate {Li(λ)}n +i=1. +2: Search for λ∗ satisfying formula (5). +3: return Cλ∗(xn+1) +Next, we introduce the definition of (α, δ)-loss-controlling +set predictors and then prove our theoretical conclusion about +CLCP. +Definition 1. Given a loss function L : Y × Y′ → R and a +random sample (X, Y ) ∈ X ×Y, a random set-valued function +C whose realization is in the space of functions X → Y′ is a +(α, δ)-loss-controlling set predictor if it satisfies that +P +� +L +� +Y, C(X) +� +≤ α +� +≥ 1 − δ, +where the randomness is both from C and (X, Y ). +After all these preparations, we can prove in Theorem 1 +that Cλ∗ constructed by CLCP is a (α, δ)-loss-controlling set +predictor. +Theorem 1. Suppose {(Xi, Yi)}n+1 +i=1 are n + 1 data drawn +exchangeably from PXY on X × Y, Cλ : X → Y′ is a set- +valued function satisfying formula (1) with the parameter λ +taking values from a discrete set Λ ⊂ R , L : Y × Y′ → R +is a loss function satisfying formula (2) and Li(λ) is defined +as formula (4). For any preset α ∈ R, if L also satisfies the +following conditions, +min +λ max +i +Li(λ) < α, +max +λ +min +i +Li(λ) > α, +(6) +then for any δ ∈ (0, 1), we have +P +� +L +� +Yn+1, Cλ∗(Xn+1) +� +≤ α +� +≥ 1 − δ, +(7) +where λ∗ is defined as formula (5). +Proof. Let Q(n+1) +1−δ (λ) be the 1 − δ quantile of {Li(λ)}n+1 +i=1 , +and define ˜λ as +˜λ = min +� +λ ∈ Λ : Q(n+1) +1−δ (λ) ≤ α +� +. + +THIS WORK HAS BEEN SUBMITTED TO THE IEEE FOR POSSIBLE PUBLICATION. +4 +Similarly, let Q(n) +1−δ(λ) be the 1 − δ quantile of {Li(λ)}n +i=1 ∪ +{B}, and we have +λ∗ = min +� +λ ∈ Λ : Q(n) +1−δ(λ) ≤ α +� +. +Since B is the upper bound of Ln+1(λ), by definition, we +have +˜λ ≤ λ∗, +and the conditions introduced in formula (6) is to make ˜λ and +ˆλ well defined. This leads to +Ln+1(λ∗) ≤ Ln+1(˜λ), +(8) +as Cλ and L satisfy the nesting properties of formula (1) and +(2). +Since ˜λ is dependent on the whole dataset {(Xi, Yi)}n+1 +i=1 , +{Li(˜λ)}n+1 +i=1 are exchangeable variables, which leads to +P +� +Ln+1(˜λ) ≤ Q(n+1) +1−δ (˜λ) +� +≥ 1 − δ, +(9) +as Q(n+1) +1−δ (˜λ) is just the corresponding 1−δ quantile (See the +proof of Lemma 1 in [12]). +Combining the definition of ˜λ, formula (8) and (9), we have +P +� +Ln+1(λ∗) ≤ α +� +≥ 1 − δ, +which completes the proof. +At the end of this section, we show that CP can be seen +as a special case of CLCP from the following viewpoint. +Suppose Cλ is constructed by a nonconformity score A, which +is defined as +Cλ(x) = {y : A(x, y) ≤ λ}, +and L is the miscoverage loss such that +Li(λ) = L(yi, Cλ(xi)) = I{yi /∈ Cλ(xi)}, +where I is the indicator function. In this case, Q(n) +1−δ(λ) can +only be 0 or 1 as the loss can only be these two numbers. +Besides, only α ∈ [0, 1) is meaningful, which means that +one wants to control the miscoverage. For CLCP, let Λ be +an arithmetic sequence whose common difference, minimum +and maximum are ∆, λmin and λmax respectively and set +α = 0. By definition, λ∗ can be written as +λ∗ = min +� +λ ∈ Λ : +1 +n + 1 +n +� +i=1 +I{ai ≤ λ} ≥ 1 − δ +� += min +� +λ ∈ Λ : 1 +n +n +� +i=1 +I{ai ≤ λ} ≥ ⌈(1 − δ)(n + 1)⌉ +n +� +, +where ai = A(xi, yi) is the nonconformity score of the ith +calibration data for CP. In comparison, referring to [15], the +optimal ˆλ for CP is +ˆλ = inf +� +λ ∈ R : 1 +n +n +� +i=1 +I{ai ≤ λ} ≥ ⌈(1 − δ)(n + 1)⌉ +n +� +. +Therefore, if λmin < ai < λmax for each i, we have +|λ∗ − ˆλ| ≤ ∆, +which implies that the prediction sets of CP and CLCP are +nearly the same if ∆ is small enough. In summary, if Cλ +and L have special forms and Λ includes the upper and lower +bounds of nonconformity scores with ∆ being small enough +to be ignored, CP can be seen as a special case of CLCP. +IV. EXPERIMENTS +This section conducts the experiments to empirically test the +approach of CLCP. First, we built CLCP for the classification +problem with a class-varying loss introduced in [16]. Then, +we focus on two types of weather forecasting applications, +which can be seen as point-wise classification and point- +wise regression problems respectively. All experiments were +coded in Python [21]. The statistical learning methods used in +Section IV-A were implemented using Scikit-learn [22] and +the deep learning methods used in Section IV-B and Section +IV-C were implemented with Pytorch [23]. +A. CLCP for classification with a class-varying loss +We collected 20 binary or multiclass classification datasets +from UCI repositories [24] whose information is summarized +in Table I. The problem is to make the prediction sets of labels +controlling the following loss +L(y, C) = LyI{y /∈ C}, +where Ly is the loss for y being not in the prediction set. +The loss for each label is generated uniformly on (0, 1) like +[16]. Support vector machine (SVM) [25], neural network +(NN) [26] and random forests (RF) [27] were employed as +the underlying algorithms separately to construct prediction +sets based on CLCP. The prediction set Cλ is constructed as +Cλ(x) = {k : ˆfk(x) ≥ 1 − λ}, +where ˆfk is the estimated probability of the observation being +kth class by the corresponding underlying algorithm. For each +dataset, we used 20% of the data for testing and 80% and 20% +of the remaining data for training and calibration respectively. +Based on the training data, we selected the meta-parameters +with three-fold cross-validation and used the optimal meta- +parameters to train the classifiers. The regularization parameter +of SVM was selected from {0.001, 0.01, 0.1, 1, 10, 100}, and +the learning rate and the epochs of NN were selected from +{0.001, 0.0001} and {200, 500, 1000}. The number of trees +of RF were selected from {100, 300, 500} and the partition +criterion was either gini or entropy. After training, we used the +trained classifiers and the calibration data to search for λ∗ with +Algorithm 1 and construct the final set predictors. All of the +features were normalized to [0, 1] by min–max normalization +and for each dataset, the experiments were conducted 10 times +and the average results were recorded. +The bar plots in Fig. 1 and Fig. 2 show the experimental +results for 20 public datasets with δ ∈ {0.05, 0.1, 0.15, 0.2} +and α ∈ {0.1, 0.2}. The results in Fig. 1 concern about the + +THIS WORK HAS BEEN SUBMITTED TO THE IEEE FOR POSSIBLE PUBLICATION. +5 +TABLE I +DATASETS FROM UCI REPOSITORIES +Dataset +Examples +Dimensionality +Classes +bc-wisc-diag +569 +30 +2 +car +1728 +6 +4 +chess-kr-kp +3196 +36 +2 +contrac +1473 +9 +3 +credit-a +690 +15 +2 +credit-g +1000 +20 +2 +ctg-10classes +2126 +21 +10 +ctg-3classes +2126 +21 +3 +haberman +306 +3 +2 +optical +5620 +62 +10 +phishing-web +11055 +30 +2 +st-image +2310 +18 +7 +st-landsat +6435 +36 +6 +tic-tac-toe +958 +9 +2 +wall-following +5456 +24 +4 +waveform +5000 +21 +3 +waveform-noise +5000 +40 +3 +wilt +4839 +5 +2 +wine-quality-red +1599 +11 +6 +wine-quality-white +4898 +11 +7 +frequency of the prediction losses being more than α on test +set, which is the estimated probability of +P +� +L +� +Yn+1, Cλ∗(Xn+1) +� +> α +� +, +which should be near or lower than δ empirically due to +formula (7). The bar plots of Fig. 1 demonstrate that the +frequency of the prediction losses is near or below δ, which +verifies the conclusion of Theorem 1. +The bar plots of Fig. 2 show the average sizes of prediction +sets for different δ, describing the informational efficiency of +the prediction sets. Changing δ can effectively change the +average size of prediction sets and changing α may slightly +change average size (such as the results for wine-quality-red). +Although many prediction sets are meaningful with average +sizes being near 1, the prediction sets for the dataset contrac +may be not usefull, since no matter how to change δ and α, +the average sizes of the prediction sets are all near or above +2, whereas the number of classes of contrac is 3. Thus, how +to construct efficient prediction sets in the learning framework +of CLCP is worth exploring for further researches. +Combining Fig. 1 and Fig. 2, we observe that different +classifiers can perform differently for different datasets, which +indicates that the underlying algorithm affects the performance +and the model selection approach is necessary for CLCP. +B. CLCP for high-impact weather forecasting +The remaining experiments apply CLCP to weather fore- +casting problems. Here we concentrate on postprocessing of +the forecasts made by numerical weather prediction (NWP) +models [28] [29]. NWP models use equations of atmospheric +dynamics and estimations of current weather conditions to do +weather forecasting, which is the mainstream weather fore- +casting technique nowadays especially for forecasting beyond +12 hours. Many errors affect the performance of NWP models, +such as the estimation errors of initial conditions and the +approximation errors of NWP models, leading to the research +topic about postprocessing the forecasts of NWP models. Most +postprocessing methods are built on some learning process, +which takes the forecasts of NWP models as inputs and the +observations of weather elements or events as outputs. +In this paper, we use CLCP to postprocess the ensemble +forecasts with the control forecast and 50 perturbed forecasts +issued by the NWP model from European Centre for Medium- +Range Weather Forecasts (ECMWF) [30], which are obtained +from the THORPEX Interactive Grand Global Ensemble +(TIGGE) dataset [31]. We focus on 2-m maximum temperature +and minimum temperature between the forecast lead times +of 12nd hour and 36th hour with the forecasts initialized at +0000 UTC. The forecast fields are grided with the resolution +of 0.5◦ × 0.5◦ and the corresponding label fields with the +same resolution are extracted from the ERA5 reanalysis data +[32]. The area ranges from 109◦E to 122◦E in longitude and +from 29◦N to 42◦N in latitude, covering the main parts of +North China, East China and Central China, whose grid size +is 27 × 27. The ECMWF forecast data and ERA5 reanalysis +data are collected from 2007 to 2020 (14 years). +We first consider high-impact weather forecasting, which +is to forecast whether a high-impact weather exists for each +grid and can be seen as a point-wise classification problem or +image segmentation problem for computer vision. The high- +impact weather we consider is whether the 2-m maximum +temperature is above 35 °C or the 2-m minimum temperature +is below −15 °C for each grid. These two cases are treated +as high temperature weather or low temperature weather in +China, which make meteorological observatories issue high +temperature warning or low temperature warning respectively. +The prediction sets and the loss function used for high- +impact weather forecasting are the same as those for image +segmentation in [15]. Taking the ensemble forecast fields of +the NWP model as input x, the corresponding label y is a set +of grids having high-impact weather, which can be seen as +a segmentation problem for high-impact weather. Therefore, +we first train a segmentation neural network f(x), where +f(p,q)(x) is the estimated probability of the grid (p, q) having +high-impact weather. Then the set-valued function Cλ can be +constructed as +Cλ(x) = {(p, q) : f(p,q)(x) ≥ 1 − λ}, +(10) +and the loss function is +L(y, C) = 1 − |y ∩ C| +|y| +, +(11) +which measures the ratio of the prediction sets failing to do the +warning. We use CLCP with the prediction set and the loss +function above to do high temperature and low temperature +forecasting respectively. +1) Dataset for high temperature forecasting: The reanalysis +fields of 2-m maximum temperature were collected from +ERA5 and the label fields were calculated based on weather +the 2-m maximum temperature is above 35 °C. To make the +loss function take finite values, we only collected the data +whose label fields have at least one high temperature grid +to do this empirical study, which resulted in 1200 samples +in total, i.e., 1200 ensemble forecasts from the NWP model +of ECMWF and corresponding label fields calculated from + +THIS WORK HAS BEEN SUBMITTED TO THE IEEE FOR POSSIBLE PUBLICATION. +6 +Fig. 1. Bar plots of the frequencies of the prediction losses being more than α vs. δ = 0.05, 0.1, 0.15, 0.2 on test data for classification with a class-varying +loss. The first row corresponds to α = 0.1 and the second row corresponds to α = 0.2. Different columns represent different classifiers. All bars are near or +below the preset δ, which confirms the controlling guarantee of CLCP empirically. +Fig. 2. Bar plots of the average sizes of prediction sets vs. δ = 0.05, 0.1, 0.15, 0.2 on test data for classification with a class-varying loss. The first row +corresponds to α = 0.1 and the second row corresponds to α = 0.2. Different columns represent different classifiers. The plots demonstrate the information +in prediction sets. In general, large δ leads to small average size and different classifiers have different informational efficiency. +ERA5. We name this dataset as HighTemp. +2) Dataset for low temperature forecasting: The dataset +for testing CLCP for low temperature weather forecasting +was constructed in a similar way. The reanalysis fields of +2-m minimum temperature were collected from ERA5 and +the label fields were calculated based on weather the 2-m +minimum temperature is below −15 °C. We only collected +the data whose label fields have at least one low temperature +grid to do this empirical study, which resulted in 1233 samples +in total. We name this dataset as LowTemp. +For each dataset, the same process was used to conduct the +experiment as Section IV-A , i.e., all forecasts from the NWP +model were normalized to [0, 1] by min–max normalization, +and we used 20% of the data for testing and 80% and 20% +of the remaining data for training and calibration respectively. +We employed two fully convolutional neural networks [33] +for binary image segmentation as our underlying algorithms. +One was U-Net [34] with the same structure as that in [35], +whose numbers of hidden feature maps were all set to 32. The +other was the naive deep neural network (nDNN) with the +same encoder-decoder structure as the U-Net without skip- +connections, i.e., the U-Net removing skip-connections. We + +α = 0.1 I Classifier = SVM +α = 0.1 I Classifier = NN +α = 0.1 Classifier = RF +0.30 +0.25 +0.20 +Dataset + bc-wisc-diag +car +chess-kr-kp +contrac +credit-a +0.05 +credit-g +ctg-10classes +ctg-3classes +0.00 +haberman +α = 0.2 I Classifier = SVM +α = 0.2 I Classifier = NN +α = 0.2 I Classifier = RF +optical +0.30 +phishing-web + st-image +st-landsat +0.25 +tic-tac-toe +α +wall-following +waveform +waveform-noise +wilt +wine-quality-red +wine-quality-white +0.05 +0.00 +0.15 +0.15 +0.2 +0.1 +0.15 +0.05 +0.1 +0.05 +0.2 +0.05 +0.1 +0.2 +6 +6 +6α = 0.1 I Classifier = SVM +α = 0.1 I Classifier = NN +α = 0.1 I Classifier = RF +3.0 +2.5 +Dataset +1.5 +bc-wisc-diag +car +chess-kr-kp +1.0 +contrac +credit-a +0.5 +credit-g +ctg-10classes +ctg-3classes +0.0 +haberman +α = 0.2 I Classifier = SVM +α = 0.2 I Classifier = NN +α = 0.2 I Classifier = RF +optical +phishing-web +3.0 +st-image +st-landsat +tic-tac-toe +2.5 + wall-following +waveform +2.0 +waveform-noise +Average s +wilt +wine-quality-red +1.5 +wine-quality-white +1.0 +0.5 +0.0 +0.05 +0.1 +0.15 +0.2 +0.05 +0.1 +0.15 +0.2 +0.05 +0.1 +0.15 +0.2 +6 +6 +6THIS WORK HAS BEEN SUBMITTED TO THE IEEE FOR POSSIBLE PUBLICATION. +7 +Fig. 3. Bar plots of the frequencies of the prediction losses being more than α vs. δ = 0.05, 0.1, 0.15, 0.2 on test data for high-impact weather forecasting. +The first row corresponds to HighTemp and the second row corresponds to LowTemp. Different columns represent different α. All bars are near or below +the preset δ, which confirms the controlling guarantee of CLCP empirically. +Fig. 4. +Boxen plots of the prediction losses vs. δ = 0.05, 0.1, 0.15, 0.2 on test data for high-impact weather forecasting. The first row corresponds to +HighTemp and the second row corresponds to LowTemp. Different columns represent different α. The loss distributions are controlled by α and δ properly +to obtain the empirical validity in Fig. 3. +use these two neural networks to show that the designation of +the underlying algorithm is necessary for better performance, +as U-Net fuses multi-scale features and nDNN does not. The +data for training U-Net and DNN were further partitioned +to the validation part (10%) for model selection and proper +training part (90%) for updating the parameters. Adam op- +timization [36] was used for training. The learning rate was +set to 0.0001 and the number of epochs was set to 50. After +training 50 epochs, the model with lowest binary cross entropy +on validation data was used for formula (10) to construct +prediction sets, where λ is search from 1 to 0 with step size +0.01. The experiments of using CLCP for the loss function +as formula (11) were conducted 10 times and the prediction +results on test set are shown in Fig. 3, Fig. 4 and Fig. 5. +Fig. 3 also shows the bar plots of the frequencies of the +prediction losses being more than α for δ = 0.05, 0.1, 0.15 +and 0.2. Four column stands for the cases where α += +0.05, 0.1, 0.15 and 0.2 respectively. It can be seen that for +the two datasets HighTemp and LowTemp, all bars are near +or below the preset δ, which verifies formula (7) empirically. +Fig. 4 further shows the distributions of the losses for different +δ and different α using boxen plots, which contain more +information than box plots by drawing narrow boxes for tails. +It can be seen that larger α and δ lead to larger losses, which +is reasonable since large α and δ relax the constraint on +prediction losses. We measure the informational efficiency of +the prediction set Cλ∗(x) using its normalized size defined +as |Cλ∗(x)|/PQ, where P and Q are the numbers of the +vertical and the horizontal grids respectively. The distributions +of normalized sizes in Fig. 5 show that U-Net is more +informationally efficient than nDNN, which indicates that +designation of the underlying algorithm is important for CLCP. +Different α and δ lead to different normalized sizes, implying +the trade-off among the preset loss bound α, confidence level +1 − δ and informational efficiency of the prediction sets. +By choosing α and δ properly, the prediction sets of CLCP + +Dataset = HighTemp | α = 0.05 +Dataset = HighTemp | α = 0.15 +Dataset = HighTemp I α = 0.2 +Dataset = HighTemp I α = 0.1 +0.30 +0.25 +0.05 +0.00 +Model +Dataset = LowTemp I α = 0.05 +Dataset = LowTemp I α = 0.1 +Dataset = LowTemp I α = 0.15 +Dataset = LowTemp I α = 0.2 +nDNN +0.30 +U-Net +0.25 +0.05 +0.00 +0.05 +0.1 +0.15 +0.2 +0.05 +0.1 +0.15 +0.2 +0.05 +0.1 +0.15 +0.05 +0.1 +0.15 +0.2 +0.2 +6 +6 +6Dataset = HighTemp I α = 0.05 +Dataset = HighTemp I α = 0.1 +Dataset = HighTemp I α = 0.15 +Dataset = HighTemp I α = 0.2 +1.0 +0.8 +0.6 +Loss +0.4 +0.2 +0.0 +Model +Dataset = LowTemp I α = 0.05 +Dataset = LowTemp I α = 0.1 +Dataset = LowTemp I α = 0.15 +Dataset = LowTemp I α = 0.2 +nDNN +1.0 +U-Net +0.8 +0.6 +Loss +0.4 +0.2 +0.0 +0.05 +0.1 +0.15 +0.2 +0.05 +0.1 +0.15 +0.2 +0.05 +0.1 +0.15 +0.2 +0.05 +0.1 +0.15 +0.2 +6 +6 +6THIS WORK HAS BEEN SUBMITTED TO THE IEEE FOR POSSIBLE PUBLICATION. +8 +Fig. 5. Boxen plots for the distributions of normalized sizes of prediction sets vs. δ = 0.05, 0.1, 0.15, 0.2 on test data for high-impact weather forecasting.. +The first row corresponds to HighTemp and the second row corresponds to LowTemp. Different columns represent different α. U-Net performs better than +nDNN, which indicates the importance of careful designation of the underlying algorithm. +can have reasonable sizes. Also, we can see that forecasting +low temperature is somehow easier than high temperature +with the fact that for the same α and δ, the normalized +sizes of forecasting low temperature are distributed lower +than the ones of forecasting high temperature, indicating the +need of designation of the underlying algorithms to improve +performance for forecasting high temperature. +C. CLCP for maximum temperature and minimum tempera- +ture forecasting +This section focuses on using CLCP to forecast the 2-m +maximum temperature or minimum temperature value for each +grid, which is a point-wise regression problem or image-to- +image regression problem. To construct the prediction sets, +we follow the procedure proposed in [37] and train the neural +network with 3 output channels jointly predicting the point- +wise 0.05, 0.5 and 0.95 quantiles of the fields using quantile +regression [38] [37], which are denoted by f 0.05(x), f 0.5(x) +and f 0.95(x). Then the prediction set Cλ(x) is equal to +� +y : y(p,q) ∈ [f 0.5 +(p,q)(x)−λ∆− +(p,q)(x), f 0.5 +(p,q)(x)+λ∆+ +(p,q)(x)] +� +, +where +∆−(x) = max{f 0.5(x) − f 0.05(x), 10−6}, +∆+(x) = max{f 0.95(x) − f 0.5(x), 10−6}, +and max is a point-wise operator making ∆− and ∆+ at least +10−6. This prediction set is a prediction band for the output +field, whose prediction interval at grid (p, q) is +[f 0.5 +(p,q)(x) − λ∆− +(p,q)(x), f 0.5 +(p,q)(x) + λ∆+ +(p,q)(x)] +(12) +with the point-wise width being an increasing function of λ. +This construction was proposed in [37] for image-to-image +regression and we use the same loss function in [37] measuring +miscoverage rate of a prediction band C for a field y, which +can be formalized as +L(y, C) = +1 +PQ +��� +� +(p, q) : y(p,q) /∈ C(p,q) +����, +(13) +where C(p,q) is the prediction interval at grid (p, q) for +prediction band C. +All of the data collected from 2007 to 2020 were used, lead- +ing to 4945 samples for each forecasting application and the +datasets are named as MaxTemp and MinTemp respectively. +The experimental designation is the same as that in Section +IV-B, except that we also normalized the label for each grid to +[0, 1] by min–max normalization, used quantile loss for model +selection and we searched for λ∗ with two steps. First we +found two values λ1 and λ2 from {100, 10, 1, 0.1, 0.01, ...} +such that Q(n) +1−δ(λ1) ≤ α and Q(n) +1−δ(λ2) > α. Then we +searched for λ∗ from 100 values starting with λ1 and ending +with λ2 using a common step size. The experimental results +are recorded in Fig. 6, Fig 7 and Fig 8. +Although the set predictors and the loss function used in +this section are different from those in Section IV-B, the +experimental results and conclusions are similar. From Fig. 6, +we can see that the frequencies of the prediction losses being +more than α are controlled by δ, which also verifies formula +(7) empirically. Larger α and δ lead to larger losses, which is +shown in Fig. 7. Here we use the following average interval +length +1 +PQ +P +� +p=1 +Q +� +q=1 +λ∗(∆+ +(p,q)(x) − ∆− +(p,q)(x)) +(14) +to measure the informational efficiency of the prediction set +Cλ∗(x) and Fig. 8 also depicts the trade-off among the +preset loss bound α, confidence level 1 − δ and informational +efficiency of the prediction sets and indicates that better des- +ignation of underlying algorithms lead to better performance. +V. CONCLUSION +This paper extends conformal prediction to the situation +where the value of a loss function needs to be controlled, + +Dataset = HighTemp I α = 0.05 +Dataset = HighTemp I α = 0.1 +Dataset = HighTemp I α = 0.15 +Dataset = HighTemp I α = 0.2 +1.0 +0.8 +I Size +Normalized +0.6 +0.4 +0.2 +0.0 +Model +Dataset = LowTemp I α = 0.05 +Dataset = LowTemp I α = 0.1 +Dataset = LowTemp I α = 0.15 +Dataset = LowTemp I α = 0.2 +nDNN +1.0 +U-Net +0.8 +0.6 +0.4 +0.2 +0.0 +0.05 +0.1 +0.15 +0.2 +0.05 +0.1 +0.15 +0.2 +0.05 +0.1 +0.15 +0.2 +0.1 +0.15 +0.2 +0.05 +6 +6THIS WORK HAS BEEN SUBMITTED TO THE IEEE FOR POSSIBLE PUBLICATION. +9 +Fig. 6. +Bar plots of the frequencies of the prediction losses being more than α vs. δ = 0.05, 0.1, 0.15, 0.2 on test data for maximum temperature and +minimum temperature forecasting. The first row corresponds to MaxTemp and the second row corresponds to MinTemp. Different columns represent different +α. All bars are near or below the preset δ, which confirms the controlling guarantee of CLCP empirically. +Fig. 7. Boxen plots of the prediction losses vs. δ = 0.05, 0.1, 0.15, 0.2 on test data for maximum temperature and minimum temperature forecasting. The +first row corresponds to MaxTemp and the second row corresponds to MinTemp. Different columns represent different α. The loss distributions are controlled +by α and δ properly to obtain the empirical validity in Fig. 6. +which is inspired by risk-controlling prediction sets and con- +formal risk control approaches. The loss-controlling guarantee +is proved in theory with the assumption of exchangeability +and is empirically verified for different kinds of applications +including classification with a class-varying loss and weather +forecasting. Different from conformal prediction, conformal +loss-controlling prediction approach proposed in this paper +has two preset parameters α and δ, which guarantees that the +prediction loss is not greater than α with confidence 1−δ. Both +parameters impose restrictions on prediction sets and should +be set based on specific applications. Despite loss-controlling +guarantee, informational efficiency of the prediction sets built +by conformal loss-controlling prediction is highly related to +underlying algorithms, which has been shown in empirical +studies. Since this is a rather new topic, the underlying +algorithms and the way of constructing set predictors are +inherited from conformal risk control. This leaves the im- +portant question on how to build informationally efficient set +predictors in an optimal way, which is one of our further +researches in the future. +REFERENCES +[1] V. Balasubramanian, S.-S. Ho, and V. Vovk, Conformal prediction +for reliable machine learning: theory, adaptations and applications. +Newnes, 2014. +[2] C. Li, G. Tang, X. Xue, A. Saeed, and X. Hu, “Short-term wind speed +interval prediction based on ensemble gru model,” IEEE Transactions +on Sustainable Energy, vol. 11, no. 3, pp. 1370–1380, 2019. +[3] P. Wang, P. Wang, D. Wang, and B. Xue, “A conformal regressor +with random forests for tropical cyclone intensity estimation,” IEEE +Transactions on Geoscience and Remote Sensing, vol. 60, pp. 1–14, +2021. +[4] V. Vovk, A. Gammerman, and G. Shafer, Algorithmic learning in a +random world. +Springer Science & Business Media, 2005. +[5] A. N. Angelopoulos and S. Bates, “A gentle introduction to confor- +mal prediction and distribution-free uncertainty quantification,” arXiv +preprint arXiv:2107.07511, 2021. +[6] M. Fontana, G. Zeni, and S. 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Bassett Jr, “Regression quantiles,” Econometrica: +Journal of the Econometric Society, pp. 33–50, 1978. + diff --git a/E9E0T4oBgHgl3EQfhAEU/content/tmp_files/load_file.txt b/E9E0T4oBgHgl3EQfhAEU/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..86d3b31b68416e4b61167714301d47ae1c932d46 --- /dev/null +++ b/E9E0T4oBgHgl3EQfhAEU/content/tmp_files/load_file.txt @@ -0,0 +1,1062 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf,len=1061 +page_content='THIS WORK HAS BEEN SUBMITTED TO THE IEEE FOR POSSIBLE PUBLICATION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' 1 Conformal Loss-Controlling Prediction Di Wang, Ping Wang, Zhong Ji, Xiaojun Yang, Hongyue Li Abstract—Conformal prediction is a learning framework con- trolling prediction coverage of prediction sets, which can be built on any learning algorithm for point prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' This work proposes a learning framework named conformal loss-controlling prediction, which extends conformal prediction to the situation where the value of a loss function needs to be controlled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Different from existing works about risk-controlling prediction sets and conformal risk control with the purpose of controlling the expected values of loss functions, the proposed approach in this paper focuses on the loss for any test object, which is an extension of conformal prediction from miscoverage loss to some general loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' The controlling guarantee is proved under the assumption of exchangeability of data in finite-sample cases and the framework is tested empirically for classification with a class- varying loss and statistical postprocessing of numerical weather forecasting applications, which are introduced as point-wise classification and point-wise regression problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' All theoretical analysis and experimental results confirm the effectiveness of our loss-controlling approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Index Terms—Conformal prediction, Loss-controlling predic- tion, Finite-sample guarantee, Weather forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' INTRODUCTION Prediction sets convey uncertainty or confidence information for users, which is more preferred than prediction points, especially for sensitive applications such as medicine, finance and weather forecasting [1] [2] [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Conformal prediction (CP) is a learning framework providing prediction sets for test labels, which guarantees the finite-sample coverage at a user-preset level under the assumption of exchangeability of data samples [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' This property of validity has been proved both theoretically and empirically in many works and applied to many areas [5] [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Besides, many researches extend CP to more general cases, such as conformal prediction for multi-label learning [7] [8], functional data [9] [10], few-shot learning [11] and distribution shift [12] [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' However, existing CP methods mainly make promise about the coverage of prediction sets, which limits its use to other broad applications concerning controlling general losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' To tackle this issue, two works have been proposed recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' [14] employs DeepSets to estimate the expected value or This work has been submitted to the IEEE for possible publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Copyright may be transferred without notice, after which this version may no longer be accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' This work was supported by the National Natural Science Foundation of China under Grant 62106169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' (Corresponding author: Hongyue Li) Di Wang and Zhong Ji are with School of Electrical and Information Engi- neering, Tianjin University, Tianjin 300072, China, and also with Tianjin Key Laboratory of Brain-inspired Intelligence Technology, School of Electrical and Information Engineering, Tianjin University, Tianjin 300072, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' (email: wangdi2015@tju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' jizhong@tju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Ping Wang and Hongyue Li are with School of Electrical and In- formation Engineering, Tianjin University, Tianjin 300072, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' (email: wangps@tju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' lihongyue@tju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Xiaojun Yang is with Tianjin Meteorological Observatory, Tianjin 300074, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' (email: boluo0127@yeah.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='net) the cumulative distribution function of the number of false positives, and then use calibration data to control the number of false positives of prediction sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Conformal risk control (CRC) [15] extends CP to prediction tasks of controlling the expected value of a general loss based on finding the optimal parameter for nested prediction sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' The spirit is to employ calibration data to obtain the information of the upper bound of the expected value of the loss function at hand and control the expected value for the test object, whose main idea was originally proposed from their pioneer work named risk- controlling prediction sets [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' This paper extends CP to the situation where the value of a general loss function needs to be controlled, which has been not considered in the literature to our best knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' This situation is reasonable and practical since one may only care about the loss value for a specific test object, just like the coverage guarantee made by CP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Our approach is similar to CRC with the main difference being that we focus on finding the optimal parameter for nested prediction sets to control the loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Therefore, we also concentrate on inductive conformal prediction [17] or split conformal prediction [18] process like CRC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Recall that inductive conformal prediction makes the cov- erage guarantee as follows, P � Yn+1 ∈ C(n) 1−δ(Xn+1) � ≥ 1 − δ, where δ is the significance level preset by users, C(n) 1−δ is the set predictor made by CP based on n calibration data {(Xi, Yi)}n i=1, (Xn+1, Yn+1) is the test feature-response pair, and the randomness is from both {(Xi, Yi)}n i=1 and (Xn+1, Yn+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' By comparison, conformal loss-controlling prediction (CLCP), the learning framework proposed in this paper, provides the controlling guarantee as follows, P � L � Yn+1, Cλ∗(Xn+1) � ≤ α � ≥ 1 − δ, where L is a loss function satisfying some monotonic con- ditions as in [15], α is the preset level of loss, Cλ(·) is the prediction set usually constructed by an underlying algorithm and a parameter λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' The optimal λ∗ is obtained based on α, δ and calibration data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' The controlling guarantee needs two levels α and δ to be chosen by users, which is similar with that in [16], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=', CLCP guarantees that the prediction loss is not greater than α with high probability 1 − δ when δ is small such as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' If L is replaced by false positive for multi- label classification, the controlling guarantee above is also the (α, δ)-FP validity defined in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='2 in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' We prove the controlling guarantee for distribution-free and finite-sample settings with the assumption of exchangeability arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='02424v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='LG] 6 Jan 2023 THIS WORK HAS BEEN SUBMITTED TO THE IEEE FOR POSSIBLE PUBLICATION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' 2 of data samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' The main idea is that we find the λ∗ to make the 1 − δ quantile of the loss values on calibration data not greater than α, which is inspired by CRC focusing on making the mean of the loss values not greater than α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Furthermore, we test our approach in classification with a class-varying loss introduced in [16], and postprocessing of numerical weather forecasts, which we consider as point-wise classification and point-wise regression problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' The experimental results em- pirically confirm the theoretical guarantee we prove in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' In summary, the main contributions of this paper are: A learning framework named conformal loss-controlling prediction (CLCP) is proposed for controlling the pre- diction loss for the test object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' The approach is simple to implement and can be built on any machine learning algorithm for point prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' The controlling guarantee is proved mathematically for finite-sample cases with the exchangeability assumption, without any further assumption for data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' The controlling guarantee is empirically verified by clas- sification with a class-varying loss and weather forecast- ing problems, which confirms the effectiveness of CLCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Section II reviews inductive conformal prediction and conformal risk control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Section III introduces conformal loss-controlling pre- diction and its theoretical guarantee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Section IV conducts experiments to test the proposed method and the conclusions are drawn in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' INDUCTIVE CONFORMAL PREDICTION AND CONFORMAL RISK CONTROL This section reviews inductive conformal prediction and conformal risk control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Throughout this paper, {(Xi, Yi)}n+1 i=1 denotes n + 1 data drawn exchangeably from PXY on X × Y, where {(Xi, Yi)}n i=1 is the calibration dataset and (Xn+1, Yn+1) is the test object-response pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' We use lower- case letter (xi, yi) to represent the realization of (Xi, Yi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' The set-valued function and loss function considered in this paper are the same as those in [15] and [16], which we formally introduce as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Let Cλ : X → Y′ be a set- valued function with a parameter λ ∈ R, where Y′ represents some space of sets and R is the set of real numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Taking single-label classification for example, Y′ can be the power set of Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' For binary image segmentation, Y′ can be equal to Y as the space of all possible results of image segmentation, where the sets here stand for all of the pixels of positive class for the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' We also introduce the nesting property for prediction sets and losses as in [16] as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' For each realization of input object x, we assume that Cλ(x) satisfies the following nesting property: λ1 < λ2 =⇒ Cλ1(x) ⊆ Cλ2(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' (1) Let L : Y × Y′ → R be a loss function respecting the nesting property for each realization of response y: C1 ⊆ C2 ⊆ Y′ =⇒ L(y, C2) ≤ L(y, C1) ≤ B, (2) where B is the upper bound of the loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Inductive Conformal Prediction Inductive conformal prediction (ICP) is a computationally efficient version of the original conformal prediction approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' It starts with any measurable function named nonconformity measure A : X × Y → R and obtain n nonconformity scores as Ai = A(Xi, Yi), for i = 1, · · · , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Then, with the exchangeable assumption and a preset δ ∈ (0, 1), one can conclude that P � A(Xn+1, Yn+1) ≤ Q(n) 1−δ � ≥ 1 − δ, where Q(n) 1−δ is the 1 − δ quantile of {Ai}n i=1 ∪ {∞} [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Therefore, the prediction set made by ICP is C(n) 1−δ(Xn+1) = {y : A(Xn+1, y) ≤ Q(n) 1−δ}, which satisfies P � Yn+1 ∈ C(n) 1−δ(Xn+1) � ≥ 1 − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' The nonconformity measure A is often defined based on a point prediction model ˆf learned from some other training samples, each of which is also drawn from PXY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Here is an example of constructing prediction sets with CP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' For a classification problem with K classes, one can first train a classifier ˆf : X → [0, 1]K with the ith output being the estimation of the probability of the ith class, and calculate the nonconformity scores as A(x, y) = 1 − ˆfk(x), where ˆfk is the kth output of ˆf(x), if y stands for the kth class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Therefore, the corresponding prediction set for an input object x is C(n) 1−δ(x) = {k : ˆfk(x) ≥ 1 − Q(n) 1−δ}, which indicates that k ∈ C(n) 1−δ(x) if the estimated probability of kth class is not less than 1 − Q(n) 1−δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Conformal Risk Control Different from conformal prediction, CRC starts with a set- valued function with the nesting property, whose approach is inspired by nested conformal prediction [19] and was first proposed in the researches about risk-controlling prediction sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Assume one has a way of constructing a set-valued function Cλ with the nesting property of formula (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Given a loss function L with the nesting property of formula (2), the purpose of CRC is to find λ∗ such that E � L(Yn+1, Cλ∗(Xn+1)) � ≤ α, (3) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=', the expected loss or the risk is not greater than α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' To do so, CRC first calculates Li as Li(λ) = L(Yi, Cλ(Xi)), (4) THIS WORK HAS BEEN SUBMITTED TO THE IEEE FOR POSSIBLE PUBLICATION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' 3 with the fact that Li(λ) is a monotone decreasing function of λ based on the nesting properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Then, CRC searches for λ∗ using the following equation, λ∗ = inf � λ : n n + 1 ˆRn(λ) + B n + 1 ≤ α � , where ˆRn(λ) = (L1(λ) + · · · + Ln(λ))/n is an estimation of the risk on calibration data and B is introduced to make the estimation not overconfident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Also, to make λ∗ well defined, one should assume that ˆRn(λ) is right continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' These two steps of CRC are too simple that one may surprise about its theoretical conclusion that with the assump- tion of exchangeability of data samples, the prediction set Cλ∗(Xn+1) obtained by CRC satisfies formula (3), which has been also proved empirically in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' CRC extends CP from controlling the expected value of miscoverage loss to some general loss, which can be applied to the cases where Y is beyond real numbers or vectors, such as images, fields and even graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' After tackling the theoretical issue, the problem for CRC is how to construct Cλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Here, we also give an example of a classification problem with K classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' In fact, with the same notations of the example in Section II-A, CRC can construct the prediction set as Cλ(x) = {k : ˆfk(x) ≥ 1 − λ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Therefore, as long as L satisfies formula (2), such as L is the indicator of miscoverage, CRC guarantees to control the risk as formula (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' CONFORMAL LOSS-CONTROLLING PREDICTION AND ITS THEORETICAL ANALYSIS This section introduces the approach of CLCP and its theoretical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' CLCP also has two steps like CRC, and the main difference between them is that CLCP focuses on whether the estimation of the 1 − δ quantile of the losses is not greater than α while CRC concentrates on whether the mean of the losses not greater than α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' The controlling of the 1−δ quantile of the losses makes CLCP be able to control the value of a general loss by employing the probability inequation derived from the exchangeability assumption, which is also employed by ICP if the loss is seen as the nonconformity score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Suppose one has a way of constructing a set-valued function Cλ with the nesting property of formula (1), which can be the same as that used in CRC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Here, we assume that the parameter λ is selected from a discrete set Λ, such as from 0 to 1 with a step size 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='01, which avoids us from the assumption of right continuous for the loss function in theoretical analysis, and is also reasonable since we actually search for λ∗ with some step size in practice [15] [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Besides, the latest paper about risk-controlling prediction also makes this discrete assumption for general cases [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' After determining Cλ(·) and Λ, CLCP first calculates Li(λ) on calibration data as formula (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Then, for any preset α ∈ R and δ ∈ (0, 1), CLCP searches for λ∗ such that λ∗ = min � λ ∈ Λ : Q(n) 1−δ(λ) ≤ α � , (5) with Q(n) 1−δ(λ) being the 1 − δ quantile of {Li(λ)}n i=1 ∪ {B}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' The approach of CLCP is summarised in Algorithm 1, which is easy to implement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Algorithm 1 Conformal Loss-Controlling Prediction Input: Calibration dataset {(xi, yi)}n i=1, test input object xn+1, the set predictor Cλ satisfies formula (1), the loss function L satisfies formula (2), preset α ∈ R and δ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Output: Predictive set for yn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' 1: Based on formula (4), calculate {Li(λ)}n i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' 2: Search for λ∗ satisfying formula (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' 3: return Cλ∗(xn+1) Next, we introduce the definition of (α, δ)-loss-controlling set predictors and then prove our theoretical conclusion about CLCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Given a loss function L : Y × Y′ → R and a random sample (X, Y ) ∈ X ×Y, a random set-valued function C whose realization is in the space of functions X → Y′ is a (α, δ)-loss-controlling set predictor if it satisfies that P � L � Y, C(X) � ≤ α � ≥ 1 − δ, where the randomness is both from C and (X, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' After all these preparations, we can prove in Theorem 1 that Cλ∗ constructed by CLCP is a (α, δ)-loss-controlling set predictor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Suppose {(Xi, Yi)}n+1 i=1 are n + 1 data drawn exchangeably from PXY on X × Y, Cλ : X → Y′ is a set- valued function satisfying formula (1) with the parameter λ taking values from a discrete set Λ ⊂ R , L : Y × Y′ → R is a loss function satisfying formula (2) and Li(λ) is defined as formula (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' For any preset α ∈ R, if L also satisfies the following conditions, min λ max i Li(λ) < α, max λ min i Li(λ) > α, (6) then for any δ ∈ (0, 1), we have P � L � Yn+1, Cλ∗(Xn+1) � ≤ α � ≥ 1 − δ, (7) where λ∗ is defined as formula (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Let Q(n+1) 1−δ (λ) be the 1 − δ quantile of {Li(λ)}n+1 i=1 , and define ˜λ as ˜λ = min � λ ∈ Λ : Q(n+1) 1−δ (λ) ≤ α � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' THIS WORK HAS BEEN SUBMITTED TO THE IEEE FOR POSSIBLE PUBLICATION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' 4 Similarly, let Q(n) 1−δ(λ) be the 1 − δ quantile of {Li(λ)}n i=1 ∪ {B}, and we have λ∗ = min � λ ∈ Λ : Q(n) 1−δ(λ) ≤ α � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Since B is the upper bound of Ln+1(λ), by definition, we have ˜λ ≤ λ∗, and the conditions introduced in formula (6) is to make ˜λ and ˆλ well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' This leads to Ln+1(λ∗) ≤ Ln+1(˜λ), (8) as Cλ and L satisfy the nesting properties of formula (1) and (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Since ˜λ is dependent on the whole dataset {(Xi, Yi)}n+1 i=1 , {Li(˜λ)}n+1 i=1 are exchangeable variables, which leads to P � Ln+1(˜λ) ≤ Q(n+1) 1−δ (˜λ) � ≥ 1 − δ, (9) as Q(n+1) 1−δ (˜λ) is just the corresponding 1−δ quantile (See the proof of Lemma 1 in [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Combining the definition of ˜λ, formula (8) and (9), we have P � Ln+1(λ∗) ≤ α � ≥ 1 − δ, which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' At the end of this section, we show that CP can be seen as a special case of CLCP from the following viewpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Suppose Cλ is constructed by a nonconformity score A, which is defined as Cλ(x) = {y : A(x, y) ≤ λ}, and L is the miscoverage loss such that Li(λ) = L(yi, Cλ(xi)) = I{yi /∈ Cλ(xi)}, where I is the indicator function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' In this case, Q(n) 1−δ(λ) can only be 0 or 1 as the loss can only be these two numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Besides, only α ∈ [0, 1) is meaningful, which means that one wants to control the miscoverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' For CLCP, let Λ be an arithmetic sequence whose common difference, minimum and maximum are ∆, λmin and λmax respectively and set α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' By definition, λ∗ can be written as λ∗ = min � λ ∈ Λ : 1 n + 1 n � i=1 I{ai ≤ λ} ≥ 1 − δ � = min � λ ∈ Λ : 1 n n � i=1 I{ai ≤ λ} ≥ ⌈(1 − δ)(n + 1)⌉ n � , where ai = A(xi, yi) is the nonconformity score of the ith calibration data for CP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' In comparison, referring to [15], the optimal ˆλ for CP is ˆλ = inf � λ ∈ R : 1 n n � i=1 I{ai ≤ λ} ≥ ⌈(1 − δ)(n + 1)⌉ n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Therefore, if λmin < ai < λmax for each i, we have |λ∗ − ˆλ| ≤ ∆, which implies that the prediction sets of CP and CLCP are nearly the same if ∆ is small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' In summary, if Cλ and L have special forms and Λ includes the upper and lower bounds of nonconformity scores with ∆ being small enough to be ignored, CP can be seen as a special case of CLCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' EXPERIMENTS This section conducts the experiments to empirically test the approach of CLCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' First, we built CLCP for the classification problem with a class-varying loss introduced in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Then, we focus on two types of weather forecasting applications, which can be seen as point-wise classification and point- wise regression problems respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' All experiments were coded in Python [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' The statistical learning methods used in Section IV-A were implemented using Scikit-learn [22] and the deep learning methods used in Section IV-B and Section IV-C were implemented with Pytorch [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' CLCP for classification with a class-varying loss We collected 20 binary or multiclass classification datasets from UCI repositories [24] whose information is summarized in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' The problem is to make the prediction sets of labels controlling the following loss L(y, C) = LyI{y /∈ C}, where Ly is the loss for y being not in the prediction set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' The loss for each label is generated uniformly on (0, 1) like [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Support vector machine (SVM) [25], neural network (NN) [26] and random forests (RF) [27] were employed as the underlying algorithms separately to construct prediction sets based on CLCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' The prediction set Cλ is constructed as Cλ(x) = {k : ˆfk(x) ≥ 1 − λ}, where ˆfk is the estimated probability of the observation being kth class by the corresponding underlying algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' For each dataset, we used 20% of the data for testing and 80% and 20% of the remaining data for training and calibration respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Based on the training data, we selected the meta-parameters with three-fold cross-validation and used the optimal meta- parameters to train the classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' The regularization parameter of SVM was selected from {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='001, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='1, 1, 10, 100}, and the learning rate and the epochs of NN were selected from {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='001, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='0001} and {200, 500, 1000}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' The number of trees of RF were selected from {100, 300, 500} and the partition criterion was either gini or entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' After training, we used the trained classifiers and the calibration data to search for λ∗ with Algorithm 1 and construct the final set predictors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' All of the features were normalized to [0, 1] by min–max normalization and for each dataset, the experiments were conducted 10 times and the average results were recorded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' The bar plots in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' 1 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' 2 show the experimental results for 20 public datasets with δ ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='15, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='2} and α ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' The results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' 1 concern about the THIS WORK HAS BEEN SUBMITTED TO THE IEEE FOR POSSIBLE PUBLICATION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='TABLE I ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='DATASETS FROM UCI REPOSITORIES ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='Dataset ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='Examples ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='Dimensionality ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='Classes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='bc-wisc-diag ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='569 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='car ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='wine-quality-white ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='4898 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='frequency of the prediction losses being more than α on test ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='set,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' which is the estimated probability of P � L � Yn+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Cλ∗(Xn+1) � > α � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' which should be near or lower than δ empirically due to formula (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' The bar plots of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' 1 demonstrate that the frequency of the prediction losses is near or below δ, which verifies the conclusion of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' The bar plots of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' 2 show the average sizes of prediction sets for different δ, describing the informational efficiency of the prediction sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Changing δ can effectively change the average size of prediction sets and changing α may slightly change average size (such as the results for wine-quality-red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Although many prediction sets are meaningful with average sizes being near 1, the prediction sets for the dataset contrac may be not usefull, since no matter how to change δ and α, the average sizes of the prediction sets are all near or above 2, whereas the number of classes of contrac is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Thus, how to construct efficient prediction sets in the learning framework of CLCP is worth exploring for further researches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Combining Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' 1 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' 2, we observe that different classifiers can perform differently for different datasets, which indicates that the underlying algorithm affects the performance and the model selection approach is necessary for CLCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' CLCP for high-impact weather forecasting The remaining experiments apply CLCP to weather fore- casting problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Here we concentrate on postprocessing of the forecasts made by numerical weather prediction (NWP) models [28] [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' NWP models use equations of atmospheric dynamics and estimations of current weather conditions to do weather forecasting, which is the mainstream weather fore- casting technique nowadays especially for forecasting beyond 12 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Many errors affect the performance of NWP models, such as the estimation errors of initial conditions and the approximation errors of NWP models, leading to the research topic about postprocessing the forecasts of NWP models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Most postprocessing methods are built on some learning process, which takes the forecasts of NWP models as inputs and the observations of weather elements or events as outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' In this paper, we use CLCP to postprocess the ensemble forecasts with the control forecast and 50 perturbed forecasts issued by the NWP model from European Centre for Medium- Range Weather Forecasts (ECMWF) [30], which are obtained from the THORPEX Interactive Grand Global Ensemble (TIGGE) dataset [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' We focus on 2-m maximum temperature and minimum temperature between the forecast lead times of 12nd hour and 36th hour with the forecasts initialized at 0000 UTC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' The forecast fields are grided with the resolution of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='5◦ × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='5◦ and the corresponding label fields with the same resolution are extracted from the ERA5 reanalysis data [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' The area ranges from 109◦E to 122◦E in longitude and from 29◦N to 42◦N in latitude, covering the main parts of North China, East China and Central China, whose grid size is 27 × 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' The ECMWF forecast data and ERA5 reanalysis data are collected from 2007 to 2020 (14 years).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' We first consider high-impact weather forecasting, which is to forecast whether a high-impact weather exists for each grid and can be seen as a point-wise classification problem or image segmentation problem for computer vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' The high- impact weather we consider is whether the 2-m maximum temperature is above 35 °C or the 2-m minimum temperature is below −15 °C for each grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' These two cases are treated as high temperature weather or low temperature weather in China, which make meteorological observatories issue high temperature warning or low temperature warning respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' The prediction sets and the loss function used for high- impact weather forecasting are the same as those for image segmentation in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Taking the ensemble forecast fields of the NWP model as input x, the corresponding label y is a set of grids having high-impact weather, which can be seen as a segmentation problem for high-impact weather.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Therefore, we first train a segmentation neural network f(x), where f(p,q)(x) is the estimated probability of the grid (p, q) having high-impact weather.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Then the set-valued function Cλ can be constructed as Cλ(x) = {(p, q) : f(p,q)(x) ≥ 1 − λ}, (10) and the loss function is L(y, C) = 1 − |y ∩ C| |y| , (11) which measures the ratio of the prediction sets failing to do the warning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' We use CLCP with the prediction set and the loss function above to do high temperature and low temperature forecasting respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' 1) Dataset for high temperature forecasting: The reanalysis fields of 2-m maximum temperature were collected from ERA5 and the label fields were calculated based on weather the 2-m maximum temperature is above 35 °C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' To make the loss function take finite values, we only collected the data whose label fields have at least one high temperature grid to do this empirical study, which resulted in 1200 samples in total, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=', 1200 ensemble forecasts from the NWP model of ECMWF and corresponding label fields calculated from THIS WORK HAS BEEN SUBMITTED TO THE IEEE FOR POSSIBLE PUBLICATION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' 6 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Bar plots of the frequencies of the prediction losses being more than α vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='15, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='2 on test data for classification with a class-varying loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' The first row corresponds to α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='1 and the second row corresponds to α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Different columns represent different classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' All bars are near or below the preset δ, which confirms the controlling guarantee of CLCP empirically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Bar plots of the average sizes of prediction sets vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='15, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='2 on test data for classification with a class-varying loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' The first row corresponds to α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='1 and the second row corresponds to α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Different columns represent different classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' The plots demonstrate the information in prediction sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' In general, large δ leads to small average size and different classifiers have different informational efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' ERA5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' We name this dataset as HighTemp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' 2) Dataset for low temperature forecasting: The dataset for testing CLCP for low temperature weather forecasting was constructed in a similar way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' The reanalysis fields of 2-m minimum temperature were collected from ERA5 and the label fields were calculated based on weather the 2-m minimum temperature is below −15 °C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' We only collected the data whose label fields have at least one low temperature grid to do this empirical study, which resulted in 1233 samples in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' We name this dataset as LowTemp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' For each dataset, the same process was used to conduct the experiment as Section IV-A , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=', all forecasts from the NWP model were normalized to [0, 1] by min–max normalization, and we used 20% of the data for testing and 80% and 20% of the remaining data for training and calibration respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' We employed two fully convolutional neural networks [33] for binary image segmentation as our underlying algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' One was U-Net [34] with the same structure as that in [35], whose numbers of hidden feature maps were all set to 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' The other was the naive deep neural network (nDNN) with the same encoder-decoder structure as the U-Net without skip- connections, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=', the U-Net removing skip-connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' We α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='1 I Classifier = SVM α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='1 I Classifier = NN α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='1 Classifier = RF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='20 Dataset bc-wisc-diag car chess-kr-kp contrac credit-a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='05 credit-g ctg-10classes ctg-3classes 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='00 haberman α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='2 I Classifier = SVM α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='2 I Classifier = NN α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='2 I Classifier = RF optical 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='30 phishing-web st-image st-landsat 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='25 tic-tac-toe α wall-following waveform waveform-noise wilt wine-quality-red wine-quality-white 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='2 6 6 6α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='1 I Classifier = SVM α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='1 I Classifier = NN α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='1 I Classifier = RF 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='5 Dataset 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='5 bc-wisc-diag car chess-kr-kp 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='0 contrac credit-a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='5 credit-g ctg-10classes ctg-3classes 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='0 haberman α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='2 I Classifier = SVM α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='2 I Classifier = NN α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='2 I Classifier = RF optical phishing-web 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='0 st-image st-landsat tic-tac-toe 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='5 wall-following waveform 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='0 waveform-noise Average s wilt wine-quality-red 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='5 wine-quality-white 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='2 6 6 6THIS WORK HAS BEEN SUBMITTED TO THE IEEE FOR POSSIBLE PUBLICATION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' 7 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Bar plots of the frequencies of the prediction losses being more than α vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='15, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='2 on test data for high-impact weather forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' The first row corresponds to HighTemp and the second row corresponds to LowTemp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Different columns represent different α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' All bars are near or below the preset δ, which confirms the controlling guarantee of CLCP empirically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Boxen plots of the prediction losses vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='15, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='2 on test data for high-impact weather forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' The first row corresponds to HighTemp and the second row corresponds to LowTemp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Different columns represent different α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' The loss distributions are controlled by α and δ properly to obtain the empirical validity in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' use these two neural networks to show that the designation of the underlying algorithm is necessary for better performance, as U-Net fuses multi-scale features and nDNN does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' The data for training U-Net and DNN were further partitioned to the validation part (10%) for model selection and proper training part (90%) for updating the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Adam op- timization [36] was used for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' The learning rate was set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='0001 and the number of epochs was set to 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' After training 50 epochs, the model with lowest binary cross entropy on validation data was used for formula (10) to construct prediction sets, where λ is search from 1 to 0 with step size 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' The experiments of using CLCP for the loss function as formula (11) were conducted 10 times and the prediction results on test set are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' 3, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' 4 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' 3 also shows the bar plots of the frequencies of the prediction losses being more than α for δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='15 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Four column stands for the cases where α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='15 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' It can be seen that for the two datasets HighTemp and LowTemp, all bars are near or below the preset δ, which verifies formula (7) empirically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' 4 further shows the distributions of the losses for different δ and different α using boxen plots, which contain more information than box plots by drawing narrow boxes for tails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' It can be seen that larger α and δ lead to larger losses, which is reasonable since large α and δ relax the constraint on prediction losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' We measure the informational efficiency of the prediction set Cλ∗(x) using its normalized size defined as |Cλ∗(x)|/PQ, where P and Q are the numbers of the vertical and the horizontal grids respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' The distributions of normalized sizes in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' 5 show that U-Net is more informationally efficient than nDNN, which indicates that designation of the underlying algorithm is important for CLCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Different α and δ lead to different normalized sizes, implying the trade-off among the preset loss bound α, confidence level 1 − δ and informational efficiency of the prediction sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' By choosing α and δ properly, the prediction sets of CLCP Dataset = HighTemp | α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='05 Dataset = HighTemp | α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='15 Dataset = HighTemp I α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='2 Dataset = HighTemp I α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='00 Model Dataset = LowTemp I α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='05 Dataset = LowTemp I α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='1 Dataset = LowTemp I α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='15 Dataset = LowTemp I α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='2 nDNN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='30 U-Net 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='2 6 6 6Dataset = HighTemp I α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='05 Dataset = HighTemp I α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='1 Dataset = HighTemp I α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='15 Dataset = HighTemp I α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='6 Loss 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='0 Model Dataset = LowTemp I α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='05 Dataset = LowTemp I α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='1 Dataset = LowTemp I α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='15 Dataset = LowTemp I α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='2 nDNN 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='0 U-Net 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='6 Loss 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='2 6 6 6THIS WORK HAS BEEN SUBMITTED TO THE IEEE FOR POSSIBLE PUBLICATION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' 8 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Boxen plots for the distributions of normalized sizes of prediction sets vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='15, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='2 on test data for high-impact weather forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='. The first row corresponds to HighTemp and the second row corresponds to LowTemp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Different columns represent different α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' U-Net performs better than nDNN, which indicates the importance of careful designation of the underlying algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' can have reasonable sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Also, we can see that forecasting low temperature is somehow easier than high temperature with the fact that for the same α and δ, the normalized sizes of forecasting low temperature are distributed lower than the ones of forecasting high temperature, indicating the need of designation of the underlying algorithms to improve performance for forecasting high temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' CLCP for maximum temperature and minimum tempera- ture forecasting This section focuses on using CLCP to forecast the 2-m maximum temperature or minimum temperature value for each grid, which is a point-wise regression problem or image-to- image regression problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' To construct the prediction sets, we follow the procedure proposed in [37] and train the neural network with 3 output channels jointly predicting the point- wise 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='5 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='95 quantiles of the fields using quantile regression [38] [37], which are denoted by f 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='05(x), f 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='5(x) and f 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='95(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Then the prediction set Cλ(x) is equal to � y : y(p,q) ∈ [f 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='5 (p,q)(x)−λ∆− (p,q)(x), f 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='5 (p,q)(x)+λ∆+ (p,q)(x)] � , where ∆−(x) = max{f 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='5(x) − f 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='05(x), 10−6}, ∆+(x) = max{f 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='95(x) − f 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='5(x), 10−6}, and max is a point-wise operator making ∆− and ∆+ at least 10−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' This prediction set is a prediction band for the output field, whose prediction interval at grid (p, q) is [f 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='5 (p,q)(x) − λ∆− (p,q)(x), f 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='5 (p,q)(x) + λ∆+ (p,q)(x)] (12) with the point-wise width being an increasing function of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' This construction was proposed in [37] for image-to-image regression and we use the same loss function in [37] measuring miscoverage rate of a prediction band C for a field y, which can be formalized as L(y, C) = 1 PQ ��� � (p, q) : y(p,q) /∈ C(p,q) ����, (13) where C(p,q) is the prediction interval at grid (p, q) for prediction band C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' All of the data collected from 2007 to 2020 were used, lead- ing to 4945 samples for each forecasting application and the datasets are named as MaxTemp and MinTemp respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' The experimental designation is the same as that in Section IV-B, except that we also normalized the label for each grid to [0, 1] by min–max normalization, used quantile loss for model selection and we searched for λ∗ with two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' First we found two values λ1 and λ2 from {100, 10, 1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='01, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='} such that Q(n) 1−δ(λ1) ≤ α and Q(n) 1−δ(λ2) > α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Then we searched for λ∗ from 100 values starting with λ1 and ending with λ2 using a common step size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' The experimental results are recorded in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' 6, Fig 7 and Fig 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Although the set predictors and the loss function used in this section are different from those in Section IV-B, the experimental results and conclusions are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' 6, we can see that the frequencies of the prediction losses being more than α are controlled by δ, which also verifies formula (7) empirically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Larger α and δ lead to larger losses, which is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Here we use the following average interval length 1 PQ P � p=1 Q � q=1 λ∗(∆+ (p,q)(x) − ∆− (p,q)(x)) (14) to measure the informational efficiency of the prediction set Cλ∗(x) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' 8 also depicts the trade-off among the preset loss bound α, confidence level 1 − δ and informational efficiency of the prediction sets and indicates that better des- ignation of underlying algorithms lead to better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' CONCLUSION This paper extends conformal prediction to the situation where the value of a loss function needs to be controlled, Dataset = HighTemp I α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='05 Dataset = HighTemp I α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='1 Dataset = HighTemp I α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='15 Dataset = HighTemp I α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='8 I Size Normalized 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='0 Model Dataset = LowTemp I α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='05 Dataset = LowTemp I α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='1 Dataset = LowTemp I α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='15 Dataset = LowTemp I α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='2 nDNN 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='0 U-Net 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='15 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='05 6 6THIS WORK HAS BEEN SUBMITTED TO THE IEEE FOR POSSIBLE PUBLICATION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' 9 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Bar plots of the frequencies of the prediction losses being more than α vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='15, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='2 on test data for maximum temperature and minimum temperature forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' The first row corresponds to MaxTemp and the second row corresponds to MinTemp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Different columns represent different α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' All bars are near or below the preset δ, which confirms the controlling guarantee of CLCP empirically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Boxen plots of the prediction losses vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='15, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='2 on test data for maximum temperature and minimum temperature forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' The first row corresponds to MaxTemp and the second row corresponds to MinTemp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Different columns represent different α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' The loss distributions are controlled by α and δ properly to obtain the empirical validity in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' which is inspired by risk-controlling prediction sets and con- formal risk control approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' The loss-controlling guarantee is proved in theory with the assumption of exchangeability and is empirically verified for different kinds of applications including classification with a class-varying loss and weather forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Different from conformal prediction, conformal loss-controlling prediction approach proposed in this paper has two preset parameters α and δ, which guarantees that the prediction loss is not greater than α with confidence 1−δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Both parameters impose restrictions on prediction sets and should be set based on specific applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Despite loss-controlling guarantee, informational efficiency of the prediction sets built by conformal loss-controlling prediction is highly related to underlying algorithms, which has been shown in empirical studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Since this is a rather new topic, the underlying algorithms and the way of constructing set predictors are inherited from conformal risk control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' This leaves the im- portant question on how to build informationally efficient set predictors in an optimal way, which is one of our further researches in the future.' metadata={'source': 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distribution-free uncertainty quantification,” arXiv preprint arXiv:2107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='07511, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' [6] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Fontana, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Zeni, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Vantini, “Conformal prediction: a unified review of theory and new challenges,” Bernoulli, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' 29, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' 1–23, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Dataset = MaxTemp I α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='05 Dataset = MaxTemp I α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='1 Dataset = MaxTemp I α = 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='2 6 6THIS WORK HAS BEEN SUBMITTED TO THE IEEE FOR POSSIBLE PUBLICATION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' 10 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' Boxen plots for the distributions of average interval length vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='15, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content='2 on test data for maximum temperature and minimum temperature forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E0T4oBgHgl3EQfhAEU/content/2301.02424v1.pdf'} +page_content=' The first row 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100644 index 0000000000000000000000000000000000000000..f5cd51be4d69d544c99ff0abc270f1b149c0ab53 --- /dev/null +++ b/LNE1T4oBgHgl3EQfswVe/content/tmp_files/2301.03369v1.pdf.txt @@ -0,0 +1,1216 @@ +arXiv:2301.03369v1 [physics.data-an] 6 Jan 2023 +NOLTA, IEICE +Paper +Time series analysis using persistent +homology of distance matrix +Takashi Ichinomiya 1,2a) +1 Gifu University School of Medicine, +Yanagido 1-1, Gifu, Gifu 501-1194, Japan +2 The United Graduate School of Drug Discovery and Medical Informatic +Science, Gifu University, +Yanagido 1-1, Gifu 501-1194, Japan +a) tk1miya@gifu-u.ac.jp +Received December 06, 2022 +Abstract: The analysis of nonlinear dynamics is an important issue in numerous fields of +science. In this study, we propose a new method to analyze the time series data using persistent +homology (PH). The key idea is the application of PH to the distance matrix. Using this +method, we can obtain the topological features embedded in the trajectories. We apply this +method to the logistic map, R¨ossler system, and electrocardiogram data. The results reveal +that our method can effectively identify nonlocal characteristics of the attractor and can classify +data based on the amount of noise. +Key Words: time-series analysis, persistent homology, dynamical system +1. Introduction +The analysis of the nonlinear dynamics is a challenging problem in physics, engineering, and data +science. Numerous methods for time series analysis, such as Fourier transformation or Kalman fil- +tering, are based on the theory of linear dynamics, and the performance of these methods is limited. +To overcome their limitations, several methods, such as Koopman’s mode decomposition[1], phase +reduction[2], deep learning[3], and reservoir computing[4, 5], have been proposed. However, we often +meet the situations where these methods are not available. In Koopman’s mode decomposition we +map the finite-dimensional nonlinear dynamical system into an infinite-dimensional linear dynamical +system, and investigate the eigenvalues and eigenfunctions in this space. However, the determination +of these eigenfunctions is often difficult. Phase reduction is a powerful method to investigate the +oscillatory dynamics, but the application to non-oscillatory systems is limited. Deep learning and +reservoir computing help us to predict the state in the future, but they do not provide a rationale for +their prediction. +In this study, we propose a new method to analyze the dynamics using distance matrix. +The +distance matrix D(t, s), defined as the distance between states at time t and s, reveals considerable +information regarding the dynamics of the system. For example, if D(t, t+T) = 0 for all t, the system +exhibits a periodic motion with period T. Recurrence plot (RP) is the most useful method for the +1 +Nonlinear Theory and Its Applications, IEICE, vol. X, no. 0, pp. 1–14 +©IEICE 2023 +DOI: 10.1587/nolta.X.1 + +time-series analysis using a distance matrix [6]. In this method, the distance matrix is visualized using +R(t, s) = Θ(ǫ − D(t, s)), where Θ(x) is the Heaviside step function and ǫ is a parameter. Using this +method, periodic motion can be distinguished from chaotic trajectories. Based on an RP, recurrence +quantification analysis (RQA), wherein the dynamics are characterized by several quantities, such as +recurrence rate and determinism, was proposed. +However, an RP uses only the limited information embedded in the distance matrix. First, an RP +does not provide information on the nonlocal properties of the trajectory. An RP lists the points that +are close to each other in phase space and is useful to investigate local properties such as Lyapunov +exponents. In contrast, it is unsuitable for investigating the global structure of the trajectory. For +example, determining whether the attractor has a double scroll structure like the Lorenz system or +an oscillation-like structure similar to the R¨ossler system by RP is difficult. Another problem with +the use of an RP is that there is no clear rule to select the value of ǫ, and studies have reported that +the result of the RP is often sensitive to this choice [7]. +In this study, we propose the application of persistent homology (PH)[8], an emerging technique of +data analysis, to the analysis of the distance matrix. PH is the one of the most popular techniques +in topological data analysis (TDA). In TDA, we investigate the topological characteristics such as +number of connected components or holes embedded in the dataset. The theory of PH is still being +developed and has been successfully applied in various fields, including biophysics[9–11], material +science[12–14], and image processing[15, 16]. +There have been several proposals to apply PH to time series datasets. The favored approach for +the time-series analysis using PH is based on delay embedding [17, 18]. In this approach, we first +create the point clouds in n-dimensional space using Takens’ delay embedding [19] and subsequently +characterize the state using PH. However, this approach has several difficulties. +First, we must +determine how to embed the dataset. There is no general rule to determine the way of embedding, +though several ideas have been proposed [20–22]. Second, the computational cost increases rapidly +as the embedding dimension and the size of point cloud increases. It is known that the computation +time for PH is O(N⌈D/2⌉), where N and D represent the size of the point cloud and dimension of +the space, respectively [23]. Therefore, the computational cost increases exponentially as D increases. +In our approach, we can avoid the latter difficulty. Because the distance matrix is represented in +two-dimensional space, the computation cost of PH is considerably reduced. +The results of this +study reveal that using PH, the essential information of a dataset, such as the non-local structure of +attractors and the amount of noise, can be easily determined. +The remainder of this study is structured as follows. In Section +2, we explain our method to +investigate the distance matrix. In Section 3, we present the results of application of our method to +three different datasets. First, we present the results of the analysis of the logistic map as a typical +example of discrete time dynamics. Second, we present the results on the R¨ossler system as an example +of continuous time dynamics. Third, we discuss the results on the analysis of electrocardiogram (ECG) +data, as an example of real-world time series data. Finally, we discuss on the possible improvements +to our method in future and conclude this study. +2. Method +The general definition of PH requires further background knowledge of algebraic topology. In this +section, we explain the PH of filtered cubical complexes of degree 0, which is used in the latter part +of this study. The readers who want to know the general definition of PH can consult textbooks on +PH and TDA[24, 25]. +We consider a real-valued filtration function f : Z2 → R. The sublevel set M(θ) is defined by +M(θ) = {(x, y) ∈ Z2|f(x, y) ≤ θ}. +(1) +For example, assume that f(x, y) is given by the table shown in Fig. 1(a). In this case, M(0) is given +by the gray blocks. When we increase θ, M(θ) also grows, as shown in Fig. 1(b) and (c). +PH with degree 0 using a sublevel set of f represents the change in connected components in M(θ) +when θ is varied from −∞ to ∞. Here we say two blocks are “connected” if they share an edge. For +2 + +! +! +! +! +" +" +" +# +# +" +" +# +$ +" +# +" +" +" +# +# +" +# +! +" +$ +! +! +! +! +" +" +" +# +# +" +" +# +$ +" +# +" +" +" +# +# +" +# +! +" +$ +! +! +! +! +" +" +" +# +# +" +" +# +$ +" +# +" +" +" +# +" +" +# +! +" +$ +%&'()!"#$ +%*'()!"%$ +%+'()!"&$ +' +# +% +& +( +) +# +% +& +( +) +* +! +" +# +$ ++ +, +! +" +# +$ +! +" +# +$ +%,' +%-' +# +% +& +( +) +# +% +& +( +) +Fig. 1. +Example of persistent homology using sublevel sets, (a)–(c) represent +the M(θ) for θ = 0, 1, and 2, respectively. +The corresponding persistence +barcode and persistence diagram are shown in (d) and (e). +example, we have two components in Fig. 1(a): there is one isolated component at (x, y) = (2, 0) +and one connected component at y = 4, 0 ≤ x ≤ 3. By increasing θ to 1, these two components are +merged into one large component, and another component appears at (x, y) = (3, 2), as shown in +Fig. 1(b). Here, we do not say these two components are connected, because although they share two +corners, they do not share an edge. In this case, we say these two components are “disconnected.” +When we increase θ to 2, all three components are merged into one large component, as shown +in Fig. 1(c). +The theory of PH guarantees that we can define the “birth” and “death” of each +component. In this example, at θ = 0, two disconnected components are “born.” At θ = 1, these +components merged into one large component. Hence, we say that one of the components “dies” and +the other disconnected component at (x, y) = (3, 2) is born. Finally, at θ = 2, these components are +connected. Further increase in θ does not change the number of disconnected components. Therefore, +we have three “connected components,” often called “generators,” whose births b and deaths d are +(b, d) = (0, ∞), (0, 1), (1, 2), respectively. +There are two major visualization techniques to represent the distribution of generators. One is +“persistence barcode,” wherein we represent each generator as a “bar” from birth to death, as shown +in Fig. 1(d). This representation is intuitive when the number of generators is small. For example, +when the dataset has a periodic structure, we will obtain numerous generators that have the same +birth and death, and we can easily identify them by persistence barcode. However, when we have +more than a hundred of generators, the number of bars becomes too large, and gaining any insight +from the barcode becomes difficult. In this case, a persistence diagram (PD) is better visualization +method; herein, we make a scatter plot of birth and death, as shown in Fig. 1(e). In a PD, generators +with infinite death are generally omitted. When the number of generators becomes large, we also use +a density heatmap of generators, which is also called a PD. In the rest of this study, we use PDs to +represent the results of our PH analysis. +The PH allows us to study the local minimum and saddle points of f. For example, we consider +the case of Fig. 2. In the case of Fig. 2(a), f is a smooth function with two minima. In this case, we +can define the sublevel set M(θ) as M(θ) = {x ∈ R|f(x) ≤ θ}. If θ is smaller than the saddle value of +3 + +!"#$% +&'($% +)(* +)!* +)+* +! +! +! +Fig. 2. +Relation between the form of the filtering function f (upper) and +corresponding persistence diagram(lower). (a) When f has two local minima, +we obtain two generators, and only one has finite death. (b) When f is more +complex and has more local minima, the number of generators increases. (c) +If the “saddles” between local minima are low, the lifetime of generators de- +creases. The dashed line in the persistence diagrams indicates the line birth = +death. +f, M(θ) has two connected components, and for larger θ, M(θ) has only one connected component. +Therefore, in this case, there are two generators, and one of these generators has finite death. In +contrast, if D has numerous local minima as described in Fig. 2(b), we obtain numerous generators +with finite deaths. Therefore, the number of generators indicates the number of local minima of f. +Moreover, PH provides information about the height of the saddles. For example, we consider the +case in Fig. 2(c). Here, f has numerous local minima, but the height of the saddle is lower than in +Fig. 2(b). In this case, the connected components of M(θ) merge after only a slight increase of θ. +Therefore, the lifetime, defined as the difference between death and birth, decreases as the height of +the corresponding saddle decreases. +In this study, we used a distance matrix D for filtration. We suppose that we have time series data +xi, i = 1, 2, . . . , K, where i represents the discretized time. From this dataset, we define the distance +matrix D(i, j) as +D(i, j) = ||xi − xj||, +(2) +where ||· · · || represents L2-norm. D(i, j) is a real-valued function from (i, j) ∈ Z2, and we can apply +PH using the distance matrix. +There are several software for PH analysis, which include Gudhi [26], Phat [27], and Javaplex [28]. +In this study, we used Homcloud developed by Obayashi et al [29]. One of the advantages of Homcloud +compared with other software is that it can provide the “birth position” and “death position” of each +generator. Using this function, we can obtain the positions of the local minima and the saddles of +D(i, j). These values are useful for interpreting the result obtained by PH. +3. Results +We applied our method to three different datasets. The first example is time-series data obtained +from the logistic map, and the second one is that obtained from the R¨ossler equations. Finally, we +analyzed the dataset ECG200, as an example of a real-world dataset. +3.1 Analysis of the logistic map +In this subsection, we investigated the distance matrix of the logistic map defined by +xi+1 = axi(1 − xi), +(3) +where 0 < a < 4 is a parameter. +We calculated xi for i = 1, 2, . . . , 1000 with initial condition +x0 = 0.832 and performed a PH analysis using data at i = 801, 802, . . . , 1000. +First, we investigated the case a = 3.4, wherein xi is periodic, shown in Fig. 3(a). In this case, all +generators had birth 0 and death 0.3901. This result indicated that xi takes only two values, and +4 + +!"# +!$# +!%# +!&# +Fig. 3. +Persistence diagram for logistic maps: (a) a = 3.4, (b) a = 3.5, (c) +a = 3.6, and (d) a = 3.7. +the difference between these two is 0.3901. This is consistent with the fact that the attractor of this +system is a cycle with period 2: x2i+1 = 0.4520 and x2i = 0.8421. The death time is given by the +difference between these two values. As the period increases, the number of values that generators +can take also increases. For example, at a = 3.5, the distribution had 6 peaks, corresponding to the +fact that the period of the logistic map was 4. We show M(θ) for several θ in Fig.4 in the case of +a = 3.5. The number of peaks represents the topological information of the attractors. +Furthermore, the PD in a chaotic region provides topological information of the attractor. In the +case of a = 3.6, where the logistic map becomes chaotic, the births and deaths of generators are widely +distributed, as shown in Fig. 3(c). Herein, we found that the distribution had several characteristic +properties. First, all deaths were larger than 0.188. This property could be explained by the existence +of a gap in the attractor. At a = 3.6, x takes values from 0.3 to 0.6 and from 0.788 to 0.900, but +does not take values between 0.601 to 0.787. This property generates a the gap in the distribution of +deaths. To confirm this suggestion, we show the “death point” of generators whose death is smaller +than 0.19 as a red line in Fig. 5. In this figure, the points connected by lines give the saddles of +D(i, j). Evidently, the red lines connect the top point of the lower band and the bottom points of the +upper band, which supports our claim. +Additionally, Fig. 3(c) reveals that birth + death < 0.4695. Unfortunately, we have no theory to +explain this property. Notably, we have an upper limit on birth + death when xmin ≤ xi ≤ xmax +for all i, because birth ≥ 0 and death ≤ xmax − xmin. However, the fact that the death corresponds +to the saddle makes the problem difficult. For example, the green line in Fig. 3(e) shows the death +point pairs with the largest birth + death, 0.4694. This figure shows that one terminal of this line +is at the upper limit of the upper band, whereas the other terminal stays in the middle of the lower +band. We have no theoretical explanation why there is no saddle pair of states that provides larger +birth + death. This is a problem that should be solved in the future. +In the case of a = 3.7 shown in Fig. 3(d), the upper limit of death+birth appears to remain with +several exceptional generators, whereas the lower limit disappears. +The absence of a lower limit +indicates the elimination of the gap in x found in the case a = 3.6. Summarizing these results, the +PD gives the topological structure of the attractor, in the cases of both the periodic trajectory and +chaotic attractor. +5 + +Logistic, a=3.5 +0.8 +0.7 +0.6 +103 +0.5 +Death +0.4 +Value +102 +0.3 +0.2 +0.1 +101 +0.0 +-0.1 +0.0 +0.2 +0.4 +0.6 +0.8 +BirthLogistic, a=3.4 +0.8 +105 +0.7 +0.6 +0.5 +104 +Death +0.4 +Value +0.3 +0.2 +103 +0.1 +0.0 +-0.1 +102 +0.0 +0.2 +0.4 +0.6 +0.8 +BirthLogistic, a=3.6 +0.8 +0.7 +0.6 +0.5 +102 +Death +0.4 +Value +0.3 +0.2 +101 +0.1 +0.0 +-0.1 +0.0 +0.2 +0.4 +0.6 +0.8 +BirthLogistic, a=3.7 +0.8 +0.7 +102 +0.6 +0.5 +Death +0.4 +0.3 +0.2 +0.1 +0.0 +-0.1 +100 +0.0 +0.2 +0.4 +0.6 +0.8 +Birth(a) +(b) +(c) +(d) +(e) +Fig. 4. +Sublevel set M(θ) represented as black areas in the case of a = 3.5. +(a) θ = 0.01, (b) θ = 0.05, (c) θ = 0.20, (d) θ = 0.35, and (e) θ = 0.40, +respectively. +Fig. 5. +Time series xi for the case of a = 3.6. Red lines indicate the death +position of generators whose lifetime is smaller than 0.19. +The green line +indicates the death position of the generators with the largest birth+death, +birth = 0.0104 and death = 0.4590. +6 + +0.9 +0.8 +0.7 +× 0.6 +0.5 +0.4 +0.3 +800 +825 +850 +875 +900 +925 +950 +975 +1000 +timeM(0.40) +980 +985 +990 +995 +980 +985 +990 +995M(0.35) +980 +985 +990 +995 +980 +985 +990 +995M(0.20) +980 +985 +990 +995 +980 +985 +990 +995 +:M(0.05) +980 +985 +990 +995 +980 +985 +990 +995M(0.01) +980 +985 +990 +995 +980 +985 +990 +995!"# +!$# +Fig. 6. +(a) Persistence diagram, and (b) trajectory of R¨ossler system, with +a = 0.2, b = 1.6, c = 5.7. Red lines in (b) represent the death positions of +generators whose deaths are larger than 10.0. +3.2 Analysis of the R¨ossler system +Next, we investigated the dynamics of the R¨ossler system described by the equations +dx +dt = −y − z +(4) +dy +dt = x + ay +(5) +dz +dt = b + xz − cz. +(6) +In this study, we set a = 0.2, c = 5.7, and investigated the change in the PD by varying b between +0.1 and 1.7. +We calculated (x(t), y(t), z(t)) for 0 ≤ t ≤ 500, and used (x(t), y(t), z(t)) for t = +400, 400.1, . . . , 499.9 for the calculation of the distance matrix. +First, we began from the case with b = 1.6. In this case, the PD shown in Fig. 6(a) shows that +there are two classes of generators: generators in the first group have deaths smaller than 1.0 and +those in the second group have deaths larger than 10.0. To study the origin of the generators with +large death, we investigated the “death position” of these generators, which are indicated by the red +lines in Fig. 6(b). At this parameter, the attractor was the orbit with period 1, and the large death +value indicated the “diameter” of this orbit. +Next, we demonstrated the PD for b = 1.2 in Fig. 7(a). The PD was similar to the case of b = 1.6, +but new generators whose deaths are approximately 4.0 appeared. These generators represent the +period doubling of the attractor. To reveal the relation between period doubling and the generators, +we show the trajectories of the system with the birth and death positions of these generators in Fig. 7. +Evidently, the birth positions of these generators, which are represented by green lines, connect the +parallel part of the trajectory. These generators dies at the red lines, where these two lines diverge +along the z axis. This example suggests that the generators with large births and deaths imply the +existence of a parallel separated trajectory in the attractor. This claim holds true in the chaotic region. +For example, we show the PD at b = 0.7 in Fig. 8(a); herein, the system had a chaotic attractor. +The PD shows several clusters of generators with large lifetimes, and these generators represent the +parallel trajectories. For example, the birth of the generators surrounded by black and green ellipses +in Fig. 8(a) correspond with the black and green lines in Fig. 8(b), respectively. These birth points +are the pairs between parallel trajectories in the phase space. These examples suggest that we can +identify the nonlocal structure of the attractors using PH. +In the analysis above, we analyzed the dynamics using the snapshots of the system with an interval +∆t = 0.1. It is natural to ask whether our results are robust against the change of the interval. +We calculated the PDs for several ∆t, using the trajectory with b = 0.7, 400 ≤ t ≤ 500. Here, we +notice that the number of snapshots decreases as ∆t increases. The result is shown in Fig. 9. When +7 + +Rossler,b=1.60 +12 +103 +10 +8 +Death +6 +102 +4 +2 +0 +101 +0.0 +2.5 +5.0 +7.5 +10.0 +BirthRossler, b=1.60 +3 +2 +Z +1 +0 +5.0 +2.5 +-5.02.50.0 +0.0 +-2.5y +2.5 +5.0 +X +5.0!"# +!$# +Fig. 7. +(a): +Persistence diagram, and (b) trajectory of R¨ossler system, +a = 0.2, b = 1.2, c = 5.7. +Green and red lines in Fig. +(b) represent the +birth and death positions of generators whose deaths are between 3.0 and 4.0, +respectively. +!"# +!$# +Fig. 8. +(a) Persistence diagram, and (b) trajectory of R¨ossler system, a = +0.2, b = 0.7, c = 5.7. The birth positions of generators surrounded by black and +green ellipses in (a) correspond to the black and green lines in (b), respectively. +8 + +Rossler,b=0.70 +12 +10 +102 +8 +Death +6 +4 +101 +2 +0 +0.0 +2.5 +5.0 +7.5 +10.0 +BirthRosslerb=0.70 +10 +8 +6 +Z +4 +2 +0 +5 +5 +0 x +0 +-5 +y +-5Rossler, b=1.20 +12 +103 +10 +8 +102 +Death +6 +4 +101 +2 +0 +0.0 +2.5 +5.0 +7.5 +10.0 +BirthRossler, b=1.20 +6 +4 +Z +2 +5 +0 +0 x +5.0 +2.5 +0.0 +5.0 +-7.5!"# +!$# +!%# +Fig. 9. +Persistence diagrams for several interval of snapshots ∆t. (a) ∆t = +0.2, (b) ∆t = 0.5, and (c)∆t = 1.0, respectively. +The pararmeters of the +R¨ossler system are a = 0.2, b = 0.7, c = 5.7 +!"# +!$# +Fig. 10. +Persistence diagrams obtained using the trajectory (a) 300 ≤ t ≤ +500, and (b) 100 ≤ t ≤ 500. The parameters of the R¨ossler system are a = +0.2, = 0.7, c = 5.7, and the interval of snapshots ∆t = 0.5. +∆t = 0.2, we obtained the PD shown in Fig. 9(a), which is qualitatively consistent with the case of +∆t = 0.1 in Fig. 8(a). Further increase of ∆t produced the many “noisy” generators, which made +the structure of generators unclear. Fig. 9 (b) and (c) represent the PDs where ∆t = 0.5 and 1.0, +respectively. In the case of ∆t = 0.5, the PD has many noisy generators, but it seems similar to the +PD with ∆t = 0.1 and 0.2. In the case of ∆t = 1.0, we find no clear cluster of generators. These +“noisy” generators cannot be removed even if we use a longer time series. Fig. 10 shows the PDs +when we take longer sequences for ∆t = 0.5. Fig. 10(a) and (b) use two times and four times longer +sequences than the case of Fig. 9(b), but the “noisy” generators remain. These results suggests that +a small ∆t is required to apply our method. +3.3 Analysis of the ECG200 dataset +In this subsection, we present the analysis of the real time-series dataset ECG200, provided by Ol- +szewski et al[30]. This dataset includes the 200 ECGs of a heartbeat, which are classified into two +classes: healthy patients and patients with a myocardial infarction (MI). The dataset is divided into +100 training samples and 100 test samples. In this study, we only used the dataset for training. The +dataset was downloaded from the Time Series Classification Repository[31]. +First, we present the examples of ECG data of a healthy patient and a patient with an MI in +Figs. 11 (a) and (d). In this figure, the ECG signal of an MI patient appears flat, whereas that of +the healthy patient contains considerable noise. The corresponding PDs are shown in Figs. 11(b) +and (e). +In the case of an MI patient shown in Fig. 11 (e), the lifetime of most generators was +smaller than 0.5. In contrast, the lifetimes in Fig. 11(e) had a large variation, which suggested the +existence of noise in the data from a healthy patient. In this case, the PH gives more intuition to +us than the standard technique such as Fourier transformation. For example, we present the results +of Fourier transformation cn = � +m x(m) exp +� 2πimn +N +� +in Fig. 11(c) and (f), where N represents the +9 + +Rossler,b=0.70, 100 ≤ t ≤500 +12 +102 +10 +8 +Death +6 +101 +4 +2 +0 +100 +0.0 +2.5 +5.0 +7.5 +10.0 +BirthRossler,b=0.70,300 ≤ t ≤500 +12 +102 +10 +8 +Death +6 +101 +4 +2 +0 +100 +0.0 +2.5 +5.0 +7.5 +10.0 +BirthRossler,b=0.70,△t=1.0 +12 +102 +10 +8 +Death +6 +101 +4 +2 +0 +100 +0.0 +2.5 +5.0 +7.5 +10.0 +BirthRossler,b=0.70.△t=0.5 +12 +10 +101 +8 +Death +6 +2 +0 +100 +0.0 +2.5 +5.0 +7.5 +10.0 +BirthRossler,b=0.70,△t=0.2 +12 +10 +8 +Death +6 +101 +4 +2 +0 +100 +0.0 +2.5 +5.0 +7.5 +10.0 +Birth!"# +!$# +!%# +!&# +!'# +!(# +Fig. 11. +Examples of electrocardiogram data and corresponding persistence +diagrams and Fourier components. +Upper: (a) electrocardiogram data, (b) +persistence diagram, and (c) Fourier components obtained from a healthy pa- +tients. Lower: (d) electrocardiogram data, (e) persisntence diagram, and (f) +Fourier components obtained from a patient with myocardial infarction. +length of sequence. The difference of the coefficients between the healthy patient and the patient with +MI seems clear, but it is difficult to define a single variable that classify these two classes. Fourier +transformation is a powerful tool when the time series has some characteristic frequencies, but in this +case, there is no typical frequency that distinguishes patients with MI from healthy patients. To apply +Fourier transformation in this problem, we require more complicated methods such as the analysis +using Bag-of-SFA-Symbols[32]. +Based on this observation, we investigated whether we can use the variance in the lifetimes of the +generators as an indicator for an MI. First, we studied the distribution of the variance of lifetimes +for patients with MI and healthy patients. The result is shown in Fig. 12(a). In the case of an MI, +the distribution peaked around 0.01, whereas in the case of the healthy patients, it peaked around +0.04 for normal persons. This figure suggests that low variance in lifetimes is the signal of an MI. To +estimate the performance of the variance of lifetimes as the marker of an MI, we calculated the receiver +operating characteristic (ROC) curve and the area under the curve (AUC). The ROC curve represents +the relation between false positive rate (FPR) and true positive rate (TPR). Suppose that we judge +the ECG whose variance of lifetimes is smaller than p is positive. Then, number of true positive +(TP), false negative (FN), true negative (TN), and false positive (FP) are defined as the number of +correctly identified MI patients, misidentified MI patients, correctly identified healthy patients, and +misidentified healthy patients, respectively. TPR and FPR are defined by +TPR = +TP +TP + FN , +(7) +and +FPR = +FP +FP + TN . +(8) +The ROC curve is the plot of (FPR, TPR). AUC, defined as the area under the ROC curve, is +the standard characteristic of the performance of a quantitative diagnostic test. If AUC = 1.0, we +have no incorrect identification, whereas if AUC=0.5, the identification is equivalent to a random +identification. +In our case, the AUC was 0.811, which implies that the variance has a moderate +accuracy as an indicator of MI [33]. Compared with this result, Kirchenko et al. classified the same +dataset using the deep learning of RQA and the RP image, with AUC=0.76 and 0.92, respectively +[34]. Therefore, our method is better than analysis using RQA, but it does not improve upon the +analysis of the RP using deep learning. However, we note that the interpretation of our result is +10 + +#2: Healthy +30 +Re(Cn) +Im(Cn) +20 +10 +G +0 +-10 +-20 +-30 +0 +10 +20 +30 +40 +50 +n#1: MI +Re(Cn) +40 +Im(Cn) +20 +0 +-20 +0 +10 +20 +30 +40 +50 +n#1: MI +2.5 +2.0 +1.5 +101 +Death +Value +1.0 +0.5 +0.0 +100 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +Birth#2:healthy +1 +-2 +0 +25 +50 +75#1: MI +2 +1 +0 +-1 +-2 +0 +25 +50 +75#2: healthy +2.5 +101 +2.0 +1.5 +Death +Value +1.0 +0.5 +0.0 +100 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +Birth!"# +!$# +Fig. 12. +(a) Distribution of variance of lifetimes for MI patients and healthy +patients. (b) Receiver operating curve when we use variance of lifetime as the +indicator of MI. Area under curve = 0.811. +much easier than that obtained by deep learning. In summary, our example shows that the variance +of lifetimes is a promising signal to diagnose MI. +4. Discussion and Conclusion +In this study, we proposed a new method for time-series analysis, using PH analysis of the distance +matrix. We demonstrated the efficacy of our method to understand the structure of attractors in the +logistic map and the R¨ossler systems; additionally, we demonstrated that this method is applicable +to real-world datasets such as ECG200. Our method uses the distance matrix, which is represented +in two-dimensional space, thereby saving computational cost compared with other methods. Thus, +our method is useful for analyzing dynamics in high-dimensional systems. +However, PH is not a developed method for data analysis, and there remains room for improvement +in our method. To conclude this study, we discuss several directions for future studies. +First, combining this technique with machine learning is a promising approach. To analyze a large +real-world dataset, machine learning techniques, such as deep learning, must be used. In machine +learning, vectorizing the characteristic features is essential. However, the number of generators pro- +duced by PH is not constant, thereby rendering difficulty in the application of standard machine +learning techniques such as principal component analysis. Moreover, the generators with small life- +time are often disregarded as meaningless because they are produced by small noise, and the “sig- +nificance” of each generator must be estimated. To overcome these difficulties, numerous researchers +have proposed a variety of techniques such as persistence landscape [35], persistence images [36], and +persistence weighted Gaussian kernel [37]. These techniques combined with our proposed method will +enable the application of our method to numerous problems. +Second, investigating the information embedded in different PDs is interesting. In this study, we +did not use the PDs of degree 1, which provide characteristics of “holes” surrounded by M(θ). The +generators with degree 1 provide insights on the peaks of the distance matrix, which may give us +essential insights on the dynamics. Additionally, we can also calculate PDs using dilation-erosion +filtration [38]. In this approach, first, we make a recurrence plot, and subsequently calculate the +distance to the “boundary,” the places where black and white cells contact, for each cell. PH analysis +using this “distance to the boundary” as a filtration function provides a new quantification of the +recurrence plot. However, to generate a PD with dilation-erosion, we must determine the threshold +for the recurrence plot. The multi-parameter PH, which is currently studied intensively [39, 40], is +the PH method with several filtration functions. The application of this method will provide a way +to combine our method and dilation-erosion based PH analysis. +Finally, another future task is to mathematically investigate the relation between PD and dynamics. +In the case of RQA, it is known that the determinism has relation to the positive Lyapunov exponents. +It would be an interesting to seek the mathematical relation between the PDs and the characteristics +of the dynamical systems. +11 + +1.0 +0.8 +Rate +Positive +0.6 +0.4 +True +0.2 +0.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +FalsePositiveRate0.06 +lifetime +0.05 +0.04 +of +variance +0.03 +0.02 +0.01 +0.00 +MI +healthy +target5. Acknowledgemens +This work is financially supported by JSPS KAKENHI Grant Number JP22K19816. +References +[1] M. Budiˇsi´c, R. Mohr, and I. Mezi´c, “Applied Koopmanism,” Chaos: +An Interdisciplinary +Journal of Nonlinear Science, vol. 22, no. 4, p. 047510, 12 2012. [Online]. 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Vipond, “Multiparameter Persistence Landscapes.” Journal of Machine Learning Research, +vol. 21, no. 61, pp. 1–38, 2020. +14 + diff --git a/LNE1T4oBgHgl3EQfswVe/content/tmp_files/load_file.txt b/LNE1T4oBgHgl3EQfswVe/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..79fd95dc03f27b166d22fa2158030269e76e9cb3 --- /dev/null +++ b/LNE1T4oBgHgl3EQfswVe/content/tmp_files/load_file.txt @@ -0,0 +1,1075 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf,len=1074 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='03369v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='data-an] 6 Jan 2023 NOLTA, IEICE Paper Time series analysis using persistent homology of distance matrix Takashi Ichinomiya 1,2a) 1 Gifu University School of Medicine, Yanagido 1-1, Gifu, Gifu 501-1194, Japan 2 The United Graduate School of Drug Discovery and Medical Informatic Science, Gifu University, Yanagido 1-1, Gifu 501-1194, Japan a) tk1miya@gifu-u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='jp Received December 06, 2022 Abstract: The analysis of nonlinear dynamics is an important issue in numerous fields of science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' In this study, we propose a new method to analyze the time series data using persistent homology (PH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' The key idea is the application of PH to the distance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Using this method, we can obtain the topological features embedded in the trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' We apply this method to the logistic map, R¨ossler system, and electrocardiogram data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' The results reveal that our method can effectively identify nonlocal characteristics of the attractor and can classify data based on the amount of noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Key Words: time-series analysis, persistent homology, dynamical system 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Introduction The analysis of the nonlinear dynamics is a challenging problem in physics, engineering, and data science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Numerous methods for time series analysis, such as Fourier transformation or Kalman fil- tering, are based on the theory of linear dynamics, and the performance of these methods is limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' To overcome their limitations, several methods, such as Koopman’s mode decomposition[1], phase reduction[2], deep learning[3], and reservoir computing[4, 5], have been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' However, we often meet the situations where these methods are not available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' In Koopman’s mode decomposition we map the finite-dimensional nonlinear dynamical system into an infinite-dimensional linear dynamical system, and investigate the eigenvalues and eigenfunctions in this space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' However, the determination of these eigenfunctions is often difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Phase reduction is a powerful method to investigate the oscillatory dynamics, but the application to non-oscillatory systems is limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Deep learning and reservoir computing help us to predict the state in the future, but they do not provide a rationale for their prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' In this study, we propose a new method to analyze the dynamics using distance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' The distance matrix D(t, s), defined as the distance between states at time t and s, reveals considerable information regarding the dynamics of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' For example, if D(t, t+T) = 0 for all t, the system exhibits a periodic motion with period T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Recurrence plot (RP) is the most useful method for the 1 Nonlinear Theory and Its Applications, IEICE, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' X, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' 0, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' 1–14 ©IEICE 2023 DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='1587/nolta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='1 time-series analysis using a distance matrix [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' In this method, the distance matrix is visualized using R(t, s) = Θ(ǫ − D(t, s)), where Θ(x) is the Heaviside step function and ǫ is a parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Using this method, periodic motion can be distinguished from chaotic trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Based on an RP, recurrence quantification analysis (RQA), wherein the dynamics are characterized by several quantities, such as recurrence rate and determinism, was proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' However, an RP uses only the limited information embedded in the distance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' First, an RP does not provide information on the nonlocal properties of the trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' An RP lists the points that are close to each other in phase space and is useful to investigate local properties such as Lyapunov exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' In contrast, it is unsuitable for investigating the global structure of the trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' For example, determining whether the attractor has a double scroll structure like the Lorenz system or an oscillation-like structure similar to the R¨ossler system by RP is difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Another problem with the use of an RP is that there is no clear rule to select the value of ǫ, and studies have reported that the result of the RP is often sensitive to this choice [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' In this study, we propose the application of persistent homology (PH)[8], an emerging technique of data analysis, to the analysis of the distance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' PH is the one of the most popular techniques in topological data analysis (TDA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' In TDA, we investigate the topological characteristics such as number of connected components or holes embedded in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' The theory of PH is still being developed and has been successfully applied in various fields, including biophysics[9–11], material science[12–14], and image processing[15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' There have been several proposals to apply PH to time series datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' The favored approach for the time-series analysis using PH is based on delay embedding [17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' In this approach, we first create the point clouds in n-dimensional space using Takens’ delay embedding [19] and subsequently characterize the state using PH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' However, this approach has several difficulties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' First, we must determine how to embed the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' There is no general rule to determine the way of embedding, though several ideas have been proposed [20–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Second, the computational cost increases rapidly as the embedding dimension and the size of point cloud increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' It is known that the computation time for PH is O(N⌈D/2⌉), where N and D represent the size of the point cloud and dimension of the space, respectively [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Therefore, the computational cost increases exponentially as D increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' In our approach, we can avoid the latter difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Because the distance matrix is represented in two-dimensional space, the computation cost of PH is considerably reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' The results of this study reveal that using PH, the essential information of a dataset, such as the non-local structure of attractors and the amount of noise, can be easily determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' The remainder of this study is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' In Section 2, we explain our method to investigate the distance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' In Section 3, we present the results of application of our method to three different datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' First, we present the results of the analysis of the logistic map as a typical example of discrete time dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Second, we present the results on the R¨ossler system as an example of continuous time dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Third, we discuss the results on the analysis of electrocardiogram (ECG) data, as an example of real-world time series data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Finally, we discuss on the possible improvements to our method in future and conclude this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Method The general definition of PH requires further background knowledge of algebraic topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' In this section, we explain the PH of filtered cubical complexes of degree 0, which is used in the latter part of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' The readers who want to know the general definition of PH can consult textbooks on PH and TDA[24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' We consider a real-valued filtration function f : Z2 → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' The sublevel set M(θ) is defined by M(θ) = {(x, y) ∈ Z2|f(x, y) ≤ θ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' (1) For example, assume that f(x, y) is given by the table shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' In this case, M(0) is given by the gray blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' When we increase θ, M(θ) also grows, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' 1(b) and (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' PH with degree 0 using a sublevel set of f represents the change in connected components in M(θ) when θ is varied from −∞ to ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Here we say two blocks are “connected” if they share an edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' For 2 !' metadata={'source': 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" # !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' " $ %&\'()!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' "#$ %*\'()!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' "%$ %+\'()!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' "&$ \' # % & ( ) # % & ( ) !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' " # $ + , !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' " # $ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' " # $ %,\' %-\' # % & ( ) # % & ( ) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Example of persistent homology using sublevel sets, (a)–(c) represent the M(θ) for θ = 0, 1, and 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' The corresponding persistence barcode and persistence diagram are shown in (d) and (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' example, we have two components in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' 1(a): there is one isolated component at (x, y) = (2, 0) and one connected component at y = 4, 0 ≤ x ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' By increasing θ to 1, these two components are merged into one large component, and another component appears at (x, y) = (3, 2), as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Here, we do not say these two components are connected, because although they share two corners, they do not share an edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' In this case, we say these two components are “disconnected.” When we increase θ to 2, all three components are merged into one large component, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' 1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' The theory of PH guarantees that we can define the “birth” and “death” of each component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' In this example, at θ = 0, two disconnected components are “born.” At θ = 1, these components merged into one large component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Hence, we say that one of the components “dies” and the other disconnected component at (x, y) = (3, 2) is born.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Finally, at θ = 2, these components are connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Further increase in θ does not change the number of disconnected components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Therefore, we have three “connected components,” often called “generators,” whose births b and deaths d are (b, d) = (0, ∞), (0, 1), (1, 2), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' There are two major visualization techniques to represent the distribution of generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' One is “persistence barcode,” wherein we represent each generator as a “bar” from birth to death, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' 1(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' This representation is intuitive when the number of generators is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' For example, when the dataset has a periodic structure, we will obtain numerous generators that have the same birth and death, and we can easily identify them by persistence barcode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' However, when we have more than a hundred of generators, the number of bars becomes too large, and gaining any insight from the barcode becomes difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' In this case, a persistence diagram (PD) is better visualization method;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' herein, we make a scatter plot of birth and death, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' 1(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' In a PD, generators with infinite death are generally omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' When the number of generators becomes large, we also use a density heatmap of generators, which is also called a PD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' In the rest of this study, we use PDs to represent the results of our PH analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' The PH allows us to study the local minimum and saddle points of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' For example, we consider the case of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' In the case of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' 2(a), f is a smooth function with two minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' In this case, we can define the sublevel set M(θ) as M(θ) = {x ∈ R|f(x) ≤ θ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' If θ is smaller than the saddle value of 3 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' "#$% &\'($% )(* )!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' * )+* !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Relation between the form of the filtering function f (upper) and corresponding persistence diagram(lower).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' (a) When f has two local minima, we obtain two generators, and only one has finite death.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' (b) When f is more complex and has more local minima, the number of generators increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' (c) If the “saddles” between local minima are low, the lifetime of generators de- creases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' The dashed line in the persistence diagrams indicates the line birth = death.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' f, M(θ) has two connected components, and for larger θ, M(θ) has only one connected component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Therefore, in this case, there are two generators, and one of these generators has finite death.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' In contrast, if D has numerous local minima as described in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' 2(b), we obtain numerous generators with finite deaths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Therefore, the number of generators indicates the number of local minima of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Moreover, PH provides information about the height of the saddles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' For example, we consider the case in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' 2(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Here, f has numerous local minima, but the height of the saddle is lower than in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' In this case, the connected components of M(θ) merge after only a slight increase of θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Therefore, the lifetime, defined as the difference between death and birth, decreases as the height of the corresponding saddle decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' In this study, we used a distance matrix D for filtration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' We suppose that we have time series data xi, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' , K, where i represents the discretized time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' From this dataset, we define the distance matrix D(i, j) as D(i, j) = ||xi − xj||, (2) where ||· · · || represents L2-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' D(i, j) is a real-valued function from (i, j) ∈ Z2, and we can apply PH using the distance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' There are several software for PH analysis, which include Gudhi [26], Phat [27], and Javaplex [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' In this study, we used Homcloud developed by Obayashi et al [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' One of the advantages of Homcloud compared with other software is that it can provide the “birth position” and “death position” of each generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Using this function, we can obtain the positions of the local minima and the saddles of D(i, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' These values are useful for interpreting the result obtained by PH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Results We applied our method to three different datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' The first example is time-series data obtained from the logistic map, and the second one is that obtained from the R¨ossler equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Finally, we analyzed the dataset ECG200, as an example of a real-world dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='1 Analysis of the logistic map In this subsection, we investigated the distance matrix of the logistic map defined by xi+1 = axi(1 − xi), (3) where 0 < a < 4 is a parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' We calculated xi for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' , 1000 with initial condition x0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='832 and performed a PH analysis using data at i = 801, 802, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' , 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' First, we investigated the case a = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='4, wherein xi is periodic, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' In this case, all generators had birth 0 and death 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='3901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' This result indicated that xi takes only two values, and 4 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' "# !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='$# !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='%# !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='&# Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Persistence diagram for logistic maps: (a) a = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='4, (b) a = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='5, (c) a = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='6, and (d) a = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' the difference between these two is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='3901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' This is consistent with the fact that the attractor of this system is a cycle with period 2: x2i+1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='4520 and x2i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='8421.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' The death time is given by the difference between these two values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' As the period increases, the number of values that generators can take also increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' For example, at a = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='5, the distribution had 6 peaks, corresponding to the fact that the period of the logistic map was 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' We show M(θ) for several θ in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='4 in the case of a = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' The number of peaks represents the topological information of the attractors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Furthermore, the PD in a chaotic region provides topological information of the attractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' In the case of a = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='6, where the logistic map becomes chaotic, the births and deaths of generators are widely distributed, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' 3(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Herein, we found that the distribution had several characteristic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' First, all deaths were larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='188.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' This property could be explained by the existence of a gap in the attractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' At a = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='6, x takes values from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='3 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='6 and from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='788 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='900, but does not take values between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='601 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='787.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' This property generates a the gap in the distribution of deaths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' To confirm this suggestion, we show the “death point” of generators whose death is smaller than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='19 as a red line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' In this figure, the points connected by lines give the saddles of D(i, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Evidently, the red lines connect the top point of the lower band and the bottom points of the upper band, which supports our claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Additionally, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' 3(c) reveals that birth + death < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='4695.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Unfortunately, we have no theory to explain this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Notably, we have an upper limit on birth + death when xmin ≤ xi ≤ xmax for all i, because birth ≥ 0 and death ≤ xmax − xmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' However, the fact that the death corresponds to the saddle makes the problem difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' For example, the green line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' 3(e) shows the death point pairs with the largest birth + death, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='4694.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' This figure shows that one terminal of this line is at the upper limit of the upper band, whereas the other terminal stays in the middle of the lower band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' We have no theoretical explanation why there is no saddle pair of states that provides larger birth + death.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' This is a problem that should be solved in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' In the case of a = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='7 shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' 3(d), the upper limit of death+birth appears to remain with several exceptional generators, whereas the lower limit disappears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' The absence of a lower limit indicates the elimination of the gap in x found in the case a = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Summarizing these results, the PD gives the topological structure of the attractor, in the cases of both the periodic trajectory and chaotic attractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' 5 Logistic, a=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='6 103 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='5 Death 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='4 Value 102 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='1 101 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='8 BirthLogistic, a=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='8 105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='5 104 Death 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='4 Value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='2 103 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='1 102 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='8 BirthLogistic, a=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='5 102 Death 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='4 Value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='2 101 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='8 BirthLogistic, a=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='7 102 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='5 Death 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='1 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='8 Birth(a) (b) (c) (d) (e) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Sublevel set M(θ) represented as black areas in the case of a = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' (a) θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='01, (b) θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='05, (c) θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='20, (d) θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='35, and (e) θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='40, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Time series xi for the case of a = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Red lines indicate the death position of generators whose lifetime is smaller than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' The green line indicates the death position of the generators with the largest birth+death, birth = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='0104 and death = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='4590.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='7 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='3 800 825 850 875 900 925 950 975 1000 timeM(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='40) 980 985 990 995 980 985 990 995M(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='35) 980 985 990 995 980 985 990 995M(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='20) 980 985 990 995 980 985 990 995 :M(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='05) 980 985 990 995 980 985 990 995M(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='01) 980 985 990 995 980 985 990 995!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' "# !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='$# Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' (a) Persistence diagram, and (b) trajectory of R¨ossler system, with a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='2, b = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='6, c = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Red lines in (b) represent the death positions of generators whose deaths are larger than 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='2 Analysis of the R¨ossler system Next, we investigated the dynamics of the R¨ossler system described by the equations dx dt = −y − z (4) dy dt = x + ay (5) dz dt = b + xz − cz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' (6) In this study, we set a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='2, c = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='7, and investigated the change in the PD by varying b between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' We calculated (x(t), y(t), z(t)) for 0 ≤ t ≤ 500, and used (x(t), y(t), z(t)) for t = 400, 400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' , 499.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='9 for the calculation of the distance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' First, we began from the case with b = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' In this case, the PD shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' 6(a) shows that there are two classes of generators: generators in the first group have deaths smaller than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='0 and those in the second group have deaths larger than 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' To study the origin of the generators with large death, we investigated the “death position” of these generators, which are indicated by the red lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' 6(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' At this parameter, the attractor was the orbit with period 1, and the large death value indicated the “diameter” of this orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Next, we demonstrated the PD for b = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='2 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' 7(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' The PD was similar to the case of b = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='6, but new generators whose deaths are approximately 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='0 appeared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' These generators represent the period doubling of the attractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' To reveal the relation between period doubling and the generators, we show the trajectories of the system with the birth and death positions of these generators in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Evidently, the birth positions of these generators, which are represented by green lines, connect the parallel part of the trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' These generators dies at the red lines, where these two lines diverge along the z axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' This example suggests that the generators with large births and deaths imply the existence of a parallel separated trajectory in the attractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' This claim holds true in the chaotic region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' For example, we show the PD at b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='7 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' 8(a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' herein, the system had a chaotic attractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' The PD shows several clusters of generators with large lifetimes, and these generators represent the parallel trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' For example, the birth of the generators surrounded by black and green ellipses in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' 8(a) correspond with the black and green lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' 8(b), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' These birth points are the pairs between parallel trajectories in the phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' These examples suggest that we can identify the nonlocal structure of the attractors using PH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' In the analysis above, we analyzed the dynamics using the snapshots of the system with an interval ∆t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' It is natural to ask whether our results are robust against the change of the interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' We calculated the PDs for several ∆t, using the trajectory with b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='7, 400 ≤ t ≤ 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Here, we notice that the number of snapshots decreases as ∆t increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' The result is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' When 7 Rossler,b=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='60 12 103 10 8 Death 6 102 4 2 0 101 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='0 BirthRossler, b=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='60 3 2 Z 1 0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='5y 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='0 X 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='0!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' "# !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='$# Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' (a): Persistence diagram, and (b) trajectory of R¨ossler system, a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='2, b = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='2, c = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Green and red lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' (b) represent the birth and death positions of generators whose deaths are between 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='0 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' "# !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='$# Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' (a) Persistence diagram, and (b) trajectory of R¨ossler system, a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='2, b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='7, c = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' The birth positions of generators surrounded by black and green ellipses in (a) correspond to the black and green lines in (b), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' 8 Rossler,b=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='70 12 10 102 8 Death 6 4 101 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='0 BirthRosslerb=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='70 10 8 6 Z 4 2 0 5 5 0 x 0 5 y 5Rossler, b=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='20 12 103 10 8 102 Death 6 4 101 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='0 BirthRossler, b=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='20 6 4 Z 2 5 0 0 x 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='5!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' "# !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='$# !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='%# Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Persistence diagrams for several interval of snapshots ∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' (a) ∆t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='2, (b) ∆t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='5, and (c)∆t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' The pararmeters of the R¨ossler system are a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='2, b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='7, c = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='7 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' "# !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='$# Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Persistence diagrams obtained using the trajectory (a) 300 ≤ t ≤ 500, and (b) 100 ≤ t ≤ 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' The parameters of the R¨ossler system are a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='2, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='7, c = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='7, and the interval of snapshots ∆t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' ∆t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='2, we obtained the PD shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' 9(a), which is qualitatively consistent with the case of ∆t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='1 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' 8(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Further increase of ∆t produced the many “noisy” generators, which made the structure of generators unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' 9 (b) and (c) represent the PDs where ∆t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='5 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' In the case of ∆t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='5, the PD has many noisy generators, but it seems similar to the PD with ∆t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' In the case of ∆t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='0, we find no clear cluster of generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' These “noisy” generators cannot be removed even if we use a longer time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' 10 shows the PDs when we take longer sequences for ∆t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' 10(a) and (b) use two times and four times longer sequences than the case of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' 9(b), but the “noisy” generators remain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' These results suggests that a small ∆t is required to apply our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='3 Analysis of the ECG200 dataset In this subsection, we present the analysis of the real time-series dataset ECG200, provided by Ol- szewski et al[30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' This dataset includes the 200 ECGs of a heartbeat, which are classified into two classes: healthy patients and patients with a myocardial infarction (MI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' The dataset is divided into 100 training samples and 100 test samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' In this study, we only used the dataset for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' The dataset was downloaded from the Time Series Classification Repository[31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' First, we present the examples of ECG data of a healthy patient and a patient with an MI in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' 11 (a) and (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' In this figure, the ECG signal of an MI patient appears flat, whereas that of the healthy patient contains considerable noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' The corresponding PDs are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' 11(b) and (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' In the case of an MI patient shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' 11 (e), the lifetime of most generators was smaller than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' In contrast, the lifetimes in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' 11(e) had a large variation, which suggested the existence of noise in the data from a healthy patient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' In this case, the PH gives more intuition to us than the standard technique such as Fourier transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' For example, we present the results of Fourier transformation cn = � m x(m) exp � 2πimn N � in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' 11(c) and (f), where N represents the 9 Rossler,b=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='70, 100 ≤ t ≤500 12 102 10 8 Death 6 101 4 2 0 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='0 BirthRossler,b=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='70,300 ≤ t ≤500 12 102 10 8 Death 6 101 4 2 0 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='0 BirthRossler,b=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='70,△t=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='0 12 102 10 8 Death 6 101 4 2 0 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='0 BirthRossler,b=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='△t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='5 12 10 101 8 Death 6 2 0 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='0 BirthRossler,b=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='70,△t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='2 12 10 8 Death 6 101 4 2 0 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='0 Birth!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' "# !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='$# !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='%# !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='&# !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=" '# !" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' (# Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Examples of electrocardiogram data and corresponding persistence diagrams and Fourier components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Upper: (a) electrocardiogram data, (b) persistence diagram, and (c) Fourier components obtained from a healthy pa- tients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Lower: (d) electrocardiogram data, (e) persisntence diagram, and (f) Fourier components obtained from a patient with myocardial infarction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' length of sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' The difference of the coefficients between the healthy patient and the patient with MI seems clear, but it is difficult to define a single variable that classify these two classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Fourier transformation is a powerful tool when the time series has some characteristic frequencies, but in this case, there is no typical frequency that distinguishes patients with MI from healthy patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' To apply Fourier transformation in this problem, we require more complicated methods such as the analysis using Bag-of-SFA-Symbols[32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Based on this observation, we investigated whether we can use the variance in the lifetimes of the generators as an indicator for an MI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' First, we studied the distribution of the variance of lifetimes for patients with MI and healthy patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' The result is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' 12(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' In the case of an MI, the distribution peaked around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='01, whereas in the case of the healthy patients, it peaked around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='04 for normal persons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' This figure suggests that low variance in lifetimes is the signal of an MI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' To estimate the performance of the variance of lifetimes as the marker of an MI, we calculated the receiver operating characteristic (ROC) curve and the area under the curve (AUC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' The ROC curve represents the relation between false positive rate (FPR) and true positive rate (TPR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Suppose that we judge the ECG whose variance of lifetimes is smaller than p is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Then, number of true positive (TP), false negative (FN), true negative (TN), and false positive (FP) are defined as the number of correctly identified MI patients, misidentified MI patients, correctly identified healthy patients, and misidentified healthy patients, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' TPR and FPR are defined by TPR = TP TP + FN , (7) and FPR = FP FP + TN .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' (8) The ROC curve is the plot of (FPR, TPR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' AUC, defined as the area under the ROC curve, is the standard characteristic of the performance of a quantitative diagnostic test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' If AUC = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='0, we have no incorrect identification, whereas if AUC=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='5, the identification is equivalent to a random identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' In our case, the AUC was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='811, which implies that the variance has a moderate accuracy as an indicator of MI [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Compared with this result, Kirchenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' classified the same dataset using the deep learning of RQA and the RP image, with AUC=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='76 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='92, respectively [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Therefore, our method is better than analysis using RQA, but it does not improve upon the analysis of the RP using deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' However, we note that the interpretation of our result is 10 #2: Healthy 30 Re(Cn) Im(Cn) 20 10 G 0 10 20 30 0 10 20 30 40 50 n#1: MI Re(Cn) 40 Im(Cn) 20 0 20 0 10 20 30 40 50 n#1: MI 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='5 101 Death Value 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='0 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='5 Birth#2:healthy 1 2 0 25 50 75#1: MI 2 1 0 1 2 0 25 50 75#2: healthy 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='5 101 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='5 Death Value 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='0 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='5 Birth!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' "# !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='$# Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' (a) Distribution of variance of lifetimes for MI patients and healthy patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' (b) Receiver operating curve when we use variance of lifetime as the indicator of MI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Area under curve = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' much easier than that obtained by deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' In summary, our example shows that the variance of lifetimes is a promising signal to diagnose MI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Discussion and Conclusion In this study, we proposed a new method for time-series analysis, using PH analysis of the distance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' We demonstrated the efficacy of our method to understand the structure of attractors in the logistic map and the R¨ossler systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' additionally, we demonstrated that this method is applicable to real-world datasets such as ECG200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Our method uses the distance matrix, which is represented in two-dimensional space, thereby saving computational cost compared with other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Thus, our method is useful for analyzing dynamics in high-dimensional systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' However, PH is not a developed method for data analysis, and there remains room for improvement in our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' To conclude this study, we discuss several directions for future studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' First, combining this technique with machine learning is a promising approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' To analyze a large real-world dataset, machine learning techniques, such as deep learning, must be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' In machine learning, vectorizing the characteristic features is essential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' However, the number of generators pro- duced by PH is not constant, thereby rendering difficulty in the application of standard machine learning techniques such as principal component analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Moreover, the generators with small life- time are often disregarded as meaningless because they are produced by small noise, and the “sig- nificance” of each generator must be estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' To overcome these difficulties, numerous researchers have proposed a variety of techniques such as persistence landscape [35], persistence images [36], and persistence weighted Gaussian kernel [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' These techniques combined with our proposed method will enable the application of our method to numerous problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Second, investigating the information embedded in different PDs is interesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' In this study, we did not use the PDs of degree 1, which provide characteristics of “holes” surrounded by M(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' The generators with degree 1 provide insights on the peaks of the distance matrix, which may give us essential insights on the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Additionally, we can also calculate PDs using dilation-erosion filtration [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' In this approach, first, we make a recurrence plot, and subsequently calculate the distance to the “boundary,” the places where black and white cells contact, for each cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' PH analysis using this “distance to the boundary” as a filtration function provides a new quantification of the recurrence plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' However, to generate a PD with dilation-erosion, we must determine the threshold for the recurrence plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' The multi-parameter PH, which is currently studied intensively [39, 40], is the PH method with several filtration functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' The application of this method will provide a way to combine our method and dilation-erosion based PH analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Finally, another future task is to mathematically investigate the relation between PD and dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' In the case of RQA, it is known that the determinism has relation to the positive Lyapunov exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' It would be an interesting to seek the mathematical relation between the PDs and the characteristics of the dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' 11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='8 Rate Positive 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='4 True 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='0 FalsePositiveRate0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='06 lifetime 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='04 of variance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='00 MI healthy target5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Acknowledgemens This work is financially supported by JSPS KAKENHI Grant Number JP22K19816.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content='1007/s00454-009-9176-0 [40] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' Vipond, “Multiparameter Persistence Landscapes.” Journal of Machine Learning Research, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' 21, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' 61, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' 1–38, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} +page_content=' 14' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfswVe/content/2301.03369v1.pdf'} diff --git a/MtE2T4oBgHgl3EQfBQa7/content/tmp_files/2301.03601v1.pdf.txt b/MtE2T4oBgHgl3EQfBQa7/content/tmp_files/2301.03601v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..2c6153c131c6f7b61e5506f3aa3f08d82d86707c --- /dev/null +++ b/MtE2T4oBgHgl3EQfBQa7/content/tmp_files/2301.03601v1.pdf.txt @@ -0,0 +1,2972 @@ +Quark-lepton Yukawa ratios and nucleon decay in +SU(5) GUTs with type-III seesaw +Stefan Antusch, Kevin Hinze, and Shaikh Saad +Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland +E-mail: stefan.antusch@unibas.ch, kevin.hinze@unibas.ch, +shaikh.saad@unibas.ch +Abstract: +We consider an extension of the Georgi-Glashow SU(5) GUT model by a +45-dimensional scalar and a 24-dimensional fermionic representation, where the latter leads +to the generation of the observed light neutrino masses via a combination of a type I and a +type III seesaw mechanism. Within this scenario, we investigate the viability of predictions +for the ratios between the charged lepton and down-type quark Yukawa couplings, focusing +on the second and third family. Such predictions can emerge when the relevant entries of the +Yukawa matrices are generated from single joint GUT operators (i.e. under the condition of +single operator dominance). We show that three combinations are viable, (i) yτ/yb = 3/2, +yµ/ys = 9/2, (ii) yτ/yb = 2, yµ/ys = 9/2, and (iii) yτ/yb = 2, yµ/ys = 6. We extend these +possibilities to three toy models, accounting also for the first family masses, and calculate +their predictions for various nucleon decay rates. We also analyse how the requirement of +gauge coupling unification constrains the masses of potentially light relic states testable at +colliders. +arXiv:2301.03601v1 [hep-ph] 9 Jan 2023 + +Contents +1 +Introduction +1 +2 +GUT scenario +3 +2.1 +Particle content +3 +2.2 +Neutrino masses +4 +2.3 +Quark-lepton Yukawa ratios +5 +2.4 +Toy models +5 +3 +Numerical procedure +6 +3.1 +Implementation of the charged fermion Yukawa sector +6 +3.2 +Implementation of the neutrino sector +6 +3.3 +GUT scale parameters and low energy observables +6 +3.4 +Fitting procedure +7 +4 +Results +7 +4.1 +Benchmark points +8 +4.2 +Highest posterior densities +9 +4.2.1 +Quark-lepton mass ratios +10 +4.2.2 +Intermediate-scale particle masses +10 +4.2.3 +Nucleon decay width and GUT scale +12 +5 +Conclusion +12 +Appendices +12 +A Definition of new Yukawa couplings +12 +B Renormalization group equations +14 +B.1 +Gauge couplings +14 +B.2 +Yukawa matrices +16 +B.3 +Effective neutrino mass operator +20 +1 +Introduction +Grand Unified Theories (GUTs) [1–6] are arguably one of the most appealing extensions of +the Standard Model (SM) of particle physics. In 1974, a simple and elegant GUT based on +the unifying gauge group SU(5) was proposed by H. Georgi and S. Glashow (GG model) +[3]. +However, this model is incompatible with the current experimental data for three +main reasons. Firstly, the GG model does not allow for gauge coupling unification, which +– 1 – + +is a necessary condition for a GUT. Secondly, it predicts massless neutrinos, which is in +conflict with neutrino oscillation experiments requiring that at least two neutrino should +be massive [7]. +Thirdly, since the SM Higgs doublet is embedded into a 5-dimensional +Higgs representation of SU(5), the GG model predicts the GUT scale relation between the +charged lepton and down-type quark Yukawa matrices +Ye = Y T +d . +(1.1) +This relation in particular implies a GUT scale unification of the tau and bottom Yukawa +couplings yτ = yb, as well as a unification of the muon and strange Yukawa couplings +yµ = ys, which disagrees with the low energy data. +The first shortcoming requires extending the particle content of the minimal model by +additional GUT representations and suitably splitting the masses of their component fields +such that the running gauge couplings meet. The second shortcoming can be addressed +by introducing SU(5) representations that allow neutrino mass generation at the tree level +[8–14] or at the loop level [15–20]. +Finally, the third shortcoming can for instance be +resolved by generating the Yukawa couplings from linear combinations of the renormalisable +and higher dimensional non-renormalisable operators [21], or at the renormalizable level +by either introducing a 45-dimensional Higgs field and considering linear combinations of +couplings between the SM fermions and both the 5- as well as the 45-dimensional Higgs +field [22], or by introducing vector-like fermions which mix with the SM fermions [23–25]. +However, historically a first and very aesthetic solution for the third problem was +proposed by H. Georgi and C. Jarlskog (GJ model) in 1979 [26]. +In their model, the +particle content of the GG model is extended by a 45-dimensional Higgs field (as well as +by two 5-dimensional Higgs fields). If the 45-dimensional Higgs field couples to the SM +fermions this gives rise to the GUT scale relation +Ye = −3Y T +d . +(1.2) +Considering a linear combination of the operators giving the relations (1.1) and (1.2) would, +on the one hand, solve the shortcoming (as already mentioned above), but, on the other +hand, predictivity in the Yukawa sector would be lost. Predictivity is however maintained +if it is ensured that different generations of charged leptons and down-type quarks couple +to different Higgs fields (which can, for example, be achieved when a family symmetry is +introduced on top of the gauge symmetry). To achieve predictivity, without referring to +any particular family symmetry, the GJ model hypothesizes the following textures of the +Yukawa coupling matrices, +Yd = +� +� +� +0 +B +0 +A C +0 +0 +0 +D +� +� +� , +Y T +e = +� +� +� +0 +B +0 +A −3C +0 +0 +0 +D +� +� +� , +(1.3) +implying the GUT scale relations yτ/yb = 1, yµ/ys = −3, ye/yd = −1/3 which were at that +time compatible with the experimental data. However, the current data suggests (taking +– 2 – + +only the known SM particles into account in the renormalization group (RG) evolution) +that other ratios such as yτ/yb = 3/2, yµ/ys = 9/2 are better suited (see e.g. [27]). +Interestingly, these latter ratios can be obtained from higher dimensional operators +[28, 29]. With these higher dimensional operators at hand, models similar to the GJ model +can be build if the following two conditions are satisfied: (i) the Yukawa matrices should +be hierarchical, (ii) the 22- and 33- entry should be dominated by a single GUT operator, +a concept which is referred to as single operator dominance [28–30].1 +Following this approach, non-SUSY GUT scenarios in which neutrino masses are gen- +erated by a type I or a type II seesaw have been investigated in [27], respectively [45]. For +GUT scenarios with a type I seesaw it was shown that the GUT scale ratios yτ/yb = 3/2 +and yµ/ys = 9/2 are compatible with the experimental data. Moreover, for GUT scenarios +in which neutrino masses are generated by a type II seesaw it was found, that two combi- +nations of GUT scale relations are viable, namely (i) yτ/yb = 3/2 and yµ/ys = 9/2 and (ii) +yτ/yb = 2 and yµ/ys = 6. +In this paper we will investigate the viability of such GUT scale ratios for the case +that neutrino masses stem from a combination of a type I [46–50] and a type III [51] +seesaw mechanism. In this regard, we will consider a GUT scenario in which the particle +content of the GG model is extended by a fermionic adjoint representation as well as by +a 45-dimensional Higgs field.2 The former representation is needed to generate neutrino +masses, while the latter gives rise to operators yielding potentially viable GUT scale Yukawa +ratios. Moreover, both of these representations help to allow for gauge coupling unification. +Using the Mathematica package ProtonDecay [52] and extending the above scenario to “toy +models” we also compute the nucleon decay widths for various decay channels. Finally, we +compute the masses of the added fermion and scalar fields. +The paper is organized as follows: While the GUT scenario as well as the toy models +are introduced in Section 2, the procedure for the numerical analysis is explained in Section +3. In Section 4 the results are presented and discussed, before concluding in Section 5. +In Appendix A, definitions of the newly introduced Yukawa couplings are given, while all +relevant RGEs that we have derived are listed in Appendix B. +2 +GUT scenario +2.1 +Particle content +The SM fermions are embedded as usual into three generations of 5F i and 10F i +5F i = dc +i(3, 1, 1 +3) ⊕ ℓi(1, 2, −1 +2), +(2.1) +10F i = qi(3, 2, 1 +6) ⊕ uc +i(3, 1, −2 +3) ⊕ ec +i(1, 1, 1). +(2.2) +In the considered scenario, neutrino masses are generated via a combination of a type I and +a type III seesaw mechanism. The corresponding fermionic singlet Σc and triplet Σb (under +1For models in which the concept of single operator dominance has been applied, see e.g. [31–44]. +2A non-supersymmetric SU(5) GUT with this particle content was first considered in [13]. However, so +far it has not been studied under the assumption of single operator dominance. +– 3 – + +SU(2)L) are contained in an adjoint fermionic representation +24F = Σa(8, 1, 0) ⊕ Σb(1, 3, 0) ⊕ Σc(1, 1, 0) ⊕ Σd(3, 2, −5 +6) ⊕ Σe(3, 2, 5 +6). +(2.3) +Moreover, the GUT Higgs fields decompose under the SM gauge group as +24H = Φa(8, 1, 0) ⊕ Φb(1, 3, 0) ⊕ Φc(1, 1, 0) ⊕ Φd(3, 2, −5 +6) ⊕ Φe(3, 2, 5 +6), +(2.4) +5H = Ta(3, 1, −1 +3) ⊕ Ha(1, 2, 1 +2), +(2.5) +45H = φa(8, 2, 1 +2) ⊕ φb(6, 1, −1 +3) ⊕ φc(3, 3, −1 +3) ⊕ φd(3, 2, −7 +6) ⊕ φe(3, 1, −4 +3) +⊕ Tb(3, 1, −1 +3) ⊕ Hb(1, 2, 1 +2). +(2.6) +After the SU(5) breaking, the color triplets Ta and Tb mix to yield the mass eigenstates +t1 = cos(α)Ta +sin(α)Tb and t2 = − sin(α)Ta +cos(α)Tb. Similarly, Ha and Hb mix to form +the mass eigenstates h1 = cos(β)Ha + sin(β)Hb and h⊥ +2 = − sin(β)Ha + cos(β)Hb, where +h1 is the SM Higgs doublet. +2.2 +Neutrino masses +At tree-level the relevant GUT operators for neutrino mass generation read3 +L ⊃ YA 5F 24F 5H + YB 5F 24F 45H. +(2.7) +After the GUT symmetry breaking the following relevant terms emerge +L ⊃ −Y2ℓΣbHa − Y8ℓΣbHb − Y4ℓΣcHa − Y13ℓΣcHb − mΣbΣbΣb − mΣcΣcΣc, +(2.8) +where mΣb and mΣb are the respective masses of Σb and Σc, and where the GUT scale +relations +Y2 = − +� +3 +10 YA, +Y4 = YA, +Y8 = +√ +5 +4 YB, +and +Y13 = +√ +3 +4 YB +(2.9) +hold. After the SU(2) triplet Σb and SU(2) singlet Σc have been integrated out and the +two Higgs fields Ha and Hb have taken their vacuum expectation values (vevs) va and vb, +where v2 +a + v2 +b = v2 = (246 GeV)2, and where va = v cos(β) and vb = v sin(β), the neutrino +mass matrix mν reads +mij +ν = −(Y i +2 va + Y i +8 vb)(Y j +2 va + Y j +8 vb) +4mΣb +− (Y i +4 va + Y i +13 vb)(Y j +4 va + Y j +13 vb) +4mΣc +. +(2.10) +Since the neutrino mass matrix mν is of rank two, two massive and one massless neutrino +are predicted. +3After the GUT symmetry breaking these two GUT operators decompose into 19 SM Yukawa interac- +tions. For details see Appendix A. +– 4 – + +2.3 +Quark-lepton Yukawa ratios +With X and Y representing one or multiple Higgs fields, the charged fermion masses stem +from GUT operators of the form +Y ij +5 +: +10F i5F jX +⊃ +Y ij +d , Y ij +e +(2.11) +Y ij +10 : +10F i10F jY +⊃ +Y ij +u , +(2.12) +where Yu, Yd and Ye denote the usual SM charged fermion Yukawa matrices. Assuming +in the charged fermion Yukawa sector the concept of single operator dominance, i.e. that +each Yukawa entry is dominated by a singlet GUT operator, allows to connect the down- +type with the charged lepton Yukawa matrix via group theoretical Clebsch-Gordan (CG) +factors cij. In SU(5) GUTs, and considering up to dimension five operators, the potentially +viable CG factors are |cij| ∈ {1/6, 1/2, 2/3, 1, 3/2, 2, 3, 9/2, 6, 9, 18}. The possible GUT +operators yielding these ratios are given in [28, 29]. Moreover, if the matrix Y5 is assumed +to be of hierarchical nature and dominated by its diagonal entries, then the second and +third family down-type quark and charged lepton masses stem dominantly from the GUT +operators O2 and O3 dominating the 22 and 33 positions in Y5. +Depending on which +operators are chosen for O2 and O3, different GUT scale Yukawa ratios yτ/yb and yµ/ys +are predicted. Our numerical analysis (cf. Section 4) shows that there are only two possible +choices for the GUT scale ratio yτ/yb, namely 3/2 or 2. The former CG factor can be +complemented by a factor 9/2 for the second family, while for the latter CG factor two +different completions, yµ/ys = 6 or yµ/ys = 9/2, are possible. +2.4 +Toy models +We now extend the above motivated scenarios to three toy models which also include the first +family. For simplicity, we chose the matrix Y5 to be of diagonal nature. The double ratio +(yµyd)/(yeys) = 10.7+1.6 +−0.9, which is nearly constant under renormalization group running +(see e.g. [53]), suggests, that the the ratio yµ/ys = 9/2 is best complemented by a ratio +ye/yd = 4/9, while the best completion of the ratio yµ/ys = 6 is given by ye/yd = 1/2. +Utilizing these ratios, our three toy models relate the down-type with the charged lepton +Yukawa matrix via +Model 1: +Ye = diag +�4 +9, 9 +2, 3 +2 +� +· Y T +d , +(2.13) +Model 2: +Ye = diag +�4 +9, 9 +2, 2 +� +· Y T +d , +(2.14) +Model 3: +Ye = diag +�1 +2, 6, 2 +� +· Y T +d . +(2.15) +Moreover, for simplicity4 we assume in each toy model that Y10 is dominated by the +operator 10F 10F 5H in all entries, yielding a symmetric up-type Yukawa matrix, i.e. Yu = +4We might consider higher-dimensional operators also for Y10, for example to explain the mass hierarchy, +however since no Yukawa ratio predictions arise from this sector, we stick to the simplest case in our toy +models. +– 5 – + +Y T +u . +Finally, in all toy models neutrino masses stem from a linear combination of the +operators 5F 24F 5H and 5F 24F 45H. +3 +Numerical procedure +3.1 +Implementation of the charged fermion Yukawa sector +We implement all three toy models at the GUT scale as described in Section 2.4. In all +three models the down-type Yukawa matrix Yd is simply implemented as +Yd = diag(yd +1, yd +2, yd +3), +(3.1) +while the charged lepton Yukawa matrix Ye is implemented according to Eq. (2.13), (2.14), +and (2.15), respectively. Since Yu is symmetric we use a Takagi decomposition and imple- +ment it as +Yu = U † +uY diag +u +U ∗ +u, +(3.2) +where5 +Uu = +� +� +� +1 +0 +0 +0 cu +23 +su +23 +0 −su +23 cu +23 +� +� +� +� +� +� +cu +13 +0 su +13e−iδu +0 +1 +0 +−su +13eiδu 0 +cu +13 +� +� +� +� +� +� +cu +12 +su +12 0 +−su +12 cu +12 0 +0 +0 1 +� +� +� +� +� +� +eiβu +1 +0 +0 +0 +eiβu +2 0 +0 +0 +1 +� +� +� , +(3.3) +and where Y diag +u += diag(yu +1, yu +2, yu +3). +3.2 +Implementation of the neutrino sector +In order to simplify the analysis we assume in the neutrino sector that the Yukawa matrices +Y5 and Y6 (for the definitions of these couplings, see Appendix A) are of the form +Y5 = z1 +� +� +� +0 +1 +1 +� +� +� , +Y6 = z2 +� +� +� +1 +1 +3 +� +� +� , +(3.4) +where z1 and z2 are real parameters. Furthermore, we denote the relative phase difference +between mΣb and mΣc by γ (i.e. γ = arg(mΣb/mΣc)). This structure is motivated by CSD3 +[56] which in the case of type I seesaw has been shown to correctly describe the low-scale +neutrino observables together with a normal neutrino mass hierarchy (see e.g. [27] for a +recent work). +3.3 +GUT scale parameters and low energy observables +Each toy model contains 33 input parameters which decompose into the GUT scale MGUT, +the SU(5) gauge coupling gGUT, the masses of the added particles,6 mΦa, mΦb, mφa, mφb, +5Here we have dropped three unphysical parameters but kept the GUT phases βu +1 and βu +2 which effect +the nucleon decay widths [54, 55]. +6Note that mΣd = mΣe. +– 6 – + +mφc, mφd, mφe, mΣa, mΣb, mΣc, mΣd, mt1, mt2, mh2, the singular values yu +1, yu +2, yu +3, yd +1, +yd +2, yd +3 and angles θu +12, θu +13, θu +23 as well as phases δu, βu +1 , βu +2 of the charged fermion Yukawa +matrices, the parameters of the neutrino Yukawa couplings z1, z2, and γ, and the eigenstate +mixing angles α and β. The respective ranges of these input parameters are given by7 +MGUT < MPl, +mΦa, mΦb, mφa, mφb, mφc, mφd, mφe, mΣa, mΣb, mΣc, mΣd, mh2 ∈ [1 TeV, MGUT], +mt1, mt2 ∈ [1011 GeV, MGUT], +gGUT, yu +1, yu +2, yu +3, yd +1, yd +2, yd +3 ∈ [0, 1], +(3.5) +θu +12, θu +13, θu +23, α, β ∈ [0, π/2], +δu, βu +1 , βu +2 , γ ∈ [−π, π), +z1, z2 > 0. +These input parameters are fitted to the 22 low-scale observables (listed in Eq. (3.6)) and +the nucleon decay widths of thirteen decay channels (listed in Table I). +g1, g2, g3, +yu, yc, yt, yd, ys, yb, θCKM +12 +, θCKM +13 +, θCKM +23 +, δCKM, ye, yµ, yτ, +(3.6) +∆m2 +21, ∆m2 +31, θPMNS +12 +, θPMNS +13 +, θPMNS +23 +, δPMNS. +For the SM gauge couplings and Yukawa observables we take the experimental values from +[53], while the values for the neutrino sector are taken from NuFIT 5.1 [57]. +3.4 +Fitting procedure +After implementing the input parameters given in Eq. (3.5) at the GUT scale we compute +the RG evolution to the Z scale. For the gauge couplings we use a 2-loop running, while +we compute the running of the Yukawa matrices and the effective neutrino mass operator +at 1-loop. The nucleon decay widths are computed using the Mathematica package Proton +Decay [52] (for a description of the calculation see e.g. [27]). Taking into account all observ- +ables we compute at the low scale the χ2-function which we minimize using a differential +evolution algorithm giving us a benchmark point. With a flat prior distribution we calcu- +late 4 × 106 data points performing a Markov-chain-Monte-Carlo (MCMC) analysis using +an adaptive Metropolis-Hastings algorithm [65] which we start from this benchmark point. +These data points are finally used to compute the highest posterior density (HPD) ranges +of various quantities. +4 +Results +The results of our numerical analysis are presented in this section. We are in particular +interested in the nucleon decay predictions, the intermediate-scale particle masses as well +7Note that although we do not put any perturbativity constraints on the neutrino Yukawa couplings z1 +and z2 the fit automatically choses them to be below 1 (cf. Section 4). +– 7 – + +decay channel +τ/B [year] +Γpartial [GeV] +Reference +Proton: +p → π0 e+ +> 2.4 · 1034 +< 8.7 · 10−67 +[58] +p → π0 µ+ +> 1.6 · 1034 +< 1.3 · 10−66 +[58] +p → η0 e+ +> 1.0 · 1034 +< 2.0 · 10−66 +[59] +p → η0 µ+ +> 4.7 · 1033 +< 4.4 · 10−66 +[59] +p → K0 e+ +> 1.1 · 1033 +< 1.9 · 10−65 +[60] +p → K0 µ+ +> 3.6 · 1033 +< 5.8 · 10−66 +[61] +p → π+ ν +> 3.9 · 1032 +< 5.3 · 10−65 +[62] +p → K+ ν +> 6.6 · 1033 +< 3.2 · 10−66 +[63] +Neutron: +n → π− e+ +> 5.3 · 1033 +< 3.9 · 10−66 +[59] +n → π− µ+ +> 3.5 · 1033 +< 5.9 · 10−66 +[59] +n → π0 ν +> 1.1 · 1033 +< 1.9 · 10−65 +[62] +n → η0 ν +> 5.6 · 1032 +< 3.7 · 10−65 +[60] +n → K0 ν +> 1.2 · 1032 +< 1.7 · 10−64 +[60] +Table I: Current experimental bounds on the decay widths Γpartial, respectively lifetime +τ/B at 90 % confidence level, where B is the branching ratio for the decay channel. See +also [64] for future projections and sensitivities of various upcoming detectors. +as the low scale predictions for the charged lepton and down-type quark mass ratios. In +Section 4.1 we show the results of our minimization procedure. Starting an MCMC analysis +from these benchmark points allows us to obtain the HPD ranges of various quantities. The +results of this analysis is presented in Section 4.2. +4.1 +Benchmark points +We obtain for all three models benchmark points through a minimization of the χ2-function +as described in Section 3. In Table II the input parameters for the respective benchmark +points are listed. Moreover, the dominant pulls χ2 +i are presented in Table III. All three +models can be very well fitted to the data. The strongest (though quite small) pull is given +by the first and second family down-type quark masses. The biggest difference between the +three models is the respectively favored GUT scale. For Models 2 and 3 a GUT scale above +1017 GeV is favored, while for the benchmark point of Model 1 a GUT scale below 1016 GeV +is obtained. This also results in different results for the predicted nucleon decay rates (cf. +Section 4.2). Another difference is the preferred choice of some of the intermediate-scale +particle masses. In the presented benchmark point the mass of the fermionic field Σa is +obtained to be at the GUT scale for Model 1, at the intermediate scale for Model 2 and at +the relatively low scale (23 TeV) for Model 3. Moreover, a mass of the leptoquark φc of 1 +– 8 – + +TeV, respectively 4 TeV is obtained for Model 3, respectively Model 2, whereas for Model 1 +the mass of this field is above 106 TeV. For the HPD results of these particle masses confer +the subsequent section. +Model 1 +Model 2 +Model 3 +gGUT / 10−1 +5.94 +6.17 +6.33 +log10(MGUT / GeV) +15.6 +17.2 +17.3 +log10(mφa / GeV) +9.43 +14.0 +16.7 +log10(mφc / GeV) +9.02 +3.63 +3.00 +log10(mΣa / GeV) +15.6 +7.53 +4.36 +log10(mΣb / GeV) +14.2 +14.9 +14.7 +log10(mΣc / GeV) +13.8 +12.8 +13.2 +log10(mΣd / GeV) +14.2 +15.9 +14.1 +yu +1 / 10−6 +2.63 +2.11 +1.99 +yu +2 / 10−3 +1.46 +1.37 +1.18 +yu +3 / 10−1 +4.54 +4.26 +3.65 +yd +1 / 10−6 +6.21 +6.30 +5.46 +yd +2 / 10−4 +1.31 +1.21 +0.99 +yd +3 / 10−3 +6.64 +6.01 +5.36 +z1 / 10−1 +3.50 +9.42 +6.86 +z2 / 10−1 +1.12 +0.32 +0.50 +γ +1.85 +1.48 +1.68 +α +0.50 +1.00 +0.50 +Table II: The GUT scale input parameters of the benchmark points for all three models. +χ2 +χ2 +yd +χ2 +ys +χ2 +yb +χ2 +yµ +χ2 +yτ +χ2 +Γ(p→π0e+) +Model 1 +1.36 +0.27 +0.41 +0.06 +0.04 +0.14 +0.44 +Model 2 +0.31 +0.23 +0.02 +0.01 +0.00 +0.05 +0.00 +Model 3 +0.33 +0.17 +0.03 +0.00 +0.06 +0.07 +0.00 +Table III: The total χ2 as well as the dominant pulls χ2 +i for the benchmark points of all +three models. +4.2 +Highest posterior densities +As described in Section 3.4 we vary the input parameters listed in Eq. 3.5 around their +benchmark points (cf. Table II) using an MCMC analysis. From these generated points we +then compute the HPD intervals of various parameters and observables. +– 9 – + +Model 1 +Model 2 +Model 3 +0.14 +0.15 +0.16 +0.17 +0.18 +0.19 +ye/yd +HPD intervals for ye/yd +Model 1 +Model 2 +Model 3 +1.7 +1.8 +1.9 +2.0 +2.1 +yμ/ys +HPD intervals for yμ/ys +Model 1 +Model 2 +Model 3 +0.59 +0.60 +0.61 +0.62 +0.63 +yτ/yb +HPD intervals for yτ/yb +Figure 1: Low scale (MZ) HPD intervals for charged lepton and down-type quark Yukawa +ratios of all three families. The 1σ (2σ) HDP intervals are colored dark (light). +In Figures 1 – 4 we use the following color coding: For Model 1, 2, and 3 the HPD +intervals of various quantities are colored red, green, and blue, respectively, while the 1σ +(2σ) HPD intervals are colored dark (light). +4.2.1 +Quark-lepton mass ratios +The HPD results for the low scale charged lepton and down-type quark mass ratios are +presented in Figure 1. +The horizontal dashed line represents the current experimental +central value, whereas the white region shows the current experimental 1σ range. Clearly, +all three models are capable of reproducing viable mass ratios. This strengthens the results +of the benchmark points in the previous subsection (cf. Tables II and III). Compared to +Model 2 and 3, Model 1 gives a bit smaller predictions for the mass ratios for all three +generations. +4.2.2 +Intermediate-scale particle masses +Figure 2 shows the predicted HPD intervals of the intermediate-scale particle masses. Most +of the masses are predicted to be out of the reach of current and future colliders, because +they would either produce too much proton decay, spoil gauge coupling unification or be- +cause of the fit of the fermion masses. But interestingly, the fields Φb, φc and Σa are not +only potentially within the reach of future searches, but can also be used to distinguish be- +tween the different models: An observation of the one of the fields Φb or Σa would strongly +hint towards Model 3, while an observation of the field φc would disfavor Model 1. In fact, +the most promising lookout could be for the leptoquark φc. The upper bound of the HPD +1σ range is predicted to be 23 TeV (2.8 TeV) in Model 2 (3), whereas the upper bound of +the 2σ intervals is 175 TeV (17 TeV). In the following, we briefly state the current collider +bounds on these particles. +The scalar triplet, Φb, with zero hypercharge, residing in the 24H multiplet is expected +to be light in Model 3. Note that Φb contains a neutral Φ0 +b and a pair of singly charged +Φ± +b states. In the low-energy effective theory, a term of the form h† +1Φ2 +bh1 is allowed, where +– 10 – + +mΦa +mΦb +mTa +mTb +m ϕa +m ϕb +m ϕc +m ϕd +m ϕe +mΣa +mΣb +mΣc +mΣd +Model 1 +Model 2 +Model 3 +2 +4 +6 +8 +10 +12 +14 +16 +18 +log10(μ/GeV) +HPD intervals for intermediate-scale particle masses +Figure 2: HPD intervals of the intermediate-scale particle masses. The 1σ (2σ) HDP +intervals are colored dark (light). +h1 is the SM Higgs doublet. As a result of this coupling, the SM Higgs can decay into two +photons h0 → γγ via a one-loop diagram mediated by the Φ± +b states. Consistency with the +LHC data requires these charged states to have masses above 250 GeV [66]. +The scalar leptoquark φc, which is a triplet of SU(2)L, resides around the TeV scale +in Models 2 and 3. +In both models, its coupling to the SM fermions is dominated by +the third-generation quark and lepton. Hence, within our scenarios, its decay branching +fraction is dominated by a bτ final state. Since leptoquarks carry color, they are efficiently +produced at the LHC through gluon-initiated as well as quark-initiated processes [67]. LHC +searches of pp → bbττ from pair-produced leptoquarks rule out leptoquark masses below +1400 GeV [68, 69]. +As can be seen from Eq. (A.2), the color octet fermion Σa, which is expected to be +light in Model 3, couples, for example, to a singlet down-quark (lepton doublet) and a +super-heavy colored triplet (octet) scalar. Consequently, the lifetime of a TeV scale Σa is +expected to be large, and it behaves like a long-lived gluino that typically arises in split- +supersymmetric scenarios [70, 71]. Long-lived colored particles would hadronize, forming +so-called R-hadrons [72]. These bound states are comprised of the long-lived state and +light SM quarks or gluons, and interact with the detector material, typically inside the +calorimeters, via hadronic interactions of the light-quark constituents. Motivated by split- +supersymmetric models, R-hadrons are extensively searched for at the LHC [73, 74]. Non- +observation of any deviations of the signal from the expected background puts to a lower +– 11 – + +limit on the mass of the long-lived Σa fermion of 2000 GeV [73]. +4.2.3 +Nucleon decay width and GUT scale +Figure 3 shows the predictions for the HPD intervals of the GUT scale MGUT. Moreover, +the predicted HPD ranges for the nucleon decay widths of the various decay channels +are presented in Figure 4. The blue line segments in the latter picture indicate the current +experimental bounds at 90 % confidence level (cf. Table I). Moreover, the future constraints +on the decay widths for the decay channels p → π0e+ and n → π−e+ which will be provided +by Hyper-Kamiokande [75] are presented by orange line segments. +In Figure 3 it can be seen that Model 1 clearly predicts the GUT scale to be below 1016 +GeV. On the other hand, a much larger GUT scale is preferred by the Models 2 and 3. Since +the nucleon decay width is inversely proportional to the forth power of the GUT scale in the +case of gauge boson mediated nucleon decay, this also results in strongly different prediction +for the nucleon decay widths of the various channels as it can be seen in Figure 4. The +nucleon decay predictions for Model 1 are very close to the current bounds, the 1σ HPD +interval of the proton decay channel p → π0e+ will be fully probed by Hyper-Kamiokande. +Moreover, Hyper-Kamiokande will probe most of the 1σ HPD interval of the neutron decay +channel n → π−e+. On the other hand, the gauge boson mediated nucleon decay is highly +suppressed in Models 2 and 3 and cannot be probed by any planed experiments. Therefore, +observation of nucleon decay in the decay channels p → π0e+ and n → π−e+ would clearly +favour Model 1 over the Models 2 and 3. +5 +Conclusion +In this paper we considered an extension of the Georgi-Glashow SU(5) GUT scenario by +a 45-dimensional scalar and a 24-dimensional fermionic representation. Neutrino masses +in this scenario are generated by a combination of a type I and a type III seesaw mech- +anism. Assuming the concept of single operator dominance we investigated which GUT +scale charged lepton and down-type quark Yukawa ratios can be viable for the second and +third family and found that three combinations work: (i) yτ/yb = 3/2, yµ/ys = 9/2, (ii) +yτ/yb = 2, yµ/ys = 9/2, and (iii) yτ/yb = 2, yµ/ys = 6. Also taking into account the origin +of the first family masses we extended these possibilities to three toy models and analyzed +various of their predictions. We showed that experimental discrimination between these +models could be possible since they predict different nucleon decay rates as well as distinct +light relics. +Appendices +A +Definition of new Yukawa couplings +The Lagrangian density contains the two terms +L ⊃ YA 5i +F 24F 5H + YB 5i +F 24F 45H. +(A.1) +– 12 – + +Model 1 +Model 2 +Model 3 +15.5 +16.0 +16.5 +17.0 +17.5 +18.0 +18.5 +MGUT +HPD intervals for MGUT +Figure 3: Predicted HPD intervals of the GUT scale. The 1σ (2σ) HDP intervals are +colored dark (light). +p → π0e+ +p → π0 μ+ +p → η0e+ +p → η0 μ+ +p → K0e+ +p → K0 μ+ p → π+ν +p → K+ν +n → π-e+ +n → π- μ+ +n → π0ν +n → η0ν +n → K0ν +Model 1 +Model 2 +Model 3 +-85 +-80 +-75 +-70 +-65 +log10(Γ/GeV) +HPD intervals for decay widths Γ of different nucleon decay channels +Figure 4: +Predicted HPD intervals of the nucleon decay widths. +The 1σ (2σ) HDP +intervals are colored dark (light). For each decay channel the blue line segments represent +the current experimental constraints. The future Hyper-Kamiokande constraints for the +decay channels p → π0e+ and n → π−e+ are indicated by orange line segments. +– 13 – + +After the GUT symmetry breaking they decompose into +L = +� +2 +15YA dcΣcTa − +� +3 +10YA ℓΣcHa + YA dcΣaTa + YA ℓΣbHa+ +YA dcΣdHa + YA ℓΣeTa + +� +5 +12YB dcΣcTb + +√ +5 +4 YB ℓΣcHb+ +1 +2 +√ +2YB dcΣaTb + 1 +√ +2YB dcΣaφb + 1 +√ +2YB ℓΣaφa + 1 +√ +2YB dcΣbφc+ +√ +3 +4 YB ℓΣbHb − 1 +√ +2YB dcΣdφa − +1 +2 +√ +6YB dcΣdHb + 1 +√ +2YB ℓΣdφe− +1 +√ +2YB dcΣeφd + +1 +2 +√ +2YB ℓΣeTb − 1 +√ +2YB ℓΣeφc +≡ Y1 dcΣcTa + Y2 ℓΣcHa + Y3 dcΣaTa + Y4 ℓΣbHa+ +Y5 dcΣdHa + Y6 ℓΣeTa + Y7 dcΣcTb + Y8 ℓΣcHb+ +Y9 dcΣaTb + Y10 dcΣaφb + Y11 ℓΣaφa + Y12 dcΣbφc+ +Y13 ℓΣbHb + Y14 dcΣdφa + Y15 dcΣdHb + Y16 ℓΣdφe+ +Y17 dcΣeφd + Y18 ℓΣeTb + Y19 ℓΣeφc , +(A.2) +where we defined the Yukawa matrices YN, with N = 1, . . . , 19. +B +Renormalization group equations +Here the RGEs for the gauge and Yukawa couplings as well as for the effective neutrino +mass operator are listed. We have used the Mathematica package SARAH [76, 77] to obtain +the RGEs for the gauge and Yukawa couplings. The SM contribution for the RGE of the +effective neutrino mass operator is taken from [78]. In order to compute the new contribution +for this RGE we have used the method described therein. We use the following definition +for the Heaviside-Theta function +H(µ, m) = +� +1, for µ ≥ m, +0, for µ < m. +(B.1) +B.1 +Gauge couplings +The RGEs for gauge couplings (i, k = 1 − 3) are given by +µdgi +dµ = +βgi +1−loop +16π2 ++ +βgi +2−loop +(16π2)2 , +(B.2) +where βgi +1−loop is the 1-loop and βgi +2−loop is the 2-loop contribution given by +βgi +1−loop = +� +aSM +i ++ H(µ, m)∆ai +� +g3 +i +(B.3) +βgi +2−loop = +� +k +bSM +ik g2 +k + +� +k +∆bikg2 +k H(µ, m) + βY,SM +i ++ ∆βY +i . +(B.4) +– 14 – + +Here, aSM +i +, bSM +ik +and βY,SM +i +are the well known SM 1-loop and 2-loop coefficients as well as +Yukawa contributions [79, 80]. Moreover, the ∆βY +i are given by +∆βY +i = g3 +i +� +k +cikY T +k Y ∗ +k H2 +k, +(B.5) +where we introduced the abbreviation H2 +k = H(µ, mF )H(µ, mH) associated to each of the +Yukawa interactions, where, F and H refer to the BSM fermion and scalar appearing in +that interaction, respectively, and where the cik are given by +c1k = − +�1 +5, 3 +10, 8 +15, 9 +20, 29 +10, 17 +5 , 1 +5, 3 +10, 8 +15, 8 +15, 12 +5 , 3 +5, 9 +20, 116 +15 , 29 +10, 17 +5 , 29 +5 , 17 +5 , 51 +10 +� +, +(B.6) +c2k = − +� +0, 1 +2, 0, 11 +4 , 3 +2, 3, 0, 1 +2, 0, 0, 4, 6, 11 +4 , 4, 3 +2, 3, 3, 3, 9 +2 +� +, +(B.7) +c3k = − +�1 +2, 0, 13 +3 , 0, 2, 1, 1 +2, 0, 13 +3 , 13 +3 , 6, 3 +2, 0, 16 +3 , 2, 1, 4, 1, 3 +2 +� +. +(B.8) +Finally, the ∆ai and ∆bi are given as a sum over the 1-loop and 2-loop coefficients of the +BSM fermions and scalars, i.e. +∆ai = +� +I +∆aI +i , +∆bi = +� +I +∆bI +i , +(B.9) +where I runs over all BSM particles. The 1-loop coefficients are then given by +∆aφa +i += +�4 +5, 4 +3, 2 +� +, +∆aφb +i += +� 2 +15, 0, 5 +6 +� +, +∆aφc +i += +�1 +5, 2, 1 +2 +� +, +∆aφd +i += +�49 +30, 1 +2, 1 +3 +� +, +∆aφe +i += +�16 +15, 0, 1 +6 +� +, +∆aΦa +i += {0, 0, 1 +2}, +∆aΦb +i += +� +0, 1 +3, 0 +� +, +∆aΣa +i += {0, 0, 2}, +∆aΣb +i += +� +0, 4 +3, 0 +� +, +∆aΣd,e +i += +�5 +3, 1, 2 +3 +� +, +∆ah⊥ +i += +� 1 +10, 1 +6, 0 +� +, +∆at,t⊥ +i += +� 1 +15, 0, 1 +6 +� +, +(B.10) +whereas the 2-loop coefficients read +∆bφa +ik = +� +� +� +36 +25 +36 +5 +144 +5 +12 +5 +52 +3 +48 +18 +5 18 84 +� +� +� , +∆bφb +ik = +� +� +� +8 +75 0 +16 +3 +0 0 +0 +2 +3 0 115 +3 +� +� +� , +∆bφc +ik = +� +� +� +4 +25 +24 +5 +16 +5 +8 +5 56 32 +2 +5 12 11 +� +� +� , +∆bφd +ik = +� +� +� +2401 +150 +147 +10 +392 +15 +49 +10 +13 +2 +8 +49 +15 +3 +22 +3 +� +� +� , +∆bφe +ik = +� +� +� +1024 +75 +0 256 +15 +0 +0 +0 +32 +15 +0 +11 +3 +� +� +� , +∆bΦa +ik = +� +� +� +0 0 0 +0 0 0 +0 0 21 +� +� +� , +∆bΦb +ik = +� +� +� +0 0 0 +0 28 +3 0 +0 0 0 +� +� +� , +∆bΣa +ik = +� +� +� +0 0 0 +0 0 0 +0 0 48 +� +� +� , +∆bΣb +ik = +� +� +� +0 0 0 +0 64 +3 0 +0 0 0 +� +� +� , +∆bΣd,e +ik += +� +� +� +25 +12 +15 +4 +20 +3 +5 +4 +49 +4 +4 +5 +6 +3 +2 +38 +3 +� +� +� , +∆bh⊥ +ik = +� +� +� +9 +50 +9 +10 0 +3 +10 +13 +6 0 +0 +0 0 +� +� +� , +∆bt,t⊥ +ik += +� +� +� +4 +75 0 16 +15 +0 0 0 +2 +15 0 11 +3 +� +� +� . +(B.11) +– 15 – + +B.2 +Yukawa matrices +The RGEs of the Yukawa matrices read +µdYf +dµ = +βf +16π2 , +(B.12) +where f = {u, d, e, k} and (k = 1, . . . , 19). For the SM Yukawa matrices Yu, Yd and Ye (i.e. +f = u, d, e) the beta functions are given by +βf = βSM +f ++ δβf, +(B.13) +where βSM +f +is the SM beta function [79, 80], and where +δβf = YfT1 + +� +k +af +k(Yk)j(Y T +d Y ∗ +k )i H2 +k . +(B.14) +Here, we have defined T1 as +T1 = Y T +2 Y ∗ +2 H2 +2 + 3 +2Y T +4 Y ∗ +4 H2 +4 + 3Y T +5 Y ∗ +5 H2 +5. +(B.15) +while the af +k are given by +au +k = +� +0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 +� +, +(B.16) +ad +k = +�1 +2, 0, 4 +3, 0, 3, 0, 1 +2, 0, 4 +3, 4 +3, 0, 3 +2, 0, 8 +3, 1, 0, 2, 0, 0 +� +, +(B.17) +ae +k = +� +0, −3 +2, 0, 15 +4 , 0, 3 +2, 0, 1 +2, 0, 0, 4, 0, 3 +4, 0, 0, 3 +2, 0, 3 +2, 9 +4 +� +. +(B.18) +In order to simplify the notation, from hereon, associated to each Yukawa Yi → Yi H2 +i must +be understood. The beta function of the Yukawa matrices Y1, . . . , Y19 then read +β1 = Y1 +� +−1 +5g2 +1 − 4g2 +3 + +� +k +a1 +kY T +k Y ∗ +k +� ++ +� +YdY † +d +� +Y1 + +� +w +b1 +w +� +Y T +1 Y ∗ +w +� +Yw ++ 8 +3 +� +Y T +3 Y ∗ +9 +� +Y7 + 2 +� +Y T +6 Y ∗ +18 +� +Y7, +(B.19) +β2 = Y2 +� +− 9 +20g2 +1 − 9 +4g2 +2 + +� +k +a2 +kY T +k Y ∗ +k + T +� ++ +� +−3 +2Y T +e Y ∗ +e +� +Y2 + +� +w +b2 +w +� +Y T +2 Y ∗ +w +� +Yw ++ 3 +2 +� +Y T +4 Y ∗ +13 +� +Y8 + 3 +� +Y T +5 Y ∗ +15 +� +Y8, +(B.20) +β3 = Y3 +� +−1 +5g2 +1 − 13g2 +3 + +� +k +a3 +kY T +k Y ∗ +k +� ++ +� +YdY † +d +� +Y3 + +� +w +b3 +w +� +Y T +3 Y ∗ +w +� +Yw ++ +� +Y T +1 Y ∗ +7 +� +Y9 + 2 +� +Y T +6 Y ∗ +18 +� +Y9, +(B.21) +β4 = Y4 +� +− 9 +20g2 +1 − 33 +4 g2 +2 + +� +k +a4 +kY T +k Y ∗ +k + T +� ++ +�5 +2Y T +e Y ∗ +e +� +Y4 + +� +w +b4 +w +� +Y T +4 Y ∗ +w +� +Yw +– 16 – + ++ +� +Y T +2 Y ∗ +8 +� +Y13 + 3 +� +Y T +5 Y ∗ +15 +� +Y13, +(B.22) +β5 = Y5 +� +−29 +20g2 +1 − 9 +4g2 +2 − 8g2 +3 + +� +k +a5 +kY T +k Y ∗ +k + T +� ++ +� +3YdY † +d +� +Y5 + +� +w +b5 +w +� +Y T +5 Y ∗ +w +� +Yw ++ +� +Y T +2 Y ∗ +8 +� +Y15 + 3 +2 +� +Y T +4 Y ∗ +13 +� +Y15, +(B.23) +β6 = Y6 +� +−17 +10g2 +1 − 9 +2g2 +2 − 4g2 +3 + +� +k +a6 +kY T +k Y ∗ +k +� ++ +�1 +2Y T +e Y ∗ +e +� +Y6 + +� +w +b6 +w +� +Y T +6 Y ∗ +w +� +Yw ++ +� +Y T +1 Y ∗ +7 +� +Y18 + 8 +3 +� +Y T +3 Y ∗ +9 +� +Y18, +(B.24) +β7 = Y7 +� +−1 +5g2 +1 − 4g2 +3 + +� +k +a7 +kY T +k Y ∗ +k +� ++ +� +YdY † +d +� +Y7 + +� +w +b7 +w +� +Y T +7 Y ∗ +w +� +Yw ++ +� +2Y T +18Y ∗ +6 +� +Y1 + 8 +3 +� +Y T +9 Y ∗ +3 +� +Y1, +(B.25) +β8 = Y8 +� +− 9 +20g2 +1 − 9 +4g2 +2 + +� +k +a8 +kY T +k Y ∗ +k +� ++ +� +Y T +e Y ∗ +e +� +Y8 + +� +w +b8 +w +� +Y T +8 Y ∗ +w +� +Yw ++ 3 +2 +� +Y T +13Y ∗ +4 +� +Y2 + 3 +� +Y T +15Y ∗ +5 +� +Y2, +(B.26) +β9 = Y9 +� +−1 +5g2 +1 − 13g2 +3 + +� +k +a9 +kY T +k Y ∗ +k +� ++ +� +YdY † +d +� +Y9 + +� +w +b9 +w +� +Y T +9 Y ∗ +w +� +Yw ++ 2 +� +Y T +18Y ∗ +6 +� +Y3 + +� +Y T +7 Y ∗ +1 +� +Y3, +(B.27) +β10 = Y10 +� +−1 +5g2 +1 − 13g2 +3 + +� +k +a10 +k Y T +k10Y ∗ +k +� ++ +� +YdY † +d +� +Y10 + +� +w +b10 +w +� +Y T +10Y ∗ +w +� +Yw, +(B.28) +β11 = Y11 +� +− 9 +20g2 +1 − 9 +4g2 +2 − 9g2 +3 + +� +k +a11 +k Y T +k Y ∗ +k +� ++ +� +Y T +e Y ∗ +e +� +Y11 + +� +w +b11 +w +� +Y T +11Y ∗ +w +� +Yw, +(B.29) +β12 = Y12 +� +−1 +5g2 +1 − 6g2 +2 − 4g2 +3 + +� +k +a12 +k Y T +k Y ∗ +k +� ++ +� +YdY † +d +� +Y12 + +� +w +b12 +w +� +Y T +12Y ∗ +w +� +Yw, +(B.30) +β13 = Y13 +� +− 9 +20g2 +1 − 33 +4 g2 +2 + +� +k +a13 +k Y T +k Y ∗ +k +� ++ +�1 +2Y T +e Y ∗ +e +� +Y13 + +� +w +b13 +w +� +Y T +13Y ∗ +w +� +Yw ++ 3 +� +Y T +15Y ∗ +5 +� +Y4 + +� +Y T +8 Y ∗ +2 +� +Y4, +(B.31) +β14 = Y14 +� +−29 +20g2 +1 − 9 +4g2 +2 − 8g2 +3 + +� +k +a14 +k Y T +k Y ∗ +k +� ++ +� +YdY † +d +� +Y14 + +� +w +b14 +w +� +Y T +14Y ∗ +w +� +Yw, +(B.32) +β15 = Y15 +� +−29 +20g2 +1 − 9 +4g2 +2 − 8g2 +3 + +� +k +a15 +k Y T +k Y ∗ +k +� ++ +� +YdY † +d +� +Y15 + +� +w +b15 +w +� +Y T +15Y ∗ +w +� +Yw +– 17 – + ++ 3 +2 +� +Y T +13Y ∗ +4 +� +Y5 + +� +Y T +8 Y ∗ +2 +� +Y5, +(B.33) +β16 = Y16 +� +−17 +10g2 +1 − 9 +2g2 +2 − 4g2 +3 + +� +k +a16 +k Y T +k Y ∗ +k +� ++ +�1 +2Y T +e Y ∗ +e +� +Y16 + +� +w +b16 +w +� +Y T +16Y ∗ +w +� +Yw, +(B.34) +β17 = Y17 +� +−29 +20g2 +1 − 9 +4g2 +2 − 8g2 +3 + +� +k +a17 +k Y T +k Y ∗ +k +� ++ +� +YdY † +d +� +Y17 + +� +w +b17 +w +� +Y T +17Y ∗ +w +� +Yw, +(B.35) +β18 = Y18 +� +−17 +10g2 +1 − 9 +2g2 +2 − 4g2 +3 + +� +k +a18 +k Y T +k Y ∗ +k +� ++ +�1 +2Y T +e Y ∗ +e +� +Y18 + +� +w +b18 +w +� +Y T +18Y ∗ +w +� +Yw ++ +� +Y T +7 Y ∗ +1 +� +Y6 + 8 +3 +� +Y T +9 Y ∗ +3 +� +Y6, +(B.36) +β19 = Y19 +� +−17 +20g2 +1 − 9 +2g2 +2 − 4g2 +3 + +� +k +a19 +k Y T +k Y ∗ +k +� ++ +�1 +2Y T +e Y ∗ +e +� +Y19 + +� +w +b19 +w +� +Y T +19Y ∗ +w +� +Yw, +(B.37) +where the coefficients af +k are given by +a1 +k = {3, 1, 8 +3, 0, 0, 2, 3 +2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, +(B.38) +a2 +k = {3 +2, 5 +2, 0, 3 +2, 3, 0, 3 +2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, +(B.39) +a3 +k = {1, 0, 9 +2, 0, 0, 2, 0, 0, 1 +2, 1 +2, 1, 0, 0, 0, 0, 0, 0, 0, 0}, +(B.40) +a4 +k = {0, 1, 0, 11 +4 , 3, 0, 0, 0, 0, 0, 0, 3 +2, 1 +2, 0, 0, 0, 0, 0, 0}, +(B.41) +a5 +k = {0, 1, 0, 3 +2, 9 +2, 0, 0, 0, 0, 0, 0, 0, 0, 4 +3, 1 +2, 1 +2, 0, 0, 0}, +(B.42) +a6 +k = {1, 0, 8 +3, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1 +2, 3 +4}, +(B.43) +a7 +k = {3 +2, 1, 0, 0, 0, 0, 3, 1, 8 +3, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0}, +(B.44) +a8 +k = {3 +2, 1, 0, 0, 0, 0, 3 +2, 5 +2, 0, 0, 0, 0, 3 +2, 0, 3, 0, 0, 0, 0}, +(B.45) +a9 +k = {0, 0, 1 +2, 0, 0, 0, 1, 0, 9 +2, 1 +2, 1, 0, 0, 0, 0, 0, 0, 2, 0}, +(B.46) +a10 +k = {0, 0, 1 +2, 0, 0, 0, 0, 0, 1 +2, 19 +6 , 1, 0, 0, 0, 0, 0, 0, 0, 0}, +(B.47) +a11 +k = {0, 0, 1 +2, 0, 0, 0, 0, 0, 1 +2, 1 +2, 6, 0, 0, 1, 0, 0, 0, 0, 0}, +(B.48) +a12 +k = {0, 0, 0, 1 +2, 0, 0, 0, 0, 0, 0, 0, 4, 1 +2, 0, 0, 0, 0, 0, 1}, +(B.49) +a13 +k = {0, 0, 0, 1 +2, 0, 0, 0, 1, 0, 0, 0, 3 +2, 11 +4 , 0, 3, 0, 0, 0, 0}, +(B.50) +a14 +k = {0, 0, 0, 0, 1 +2, 0, 0, 0, 0, 0, 1, 0, 0, 5, 1 +2, 1 +2, 0, 0, 0}, +(B.51) +– 18 – + +a15 +k = {0, 0, 0, 0, 1 +2, 0, 0, 1, 0, 0, 0, 0, 3 +2, 4 +3, 9 +2, 1 +2, 0, 0, 0}, +(B.52) +a16 +k = {0, 0, 0, 0, 1 +2, 0, 0, 0, 0, 0, 0, 0, 0, 4 +3, 1 +2, 4, 0, 0, 0}, +(B.53) +a17 +k = {0, 0, 0, 0, 0, 1 +2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 1 +2, 3 +4}, +(B.54) +a18 +k = {0, 0, 0, 0, 0, 1 +2, 1, 0, 8 +3, 0, 0, 0, 0, 0, 0, 0, 1, 4, 3 +4}, +(B.55) +a19 +k = {0, 0, 0, 0, 0, 1 +2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1 +2, 4}, +(B.56) +while the coefficients bf +k read +b1 +w = {0, 0, 4 +3, 0, 1, 0, 3 +2, 0, 0, 4 +3, 0, 3 +2, 0, 8 +3, 1, 0, 2, 0, 0}, +(B.57) +b2 +w = {0, 0, 0, 3 +4, 0, 3 +2, 0, 3 +2, 0, 0, 4, 0, 3 +4, 0, 0, 3 +2, 0, 3 +2, 9 +4}, +(B.58) +b3 +w = {1 +2, 0, 0, 0, 1, 0, 1 +2, 0, 0, 4 +3, 0, 3 +2, 0, 8 +3, 1, 0, 2, 0}, +(B.59) +b4 +w = {0, 1 +2, 0, 0, 0, 3 +2, 0, 1 +2, 0, 0, 4, 0, 9 +4, 0, 0, 3 +2, 0, 3 +2, 9 +4}, +(B.60) +b5 +w = {1 +2, 0, 4 +3, 0, 0, 0, 1 +2, 0, 4 +3, 4 +3, 0, 3 +2, 0, 8 +3, 4, 0, 2, 0, 0}, +(B.61) +b6 +w = {0, 1 +2, 0, 3 +4, 0, 0, 0, 1 +2, 0, 0, 4, 0, 3 +4, 0, 0, 3 +2, 0, 7 +2, 9 +4}, +(B.62) +b7 +w = {3 +2, 0, 4 +3, 0, 1, 0, 0, 0, 4 +3, 4 +3, 0, 3 +2, 0, 8 +3, 1, 0, 2, 0, 0}, +(B.63) +b8 +w = {0, 3 +2, 0, 3 +4, 0, 3 +2, 0, 0, 0, 0, 4, 0, 3 +4, 0, 0, 3 +2, 0, 3 +2, 9 +4}, +(B.64) +b9 +w = {1 +2, 0, 4, 0, 1, 0, 1 +2, 0, 0, 4 +3, 0, 3 +2, 0, 8 +3, 1, 0, 2, 0, 0}, +(B.65) +b10 +w = {1 +2, 0, 4 +3, 0, 1, 0, 1 +2, 0, 4 +3, 0, 0, 3 +2, 0, 8 +3, 1, 0, 2, 0, 0}, +(B.66) +b11 +w = {0, 1 +2, 0, 3 +4, 0, 3 +2, 0, 1 +2, 0, 0, 0, 0, 3 +4, 0, 0, 3 +2, 0, 3 +2, 9 +4}, +(B.67) +b12 +w = {1 +2, 0, 4 +3, 0, 1, 0, 1 +2, 0, 4 +3, 4 +3, 0, 0, 0, 8 +3, 1, 0, 2, 0, 0}, +(B.68) +b13 +w = {0, 1 +2, 0, 9 +4, 0, 3 +2, 0, 1 +2, 0, 0, 4, 0, 0, 0, 0, 3 +2, 0, 3 +2, 9 +4}, +(B.69) +b14 +w = {1 +2, 0, 4 +3, 0, 1, 0, 1 +2, 0, 4 +3, 4 +3, 0, 3 +2, 0, 0, 1, 0, 2, 0, 0}, +(B.70) +b15 +w = {1 +2, 0, 4 +3, 0, 4, 0, 1 +2, 0, 4 +3, 4 +3, 0, 3 +2, 0, 8 +3, 0, 0, 2, 0, 0}, +(B.71) +b16 +w = {0, 1 +2, 0, 3 +4, 0, 3 +2, 0, 1 +2, 0, 0, 4, 0, 3 +4, 0, 0, 0, 0, 3 +2, 9 +4}, +(B.72) +b17 +w = {1 +2, 0, 4 +3, 0, 1, 0, 1 +2, 0, 4 +3, 4 +3, 0, 3 +2, 0, 8 +3, 1, 0, 0, 0, 0}, +(B.73) +b18 +w = {0, 1 +2, 0, 3 +4, 0, 7 +2, 0, 1 +2, 0, 0, 4, 0, 3 +4, 0, 0, 3 +2, 0, 0, 9 +4}, +(B.74) +b19 +w = {0, 1 +2, 0, 3 +4, 0, 3 +2, 0, 1 +2, 0, 0, 4, 0, 3 +4, 0, 0, 3 +2, 0, 3 +2, 0}. +(B.75) +– 19 – + +B.3 +Effective neutrino mass operator +The RGE for the effective neutrino mass operator reads +16π2µdκ +dµ = βSM +κ ++ ∆βκ, +(B.76) +where βSM +κ +is the SM contribution as given in [78] and ∆βκ is the correction due to the +added BSM particles. 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B 236 (1984) 221–232. +– 24 – + diff --git a/MtE2T4oBgHgl3EQfBQa7/content/tmp_files/load_file.txt b/MtE2T4oBgHgl3EQfBQa7/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..70d2d30e0e06ed4aecb4e8bb6f01770753d21c05 --- /dev/null +++ b/MtE2T4oBgHgl3EQfBQa7/content/tmp_files/load_file.txt @@ -0,0 +1,1195 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf,len=1194 +page_content='Quark-lepton Yukawa ratios and nucleon decay in SU(5) GUTs with type-III seesaw Stefan Antusch, Kevin Hinze, and Shaikh Saad Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland E-mail: stefan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='antusch@unibas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='ch, kevin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='hinze@unibas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='ch, shaikh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='saad@unibas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='ch Abstract: We consider an extension of the Georgi-Glashow SU(5) GUT model by a 45-dimensional scalar and a 24-dimensional fermionic representation, where the latter leads to the generation of the observed light neutrino masses via a combination of a type I and a type III seesaw mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Within this scenario, we investigate the viability of predictions for the ratios between the charged lepton and down-type quark Yukawa couplings, focusing on the second and third family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Such predictions can emerge when the relevant entries of the Yukawa matrices are generated from single joint GUT operators (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' under the condition of single operator dominance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' We show that three combinations are viable, (i) yτ/yb = 3/2, yµ/ys = 9/2, (ii) yτ/yb = 2, yµ/ys = 9/2, and (iii) yτ/yb = 2, yµ/ys = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' We extend these possibilities to three toy models, accounting also for the first family masses, and calculate their predictions for various nucleon decay rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' We also analyse how the requirement of gauge coupling unification constrains the masses of potentially light relic states testable at colliders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='03601v1 [hep-ph] 9 Jan 2023 Contents 1 Introduction 1 2 GUT scenario 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='1 Particle content 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2 Neutrino masses 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='3 Quark-lepton Yukawa ratios 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='4 Toy models 5 3 Numerical procedure 6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='1 Implementation of the charged fermion Yukawa sector 6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2 Implementation of the neutrino sector 6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='3 GUT scale parameters and low energy observables 6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='4 Fitting procedure 7 4 Results 7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='1 Benchmark points 8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2 Highest posterior densities 9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='1 Quark-lepton mass ratios 10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2 Intermediate-scale particle masses 10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='3 Nucleon decay width and GUT scale 12 5 Conclusion 12 Appendices 12 A Definition of new Yukawa couplings 12 B Renormalization group equations 14 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='1 Gauge couplings 14 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2 Yukawa matrices 16 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='3 Effective neutrino mass operator 20 1 Introduction Grand Unified Theories (GUTs) [1–6] are arguably one of the most appealing extensions of the Standard Model (SM) of particle physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' In 1974, a simple and elegant GUT based on the unifying gauge group SU(5) was proposed by H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Georgi and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Glashow (GG model) [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' However, this model is incompatible with the current experimental data for three main reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Firstly, the GG model does not allow for gauge coupling unification, which – 1 – is a necessary condition for a GUT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Secondly, it predicts massless neutrinos, which is in conflict with neutrino oscillation experiments requiring that at least two neutrino should be massive [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Thirdly, since the SM Higgs doublet is embedded into a 5-dimensional Higgs representation of SU(5), the GG model predicts the GUT scale relation between the charged lepton and down-type quark Yukawa matrices Ye = Y T d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='1) This relation in particular implies a GUT scale unification of the tau and bottom Yukawa couplings yτ = yb, as well as a unification of the muon and strange Yukawa couplings yµ = ys, which disagrees with the low energy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' The first shortcoming requires extending the particle content of the minimal model by additional GUT representations and suitably splitting the masses of their component fields such that the running gauge couplings meet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' The second shortcoming can be addressed by introducing SU(5) representations that allow neutrino mass generation at the tree level [8–14] or at the loop level [15–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' the third shortcoming can for instance be resolved by generating the Yukawa couplings from linear combinations of the renormalisable and higher dimensional non-renormalisable operators [21],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' or at the renormalizable level by either introducing a 45-dimensional Higgs field and considering linear combinations of couplings between the SM fermions and both the 5- as well as the 45-dimensional Higgs field [22],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' or by introducing vector-like fermions which mix with the SM fermions [23–25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' However, historically a first and very aesthetic solution for the third problem was proposed by H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Georgi and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Jarlskog (GJ model) in 1979 [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' In their model, the particle content of the GG model is extended by a 45-dimensional Higgs field (as well as by two 5-dimensional Higgs fields).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' If the 45-dimensional Higgs field couples to the SM fermions this gives rise to the GUT scale relation Ye = −3Y T d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2) Considering a linear combination of the operators giving the relations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='1) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2) would, on the one hand, solve the shortcoming (as already mentioned above), but, on the other hand, predictivity in the Yukawa sector would be lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Predictivity is however maintained if it is ensured that different generations of charged leptons and down-type quarks couple to different Higgs fields (which can, for example, be achieved when a family symmetry is introduced on top of the gauge symmetry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' To achieve predictivity, without referring to any particular family symmetry, the GJ model hypothesizes the following textures of the Yukawa coupling matrices, Yd = � � � 0 B 0 A C 0 0 0 D � � � , Y T e = � � � 0 B 0 A −3C 0 0 0 D � � � , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='3) implying the GUT scale relations yτ/yb = 1, yµ/ys = −3, ye/yd = −1/3 which were at that time compatible with the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' However, the current data suggests (taking – 2 – only the known SM particles into account in the renormalization group (RG) evolution) that other ratios such as yτ/yb = 3/2, yµ/ys = 9/2 are better suited (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' [27]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Interestingly, these latter ratios can be obtained from higher dimensional operators [28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' With these higher dimensional operators at hand, models similar to the GJ model can be build if the following two conditions are satisfied: (i) the Yukawa matrices should be hierarchical, (ii) the 22- and 33- entry should be dominated by a single GUT operator, a concept which is referred to as single operator dominance [28–30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='1 Following this approach, non-SUSY GUT scenarios in which neutrino masses are gen- erated by a type I or a type II seesaw have been investigated in [27], respectively [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' For GUT scenarios with a type I seesaw it was shown that the GUT scale ratios yτ/yb = 3/2 and yµ/ys = 9/2 are compatible with the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Moreover, for GUT scenarios in which neutrino masses are generated by a type II seesaw it was found, that two combi- nations of GUT scale relations are viable, namely (i) yτ/yb = 3/2 and yµ/ys = 9/2 and (ii) yτ/yb = 2 and yµ/ys = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' In this paper we will investigate the viability of such GUT scale ratios for the case that neutrino masses stem from a combination of a type I [46–50] and a type III [51] seesaw mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' In this regard, we will consider a GUT scenario in which the particle content of the GG model is extended by a fermionic adjoint representation as well as by a 45-dimensional Higgs field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2 The former representation is needed to generate neutrino masses, while the latter gives rise to operators yielding potentially viable GUT scale Yukawa ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Moreover, both of these representations help to allow for gauge coupling unification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Using the Mathematica package ProtonDecay [52] and extending the above scenario to “toy models” we also compute the nucleon decay widths for various decay channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Finally, we compute the masses of the added fermion and scalar fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' The paper is organized as follows: While the GUT scenario as well as the toy models are introduced in Section 2, the procedure for the numerical analysis is explained in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' In Section 4 the results are presented and discussed, before concluding in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' In Appendix A, definitions of the newly introduced Yukawa couplings are given, while all relevant RGEs that we have derived are listed in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' 2 GUT scenario 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='1 Particle content The SM fermions are embedded as usual into three generations of 5F i and 10F i 5F i = dc i(3, 1, 1 3) ⊕ ℓi(1, 2, −1 2), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='1) 10F i = qi(3, 2, 1 6) ⊕ uc i(3, 1, −2 3) ⊕ ec i(1, 1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2) In the considered scenario, neutrino masses are generated via a combination of a type I and a type III seesaw mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' The corresponding fermionic singlet Σc and triplet Σb (under 1For models in which the concept of single operator dominance has been applied, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' [31–44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' 2A non-supersymmetric SU(5) GUT with this particle content was first considered in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' However, so far it has not been studied under the assumption of single operator dominance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' – 3 – SU(2)L) are contained in an adjoint fermionic representation 24F = Σa(8, 1, 0) ⊕ Σb(1, 3, 0) ⊕ Σc(1, 1, 0) ⊕ Σd(3, 2, −5 6) ⊕ Σe(3, 2, 5 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='3) Moreover, the GUT Higgs fields decompose under the SM gauge group as 24H = Φa(8, 1, 0) ⊕ Φb(1, 3, 0) ⊕ Φc(1, 1, 0) ⊕ Φd(3, 2, −5 6) ⊕ Φe(3, 2, 5 6), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='4) 5H = Ta(3, 1, −1 3) ⊕ Ha(1, 2, 1 2), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='5) 45H = φa(8, 2, 1 2) ⊕ φb(6, 1, −1 3) ⊕ φc(3, 3, −1 3) ⊕ φd(3, 2, −7 6) ⊕ φe(3, 1, −4 3) ⊕ Tb(3, 1, −1 3) ⊕ Hb(1, 2, 1 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='6) After the SU(5) breaking, the color triplets Ta and Tb mix to yield the mass eigenstates t1 = cos(α)Ta +sin(α)Tb and t2 = − sin(α)Ta +cos(α)Tb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Similarly, Ha and Hb mix to form the mass eigenstates h1 = cos(β)Ha + sin(β)Hb and h⊥ 2 = − sin(β)Ha + cos(β)Hb, where h1 is the SM Higgs doublet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2 Neutrino masses At tree-level the relevant GUT operators for neutrino mass generation read3 L ⊃ YA 5F 24F 5H + YB 5F 24F 45H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='7) After the GUT symmetry breaking the following relevant terms emerge L ⊃ −Y2ℓΣbHa − Y8ℓΣbHb − Y4ℓΣcHa − Y13ℓΣcHb − mΣbΣbΣb − mΣcΣcΣc, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='8) where mΣb and mΣb are the respective masses of Σb and Σc, and where the GUT scale relations Y2 = − � 3 10 YA, Y4 = YA, Y8 = √ 5 4 YB, and Y13 = √ 3 4 YB (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='9) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' After the SU(2) triplet Σb and SU(2) singlet Σc have been integrated out and the two Higgs fields Ha and Hb have taken their vacuum expectation values (vevs) va and vb, where v2 a + v2 b = v2 = (246 GeV)2, and where va = v cos(β) and vb = v sin(β), the neutrino mass matrix mν reads mij ν = −(Y i 2 va + Y i 8 vb)(Y j 2 va + Y j 8 vb) 4mΣb − (Y i 4 va + Y i 13 vb)(Y j 4 va + Y j 13 vb) 4mΣc .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='10) Since the neutrino mass matrix mν is of rank two, two massive and one massless neutrino are predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' 3After the GUT symmetry breaking these two GUT operators decompose into 19 SM Yukawa interac- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' For details see Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' – 4 – 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='3 Quark-lepton Yukawa ratios With X and Y representing one or multiple Higgs fields, the charged fermion masses stem from GUT operators of the form Y ij 5 : 10F i5F jX ⊃ Y ij d , Y ij e (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='11) Y ij 10 : 10F i10F jY ⊃ Y ij u , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='12) where Yu, Yd and Ye denote the usual SM charged fermion Yukawa matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Assuming in the charged fermion Yukawa sector the concept of single operator dominance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' that each Yukawa entry is dominated by a singlet GUT operator, allows to connect the down- type with the charged lepton Yukawa matrix via group theoretical Clebsch-Gordan (CG) factors cij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' In SU(5) GUTs, and considering up to dimension five operators, the potentially viable CG factors are |cij| ∈ {1/6, 1/2, 2/3, 1, 3/2, 2, 3, 9/2, 6, 9, 18}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' The possible GUT operators yielding these ratios are given in [28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Moreover, if the matrix Y5 is assumed to be of hierarchical nature and dominated by its diagonal entries, then the second and third family down-type quark and charged lepton masses stem dominantly from the GUT operators O2 and O3 dominating the 22 and 33 positions in Y5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Depending on which operators are chosen for O2 and O3, different GUT scale Yukawa ratios yτ/yb and yµ/ys are predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Our numerical analysis (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Section 4) shows that there are only two possible choices for the GUT scale ratio yτ/yb, namely 3/2 or 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' The former CG factor can be complemented by a factor 9/2 for the second family, while for the latter CG factor two different completions, yµ/ys = 6 or yµ/ys = 9/2, are possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='4 Toy models We now extend the above motivated scenarios to three toy models which also include the first family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' For simplicity, we chose the matrix Y5 to be of diagonal nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' The double ratio (yµyd)/(yeys) = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='7+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='6 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='9, which is nearly constant under renormalization group running (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' [53]), suggests, that the the ratio yµ/ys = 9/2 is best complemented by a ratio ye/yd = 4/9, while the best completion of the ratio yµ/ys = 6 is given by ye/yd = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Utilizing these ratios, our three toy models relate the down-type with the charged lepton Yukawa matrix via Model 1: Ye = diag �4 9, 9 2, 3 2 � Y T d , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='13) Model 2: Ye = diag �4 9, 9 2, 2 � Y T d , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='14) Model 3: Ye = diag �1 2, 6, 2 � Y T d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='15) Moreover, for simplicity4 we assume in each toy model that Y10 is dominated by the operator 10F 10F 5H in all entries, yielding a symmetric up-type Yukawa matrix, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Yu = 4We might consider higher-dimensional operators also for Y10, for example to explain the mass hierarchy, however since no Yukawa ratio predictions arise from this sector, we stick to the simplest case in our toy models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' – 5 – Y T u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Finally, in all toy models neutrino masses stem from a linear combination of the operators 5F 24F 5H and 5F 24F 45H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' 3 Numerical procedure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='1 Implementation of the charged fermion Yukawa sector We implement all three toy models at the GUT scale as described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' In all three models the down-type Yukawa matrix Yd is simply implemented as Yd = diag(yd 1, yd 2, yd 3), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='1) while the charged lepton Yukawa matrix Ye is implemented according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='13), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='14), and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='15), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Since Yu is symmetric we use a Takagi decomposition and imple- ment it as Yu = U † uY diag u U ∗ u, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2) where5 Uu = � � � 1 0 0 0 cu 23 su 23 0 −su 23 cu 23 � � � � � � cu 13 0 su 13e−iδu 0 1 0 −su 13eiδu 0 cu 13 � � � � � � cu 12 su 12 0 −su 12 cu 12 0 0 0 1 � � � � � � eiβu 1 0 0 0 eiβu 2 0 0 0 1 � � � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='3) and where Y diag u = diag(yu 1, yu 2, yu 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2 Implementation of the neutrino sector In order to simplify the analysis we assume in the neutrino sector that the Yukawa matrices Y5 and Y6 (for the definitions of these couplings, see Appendix A) are of the form Y5 = z1 � � � 0 1 1 � � � , Y6 = z2 � � � 1 1 3 � � � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='4) where z1 and z2 are real parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Furthermore, we denote the relative phase difference between mΣb and mΣc by γ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' γ = arg(mΣb/mΣc)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' This structure is motivated by CSD3 [56] which in the case of type I seesaw has been shown to correctly describe the low-scale neutrino observables together with a normal neutrino mass hierarchy (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' [27] for a recent work).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='3 GUT scale parameters and low energy observables Each toy model contains 33 input parameters which decompose into the GUT scale MGUT, the SU(5) gauge coupling gGUT, the masses of the added particles,6 mΦa, mΦb, mφa, mφb, 5Here we have dropped three unphysical parameters but kept the GUT phases βu 1 and βu 2 which effect the nucleon decay widths [54, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' 6Note that mΣd = mΣe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' – 6 – mφc, mφd, mφe, mΣa, mΣb, mΣc, mΣd, mt1, mt2, mh2, the singular values yu 1, yu 2, yu 3, yd 1, yd 2, yd 3 and angles θu 12, θu 13, θu 23 as well as phases δu, βu 1 , βu 2 of the charged fermion Yukawa matrices, the parameters of the neutrino Yukawa couplings z1, z2, and γ, and the eigenstate mixing angles α and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' The respective ranges of these input parameters are given by7 MGUT < MPl, mΦa, mΦb, mφa, mφb, mφc, mφd, mφe, mΣa, mΣb, mΣc, mΣd, mh2 ∈ [1 TeV, MGUT], mt1, mt2 ∈ [1011 GeV, MGUT], gGUT, yu 1, yu 2, yu 3, yd 1, yd 2, yd 3 ∈ [0, 1], (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='5) θu 12, θu 13, θu 23, α, β ∈ [0, π/2], δu, βu 1 , βu 2 , γ ∈ [−π, π), z1, z2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' These input parameters are fitted to the 22 low-scale observables (listed in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='6)) and the nucleon decay widths of thirteen decay channels (listed in Table I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' g1, g2, g3, yu, yc, yt, yd, ys, yb, θCKM 12 , θCKM 13 , θCKM 23 , δCKM, ye, yµ, yτ, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='6) ∆m2 21, ∆m2 31, θPMNS 12 , θPMNS 13 , θPMNS 23 , δPMNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' For the SM gauge couplings and Yukawa observables we take the experimental values from [53], while the values for the neutrino sector are taken from NuFIT 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='1 [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='4 Fitting procedure After implementing the input parameters given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='5) at the GUT scale we compute the RG evolution to the Z scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' For the gauge couplings we use a 2-loop running, while we compute the running of the Yukawa matrices and the effective neutrino mass operator at 1-loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' The nucleon decay widths are computed using the Mathematica package Proton Decay [52] (for a description of the calculation see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' [27]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Taking into account all observ- ables we compute at the low scale the χ2-function which we minimize using a differential evolution algorithm giving us a benchmark point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' With a flat prior distribution we calcu- late 4 × 106 data points performing a Markov-chain-Monte-Carlo (MCMC) analysis using an adaptive Metropolis-Hastings algorithm [65] which we start from this benchmark point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' These data points are finally used to compute the highest posterior density (HPD) ranges of various quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' 4 Results The results of our numerical analysis are presented in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' We are in particular interested in the nucleon decay predictions, the intermediate-scale particle masses as well 7Note that although we do not put any perturbativity constraints on the neutrino Yukawa couplings z1 and z2 the fit automatically choses them to be below 1 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' – 7 – decay channel τ/B [year] Γpartial [GeV] Reference Proton: p → π0 e+ > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='4 · 1034 < 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='7 · 10−67 [58] p → π0 µ+ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='6 · 1034 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='3 · 10−66 [58] p → η0 e+ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='0 · 1034 < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='0 · 10−66 [59] p → η0 µ+ > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='7 · 1033 < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='4 · 10−66 [59] p → K0 e+ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='1 · 1033 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='9 · 10−65 [60] p → K0 µ+ > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='6 · 1033 < 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='8 · 10−66 [61] p → π+ ν > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='9 · 1032 < 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='3 · 10−65 [62] p → K+ ν > 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='6 · 1033 < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2 · 10−66 [63] Neutron: n → π− e+ > 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='3 · 1033 < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='9 · 10−66 [59] n → π− µ+ > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='5 · 1033 < 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='9 · 10−66 [59] n → π0 ν > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='1 · 1033 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='9 · 10−65 [62] n → η0 ν > 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='6 · 1032 < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='7 · 10−65 [60] n → K0 ν > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2 · 1032 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='7 · 10−64 [60] Table I: Current experimental bounds on the decay widths Γpartial, respectively lifetime τ/B at 90 % confidence level, where B is the branching ratio for the decay channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' See also [64] for future projections and sensitivities of various upcoming detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' as the low scale predictions for the charged lepton and down-type quark mass ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='1 we show the results of our minimization procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Starting an MCMC analysis from these benchmark points allows us to obtain the HPD ranges of various quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' The results of this analysis is presented in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='1 Benchmark points We obtain for all three models benchmark points through a minimization of the χ2-function as described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' In Table II the input parameters for the respective benchmark points are listed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Moreover, the dominant pulls χ2 i are presented in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' All three models can be very well fitted to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' The strongest (though quite small) pull is given by the first and second family down-type quark masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' The biggest difference between the three models is the respectively favored GUT scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' For Models 2 and 3 a GUT scale above 1017 GeV is favored, while for the benchmark point of Model 1 a GUT scale below 1016 GeV is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' This also results in different results for the predicted nucleon decay rates (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Another difference is the preferred choice of some of the intermediate-scale particle masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' In the presented benchmark point the mass of the fermionic field Σa is obtained to be at the GUT scale for Model 1, at the intermediate scale for Model 2 and at the relatively low scale (23 TeV) for Model 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Moreover, a mass of the leptoquark φc of 1 – 8 – TeV, respectively 4 TeV is obtained for Model 3, respectively Model 2, whereas for Model 1 the mass of this field is above 106 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' For the HPD results of these particle masses confer the subsequent section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Model 1 Model 2 Model 3 gGUT / 10−1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='94 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='17 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='33 log10(MGUT / GeV) 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='6 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='3 log10(mφa / GeV) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='43 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='0 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='7 log10(mφc / GeV) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='02 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='63 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='00 log10(mΣa / GeV) 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='6 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='53 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='36 log10(mΣb / GeV) 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='9 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='7 log10(mΣc / GeV) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='8 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='8 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2 log10(mΣd / GeV) 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='9 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='1 yu 1 / 10−6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='63 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='99 yu 2 / 10−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='46 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='37 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='18 yu 3 / 10−1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='54 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='26 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='65 yd 1 / 10−6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='21 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='30 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='46 yd 2 / 10−4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='31 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='99 yd 3 / 10−3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='64 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='01 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='36 z1 / 10−1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='50 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='42 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='86 z2 / 10−1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='50 γ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='85 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='48 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='68 α 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='50 Table II: The GUT scale input parameters of the benchmark points for all three models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' χ2 χ2 yd χ2 ys χ2 yb χ2 yµ χ2 yτ χ2 Γ(p→π0e+) Model 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='44 Model 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='00 Model 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='00 Table III: The total χ2 as well as the dominant pulls χ2 i for the benchmark points of all three models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2 Highest posterior densities As described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='4 we vary the input parameters listed in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='5 around their benchmark points (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Table II) using an MCMC analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' From these generated points we then compute the HPD intervals of various parameters and observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' – 9 – Model 1 Model 2 Model 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='19 ye/yd HPD intervals for ye/yd Model 1 Model 2 Model 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='1 yμ/ys HPD intervals for yμ/ys Model 1 Model 2 Model 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='63 yτ/yb HPD intervals for yτ/yb Figure 1: Low scale (MZ) HPD intervals for charged lepton and down-type quark Yukawa ratios of all three families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' The 1σ (2σ) HDP intervals are colored dark (light).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' In Figures 1 – 4 we use the following color coding: For Model 1, 2, and 3 the HPD intervals of various quantities are colored red, green, and blue, respectively, while the 1σ (2σ) HPD intervals are colored dark (light).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='1 Quark-lepton mass ratios The HPD results for the low scale charged lepton and down-type quark mass ratios are presented in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' The horizontal dashed line represents the current experimental central value, whereas the white region shows the current experimental 1σ range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Clearly, all three models are capable of reproducing viable mass ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' This strengthens the results of the benchmark points in the previous subsection (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Tables II and III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Compared to Model 2 and 3, Model 1 gives a bit smaller predictions for the mass ratios for all three generations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2 Intermediate-scale particle masses Figure 2 shows the predicted HPD intervals of the intermediate-scale particle masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Most of the masses are predicted to be out of the reach of current and future colliders, because they would either produce too much proton decay, spoil gauge coupling unification or be- cause of the fit of the fermion masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' But interestingly, the fields Φb, φc and Σa are not only potentially within the reach of future searches, but can also be used to distinguish be- tween the different models: An observation of the one of the fields Φb or Σa would strongly hint towards Model 3, while an observation of the field φc would disfavor Model 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' In fact, the most promising lookout could be for the leptoquark φc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' The upper bound of the HPD 1σ range is predicted to be 23 TeV (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='8 TeV) in Model 2 (3), whereas the upper bound of the 2σ intervals is 175 TeV (17 TeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' In the following, we briefly state the current collider bounds on these particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' The scalar triplet, Φb, with zero hypercharge, residing in the 24H multiplet is expected to be light in Model 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Note that Φb contains a neutral Φ0 b and a pair of singly charged Φ± b states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' In the low-energy effective theory, a term of the form h† 1Φ2 bh1 is allowed, where – 10 – mΦa mΦb mTa mTb m ϕa m ϕb m ϕc m ϕd m ϕe mΣa mΣb mΣc mΣd Model 1 Model 2 Model 3 2 4 6 8 10 12 14 16 18 log10(μ/GeV) HPD intervals for intermediate-scale particle masses Figure 2: HPD intervals of the intermediate-scale particle masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' The 1σ (2σ) HDP intervals are colored dark (light).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' h1 is the SM Higgs doublet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' As a result of this coupling, the SM Higgs can decay into two photons h0 → γγ via a one-loop diagram mediated by the Φ± b states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Consistency with the LHC data requires these charged states to have masses above 250 GeV [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' The scalar leptoquark φc, which is a triplet of SU(2)L, resides around the TeV scale in Models 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' In both models, its coupling to the SM fermions is dominated by the third-generation quark and lepton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Hence, within our scenarios, its decay branching fraction is dominated by a bτ final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Since leptoquarks carry color, they are efficiently produced at the LHC through gluon-initiated as well as quark-initiated processes [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' LHC searches of pp → bbττ from pair-produced leptoquarks rule out leptoquark masses below 1400 GeV [68, 69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' As can be seen from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2), the color octet fermion Σa, which is expected to be light in Model 3, couples, for example, to a singlet down-quark (lepton doublet) and a super-heavy colored triplet (octet) scalar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Consequently, the lifetime of a TeV scale Σa is expected to be large, and it behaves like a long-lived gluino that typically arises in split- supersymmetric scenarios [70, 71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Long-lived colored particles would hadronize, forming so-called R-hadrons [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' These bound states are comprised of the long-lived state and light SM quarks or gluons, and interact with the detector material, typically inside the calorimeters, via hadronic interactions of the light-quark constituents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Motivated by split- supersymmetric models, R-hadrons are extensively searched for at the LHC [73, 74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Non- observation of any deviations of the signal from the expected background puts to a lower – 11 – limit on the mass of the long-lived Σa fermion of 2000 GeV [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='3 Nucleon decay width and GUT scale Figure 3 shows the predictions for the HPD intervals of the GUT scale MGUT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Moreover, the predicted HPD ranges for the nucleon decay widths of the various decay channels are presented in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' The blue line segments in the latter picture indicate the current experimental bounds at 90 % confidence level (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Table I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Moreover, the future constraints on the decay widths for the decay channels p → π0e+ and n → π−e+ which will be provided by Hyper-Kamiokande [75] are presented by orange line segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' In Figure 3 it can be seen that Model 1 clearly predicts the GUT scale to be below 1016 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' On the other hand, a much larger GUT scale is preferred by the Models 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Since the nucleon decay width is inversely proportional to the forth power of the GUT scale in the case of gauge boson mediated nucleon decay, this also results in strongly different prediction for the nucleon decay widths of the various channels as it can be seen in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' The nucleon decay predictions for Model 1 are very close to the current bounds, the 1σ HPD interval of the proton decay channel p → π0e+ will be fully probed by Hyper-Kamiokande.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Moreover, Hyper-Kamiokande will probe most of the 1σ HPD interval of the neutron decay channel n → π−e+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' On the other hand, the gauge boson mediated nucleon decay is highly suppressed in Models 2 and 3 and cannot be probed by any planed experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Therefore, observation of nucleon decay in the decay channels p → π0e+ and n → π−e+ would clearly favour Model 1 over the Models 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' 5 Conclusion In this paper we considered an extension of the Georgi-Glashow SU(5) GUT scenario by a 45-dimensional scalar and a 24-dimensional fermionic representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Neutrino masses in this scenario are generated by a combination of a type I and a type III seesaw mech- anism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Assuming the concept of single operator dominance we investigated which GUT scale charged lepton and down-type quark Yukawa ratios can be viable for the second and third family and found that three combinations work: (i) yτ/yb = 3/2, yµ/ys = 9/2, (ii) yτ/yb = 2, yµ/ys = 9/2, and (iii) yτ/yb = 2, yµ/ys = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Also taking into account the origin of the first family masses we extended these possibilities to three toy models and analyzed various of their predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' We showed that experimental discrimination between these models could be possible since they predict different nucleon decay rates as well as distinct light relics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Appendices A Definition of new Yukawa couplings The Lagrangian density contains the two terms L ⊃ YA 5i F 24F 5H + YB 5i F 24F 45H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='1) – 12 – Model 1 Model 2 Model 3 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='5 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='0 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='5 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='5 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='0 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='5 MGUT HPD intervals for MGUT Figure 3: Predicted HPD intervals of the GUT scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' The 1σ (2σ) HDP intervals are colored dark (light).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' p → π0e+ p → π0 μ+ p → η0e+ p → η0 μ+ p → K0e+ p → K0 μ+ p → π+ν p → K+ν n → π-e+ n → π- μ+ n → π0ν n → η0ν n → K0ν Model 1 Model 2 Model 3 85 80 75 70 65 log10(Γ/GeV) HPD intervals for decay widths Γ of different nucleon decay channels Figure 4: Predicted HPD intervals of the nucleon decay widths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' The 1σ (2σ) HDP intervals are colored dark (light).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' For each decay channel the blue line segments represent the current experimental constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' The future Hyper-Kamiokande constraints for the decay channels p → π0e+ and n → π−e+ are indicated by orange line segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='– 13 – ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='After the GUT symmetry breaking they decompose into ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='L = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='15YA dcΣcTa − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='10YA ℓΣcHa + YA dcΣaTa + YA ℓΣbHa+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='YA dcΣdHa + YA ℓΣeTa + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='12YB dcΣcTb + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='4 YB ℓΣcHb+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2YB dcΣaTb + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2YB dcΣaφb + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2YB ℓΣaφa + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2YB dcΣbφc+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='4 YB ℓΣbHb − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2YB dcΣdφa − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='6YB dcΣdHb + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2YB ℓΣdφe− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2YB dcΣeφd + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2YB ℓΣeTb − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2YB ℓΣeφc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='≡ Y1 dcΣcTa + Y2 ℓΣcHa + Y3 dcΣaTa + Y4 ℓΣbHa+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='Y5 dcΣdHa + Y6 ℓΣeTa + Y7 dcΣcTb + Y8 ℓΣcHb+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='Y9 dcΣaTb + Y10 dcΣaφb + Y11 ℓΣaφa + Y12 dcΣbφc+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='Y13 ℓΣbHb + Y14 dcΣdφa + Y15 dcΣdHb + Y16 ℓΣdφe+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='Y17 dcΣeφd + Y18 ℓΣeTb + Y19 ℓΣeφc ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2) where we defined the Yukawa matrices YN, with N = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' , 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' B Renormalization group equations Here the RGEs for the gauge and Yukawa couplings as well as for the effective neutrino mass operator are listed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' We have used the Mathematica package SARAH [76, 77] to obtain the RGEs for the gauge and Yukawa couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' The SM contribution for the RGE of the effective neutrino mass operator is taken from [78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' In order to compute the new contribution for this RGE we have used the method described therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' We use the following definition for the Heaviside-Theta function H(µ, m) = � 1, for µ ≥ m, 0, for µ < m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='1) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='1 Gauge couplings The RGEs for gauge couplings (i, k = 1 − 3) are given by µdgi dµ = βgi 1−loop 16π2 + βgi 2−loop (16π2)2 , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2) where βgi 1−loop is the 1-loop and βgi 2−loop is the 2-loop contribution given by βgi 1−loop = � aSM i + H(µ, m)∆ai � g3 i (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='3) βgi 2−loop = � k bSM ik g2 k + � k ∆bikg2 k H(µ, m) + βY,SM i + ∆βY i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='4) – 14 – Here, aSM i , bSM ik and βY,SM i are the well known SM 1-loop and 2-loop coefficients as well as Yukawa contributions [79, 80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' Moreover, the ∆βY i are given by ∆βY i = g3 i � k cikY T k Y ∗ k H2 k, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='5) where we introduced the abbreviation H2 k = H(µ, mF )H(µ, mH) associated to each of the Yukawa interactions, where, F and H refer to the BSM fermion and scalar appearing in that interaction, respectively, and where the cik are given by c1k = − �1 5, 3 10, 8 15, 9 20, 29 10, 17 5 , 1 5, 3 10, 8 15, 8 15, 12 5 , 3 5, 9 20, 116 15 , 29 10, 17 5 , 29 5 , 17 5 , 51 10 � , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='6) c2k = − � 0, 1 2, 0, 11 4 , 3 2, 3, 0, 1 2, 0, 0, 4, 6, 11 4 , 4, 3 2, 3, 3, 3, 9 2 � , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='7) c3k = − �1 2, 0, 13 3 , 0, 2, 1, 1 2, 0, 13 3 , 13 3 , 6, 3 2, 0, 16 3 , 2, 1, 4, 1, 3 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='8) Finally, the ∆ai and ∆bi are given as a sum over the 1-loop and 2-loop coefficients of the BSM fermions and scalars, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' ∆ai = � I ∆aI i , ∆bi = � I ∆bI i , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='9) where I runs over all BSM particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' The 1-loop coefficients are then given by ∆aφa i = �4 5, 4 3, 2 � , ∆aφb i = � 2 15, 0, 5 6 � , ∆aφc i = �1 5, 2, 1 2 � , ∆aφd i = �49 30, 1 2, 1 3 � , ∆aφe i = �16 15, 0, 1 6 � , ∆aΦa i = {0, 0, 1 2}, ∆aΦb i = � 0, 1 3, 0 � , ∆aΣa i = {0, 0, 2}, ∆aΣb i = � 0, 4 3, 0 � , ∆aΣd,e i = �5 3, 1, 2 3 � , ∆ah⊥ i = � 1 10, 1 6, 0 � , ∆at,t⊥ i = � 1 15, 0, 1 6 � , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='10) whereas the 2-loop coefficients read ∆bφa ik = � � � 36 25 36 5 144 5 12 5 52 3 48 18 5 18 84 � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' ∆bφb ik = � � � 8 75 0 16 3 0 0 0 2 3 0 115 3 � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' ∆bφc ik = � � � 4 25 24 5 16 5 8 5 56 32 2 5 12 11 � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' ∆bφd ik = � � � 2401 150 147 10 392 15 49 10 13 2 8 49 15 3 22 3 � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' ∆bφe ik = � � � 1024 75 0 256 15 0 0 0 32 15 0 11 3 � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' ∆bΦa ik = � � � 0 0 0 0 0 0 0 0 21 � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' ∆bΦb ik = � � � 0 0 0 0 28 3 0 0 0 0 � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' ∆bΣa ik = � � � 0 0 0 0 0 0 0 0 48 � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' ∆bΣb ik = � � � 0 0 0 0 64 3 0 0 0 0 � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' ∆bΣd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='e ik = � � � 25 12 15 4 20 3 5 4 49 4 4 5 6 3 2 38 3 � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' ∆bh⊥ ik = � � � 9 50 9 10 0 3 10 13 6 0 0 0 0 � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' ∆bt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='t⊥ ik = � � � 4 75 0 16 15 0 0 0 2 15 0 11 3 � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='11) – 15 – B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2 Yukawa matrices The RGEs of the Yukawa matrices read µdYf dµ = βf 16π2 , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='12) where f = {u, d, e, k} and (k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' , 19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' For the SM Yukawa matrices Yu, Yd and Ye (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' f = u, d, e) the beta functions are given by βf = βSM f + δβf, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='13) where βSM f is the SM beta function [79, 80], and where δβf = YfT1 + � k af k(Yk)j(Y T d Y ∗ k )i H2 k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='14) Here, we have defined T1 as T1 = Y T 2 Y ∗ 2 H2 2 + 3 2Y T 4 Y ∗ 4 H2 4 + 3Y T 5 Y ∗ 5 H2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='15) while the af k are given by au k = � 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 � , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='16) ad k = �1 2, 0, 4 3, 0, 3, 0, 1 2, 0, 4 3, 4 3, 0, 3 2, 0, 8 3, 1, 0, 2, 0, 0 � , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='17) ae k = � 0, −3 2, 0, 15 4 , 0, 3 2, 0, 1 2, 0, 0, 4, 0, 3 4, 0, 0, 3 2, 0, 3 2, 9 4 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='18) In order to simplify the notation, from hereon, associated to each Yukawa Yi → Yi H2 i must be understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' The beta function of the Yukawa matrices Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' , Y19 then read β1 = Y1 � −1 5g2 1 − 4g2 3 + � k a1 kY T k Y ∗ k � + � YdY † d � Y1 + � w b1 w � Y T 1 Y ∗ w � Yw + 8 3 � Y T 3 Y ∗ 9 � Y7 + 2 � Y T 6 Y ∗ 18 � Y7, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='19) β2 = Y2 � − 9 20g2 1 − 9 4g2 2 + � k a2 kY T k Y ∗ k + T � + � −3 2Y T e Y ∗ e � Y2 + � w b2 w � Y T 2 Y ∗ w � Yw + 3 2 � Y T 4 Y ∗ 13 � Y8 + 3 � Y T 5 Y ∗ 15 � Y8, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='20) β3 = Y3 � −1 5g2 1 − 13g2 3 + � k a3 kY T k Y ∗ k � + � YdY † d � Y3 + � w b3 w � Y T 3 Y ∗ w � Yw + � Y T 1 Y ∗ 7 � Y9 + 2 � Y T 6 Y ∗ 18 � Y9, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='21) β4 = Y4 � − 9 20g2 1 − 33 4 g2 2 + � k a4 kY T k Y ∗ k + T � + �5 2Y T e Y ∗ e � Y4 + � w b4 w � Y T 4 Y ∗ w � Yw – 16 – + � Y T 2 Y ∗ 8 � Y13 + 3 � Y T 5 Y ∗ 15 � Y13, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='22) β5 = Y5 � −29 20g2 1 − 9 4g2 2 − 8g2 3 + � k a5 kY T k Y ∗ k + T � + � 3YdY † d � Y5 + � w b5 w � Y T 5 Y ∗ w � Yw + � Y T 2 Y ∗ 8 � Y15 + 3 2 � Y T 4 Y ∗ 13 � Y15, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='23) β6 = Y6 � −17 10g2 1 − 9 2g2 2 − 4g2 3 + � k a6 kY T k Y ∗ k � + �1 2Y T e Y ∗ e � Y6 + � w b6 w � Y T 6 Y ∗ w � Yw + � Y T 1 Y ∗ 7 � Y18 + 8 3 � Y T 3 Y ∗ 9 � Y18, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='24) β7 = Y7 � −1 5g2 1 − 4g2 3 + � k a7 kY T k Y ∗ k � + � YdY † d � Y7 + � w b7 w � Y T 7 Y ∗ w � Yw + � 2Y T 18Y ∗ 6 � Y1 + 8 3 � Y T 9 Y ∗ 3 � Y1, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='25) β8 = Y8 � − 9 20g2 1 − 9 4g2 2 + � k a8 kY T k Y ∗ k � + � Y T e Y ∗ e � Y8 + � w b8 w � Y T 8 Y ∗ w � Yw + 3 2 � Y T 13Y ∗ 4 � Y2 + 3 � Y T 15Y ∗ 5 � Y2, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='26) β9 = Y9 � −1 5g2 1 − 13g2 3 + � k a9 kY T k Y ∗ k � + � YdY † d � Y9 + � w b9 w � Y T 9 Y ∗ w � Yw + 2 � Y T 18Y ∗ 6 � Y3 + � Y T 7 Y ∗ 1 � Y3, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='27) β10 = Y10 � −1 5g2 1 − 13g2 3 + � k a10 k Y T k10Y ∗ k � + � YdY † d � Y10 + � w b10 w � Y T 10Y ∗ w � Yw, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='28) β11 = Y11 � − 9 20g2 1 − 9 4g2 2 − 9g2 3 + � k a11 k Y T k Y ∗ k � + � Y T e Y ∗ e � Y11 + � w b11 w � Y T 11Y ∗ w � Yw, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='29) β12 = Y12 � −1 5g2 1 − 6g2 2 − 4g2 3 + � k a12 k Y T k Y ∗ k � + � YdY † d � Y12 + � w b12 w � Y T 12Y ∗ w � Yw, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='30) β13 = Y13 � − 9 20g2 1 − 33 4 g2 2 + � k a13 k Y T k Y ∗ k � + �1 2Y T e Y ∗ e � Y13 + � w b13 w � Y T 13Y ∗ w � Yw + 3 � Y T 15Y ∗ 5 � Y4 + � Y T 8 Y ∗ 2 � Y4, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='31) β14 = Y14 � −29 20g2 1 − 9 4g2 2 − 8g2 3 + � k a14 k Y T k Y ∗ k � + � YdY † d � Y14 + � w b14 w � Y T 14Y ∗ w � Yw, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='32) β15 = Y15 � −29 20g2 1 − 9 4g2 2 − 8g2 3 + � k a15 k Y T k Y ∗ k � + � YdY † d � Y15 + � w b15 w � Y T 15Y ∗ w � Yw – 17 – + 3 2 � Y T 13Y ∗ 4 � Y5 + � Y T 8 Y ∗ 2 � Y5, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='33) β16 = Y16 � −17 10g2 1 − 9 2g2 2 − 4g2 3 + � k a16 k Y T k Y ∗ k � + �1 2Y T e Y ∗ e � Y16 + � w b16 w � Y T 16Y ∗ w � Yw, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='34) β17 = Y17 � −29 20g2 1 − 9 4g2 2 − 8g2 3 + � k a17 k Y T k Y ∗ k � + � YdY † d � Y17 + � w b17 w � Y T 17Y ∗ w � Yw, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='35) β18 = Y18 � −17 10g2 1 − 9 2g2 2 − 4g2 3 + � k a18 k Y T k Y ∗ k � + �1 2Y T e Y ∗ e � Y18 + � w b18 w � Y T 18Y ∗ w � Yw + � Y T 7 Y ∗ 1 � Y6 + 8 3 � Y T 9 Y ∗ 3 � Y6, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='36) β19 = Y19 � −17 20g2 1 − 9 2g2 2 − 4g2 3 + � k a19 k Y T k Y ∗ k � + �1 2Y T e Y ∗ e � Y19 + � w b19 w � Y T 19Y ∗ w � Yw, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='37) where the coefficients af k are given by a1 k = {3, 1, 8 3, 0, 0, 2, 3 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='38) a2 k = {3 2, 5 2, 0, 3 2, 3, 0, 3 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='39) a3 k = {1, 0, 9 2, 0, 0, 2, 0, 0, 1 2, 1 2, 1, 0, 0, 0, 0, 0, 0, 0, 0}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='40) a4 k = {0, 1, 0, 11 4 , 3, 0, 0, 0, 0, 0, 0, 3 2, 1 2, 0, 0, 0, 0, 0, 0}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='41) a5 k = {0, 1, 0, 3 2, 9 2, 0, 0, 0, 0, 0, 0, 0, 0, 4 3, 1 2, 1 2, 0, 0, 0}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='42) a6 k = {1, 0, 8 3, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1 2, 3 4}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='43) a7 k = {3 2, 1, 0, 0, 0, 0, 3, 1, 8 3, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='44) a8 k = {3 2, 1, 0, 0, 0, 0, 3 2, 5 2, 0, 0, 0, 0, 3 2, 0, 3, 0, 0, 0, 0}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='45) a9 k = {0, 0, 1 2, 0, 0, 0, 1, 0, 9 2, 1 2, 1, 0, 0, 0, 0, 0, 0, 2, 0}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='46) a10 k = {0, 0, 1 2, 0, 0, 0, 0, 0, 1 2, 19 6 , 1, 0, 0, 0, 0, 0, 0, 0, 0}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='47) a11 k = {0, 0, 1 2, 0, 0, 0, 0, 0, 1 2, 1 2, 6, 0, 0, 1, 0, 0, 0, 0, 0}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='48) a12 k = {0, 0, 0, 1 2, 0, 0, 0, 0, 0, 0, 0, 4, 1 2, 0, 0, 0, 0, 0, 1}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='49) a13 k = {0, 0, 0, 1 2, 0, 0, 0, 1, 0, 0, 0, 3 2, 11 4 , 0, 3, 0, 0, 0, 0}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='50) a14 k = {0, 0, 0, 0, 1 2, 0, 0, 0, 0, 0, 1, 0, 0, 5, 1 2, 1 2, 0, 0, 0}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='51) – 18 – a15 k = {0, 0, 0, 0, 1 2, 0, 0, 1, 0, 0, 0, 0, 3 2, 4 3, 9 2, 1 2, 0, 0, 0}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='52) a16 k = {0, 0, 0, 0, 1 2, 0, 0, 0, 0, 0, 0, 0, 0, 4 3, 1 2, 4, 0, 0, 0}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='53) a17 k = {0, 0, 0, 0, 0, 1 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 1 2, 3 4}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='54) a18 k = {0, 0, 0, 0, 0, 1 2, 1, 0, 8 3, 0, 0, 0, 0, 0, 0, 0, 1, 4, 3 4}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='55) a19 k = {0, 0, 0, 0, 0, 1 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1 2, 4}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='56) while the coefficients bf k read b1 w = {0, 0, 4 3, 0, 1, 0, 3 2, 0, 0, 4 3, 0, 3 2, 0, 8 3, 1, 0, 2, 0, 0}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='57) b2 w = {0, 0, 0, 3 4, 0, 3 2, 0, 3 2, 0, 0, 4, 0, 3 4, 0, 0, 3 2, 0, 3 2, 9 4}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='58) b3 w = {1 2, 0, 0, 0, 1, 0, 1 2, 0, 0, 4 3, 0, 3 2, 0, 8 3, 1, 0, 2, 0}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='59) b4 w = {0, 1 2, 0, 0, 0, 3 2, 0, 1 2, 0, 0, 4, 0, 9 4, 0, 0, 3 2, 0, 3 2, 9 4}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='60) b5 w = {1 2, 0, 4 3, 0, 0, 0, 1 2, 0, 4 3, 4 3, 0, 3 2, 0, 8 3, 4, 0, 2, 0, 0}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='61) b6 w = {0, 1 2, 0, 3 4, 0, 0, 0, 1 2, 0, 0, 4, 0, 3 4, 0, 0, 3 2, 0, 7 2, 9 4}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='62) b7 w = {3 2, 0, 4 3, 0, 1, 0, 0, 0, 4 3, 4 3, 0, 3 2, 0, 8 3, 1, 0, 2, 0, 0}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='63) b8 w = {0, 3 2, 0, 3 4, 0, 3 2, 0, 0, 0, 0, 4, 0, 3 4, 0, 0, 3 2, 0, 3 2, 9 4}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='64) b9 w = {1 2, 0, 4, 0, 1, 0, 1 2, 0, 0, 4 3, 0, 3 2, 0, 8 3, 1, 0, 2, 0, 0}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='65) b10 w = {1 2, 0, 4 3, 0, 1, 0, 1 2, 0, 4 3, 0, 0, 3 2, 0, 8 3, 1, 0, 2, 0, 0}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='66) b11 w = {0, 1 2, 0, 3 4, 0, 3 2, 0, 1 2, 0, 0, 0, 0, 3 4, 0, 0, 3 2, 0, 3 2, 9 4}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='67) b12 w = {1 2, 0, 4 3, 0, 1, 0, 1 2, 0, 4 3, 4 3, 0, 0, 0, 8 3, 1, 0, 2, 0, 0}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='68) b13 w = {0, 1 2, 0, 9 4, 0, 3 2, 0, 1 2, 0, 0, 4, 0, 0, 0, 0, 3 2, 0, 3 2, 9 4}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='69) b14 w = {1 2, 0, 4 3, 0, 1, 0, 1 2, 0, 4 3, 4 3, 0, 3 2, 0, 0, 1, 0, 2, 0, 0}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='70) b15 w = {1 2, 0, 4 3, 0, 4, 0, 1 2, 0, 4 3, 4 3, 0, 3 2, 0, 8 3, 0, 0, 2, 0, 0}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='71) b16 w = {0, 1 2, 0, 3 4, 0, 3 2, 0, 1 2, 0, 0, 4, 0, 3 4, 0, 0, 0, 0, 3 2, 9 4}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='72) b17 w = {1 2, 0, 4 3, 0, 1, 0, 1 2, 0, 4 3, 4 3, 0, 3 2, 0, 8 3, 1, 0, 0, 0, 0}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='73) b18 w = {0, 1 2, 0, 3 4, 0, 7 2, 0, 1 2, 0, 0, 4, 0, 3 4, 0, 0, 3 2, 0, 0, 9 4}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='74) b19 w = {0, 1 2, 0, 3 4, 0, 3 2, 0, 1 2, 0, 0, 4, 0, 3 4, 0, 0, 3 2, 0, 3 2, 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='75) – 19 – B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='3 Effective neutrino mass operator The RGE for the effective neutrino mass operator reads 16π2µdκ dµ = βSM κ + ∆βκ, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='76) where βSM κ is the SM contribution as given in [78] and ∆βκ is the correction due to the added BSM particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' For ∆βκ we find8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='∆βκ = κ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='�1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='2Y ∗ ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='13 + 6Y T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='15Y ∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='κ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content='77) References [1] J.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} +page_content=' – 24 –' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfBQa7/content/2301.03601v1.pdf'} diff --git a/PdAzT4oBgHgl3EQfzv5V/vector_store/index.faiss b/PdAzT4oBgHgl3EQfzv5V/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..02ab905b9f672e028a80fa2fba30f483b969d960 --- /dev/null +++ b/PdAzT4oBgHgl3EQfzv5V/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:8df995e7078a7d1f1edf9a262365349a784850eb5cfd86a3aaa2393fd0ee84db +size 3735597 diff --git a/PtAyT4oBgHgl3EQftfn_/content/tmp_files/2301.00598v1.pdf.txt b/PtAyT4oBgHgl3EQftfn_/content/tmp_files/2301.00598v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..6c65fe83592992c6d67e0b70e017e5896219b300 --- /dev/null +++ b/PtAyT4oBgHgl3EQftfn_/content/tmp_files/2301.00598v1.pdf.txt @@ -0,0 +1,1563 @@ +Double Half-Heusler Alloys X2Ni2InSb (X= +Zr/Hf) with promising Thermoelectric +Performance: Role of varying structural phases +Bhawna Sahni and Aftab Alam∗ +Department of Physics, Indian Institute of Technology, Bombay, Powai, Mumbai 400 076, +India +E-mail: aftab@phy.iitb.ac.in +Abstract +Double half-heusler alloys are the new class of compounds which can be seen as +transmuted version of two single half-heusler with higher flexibility of tuning their +properties. Here, we report a detailed study of thermoelectric (TE) properties of two +double half-heusler (HH) alloys X2Ni2InSb (X=Hf/Zr), using first-principles calcula- +tion. These alloys exhibit a rich phase diagram with the possibility of tetragonal, cubic +and solid solution phase at different temperature range. As such, a comparative study +of TE properties of all these phases is performed. The ordered phases show quite fa- +vorable electronic transport as compared to the disordered ones in both compounds. +Lattice thermal conductivity of double HH alloys is lower than their ternary counter- +part, making them most promising for TE application. Simulated band gap, obtained +using hybrid functional, of ordered phases of Hf2Ni2InSb and Zr2Ni2InSb lie in the +range 0.24-0.4 eV and 0.17-0.59 eV respectively, while for disordered phase, it is 0.05- +0.06 eV. Hf2Ni2InSb shows a reasonably high ZT value of ∼ 2.19, while Zr2Ni2InSb +yields 2.46 at high temperature for n-type conduction in tetragonal phase. The ZT +1 +arXiv:2301.00598v1 [cond-mat.mtrl-sci] 2 Jan 2023 + +value for p-type conduction is also quite promising (∼ 1.35 and ∼ 2.19 for Hf- and +Zr-based compounds). In both the compounds, electronic transport (Seebeck and elec- +trical conductivity) plays the dominant role for the high ZT-value. Keeping in mind +the promising TE performance, we propose immediate attention from experimentalists +to synthesize and cross validate our findings for these new candidate materials. +Keywords +Double half-heusler alloys, Thermoelectric, Ab-initio calculation, Carrier relaxation time, +Electronic and thermal transport +Introduction +Thermoelectric technology which enables to convert waste heat into electricity, has proven +to be extremely useful in providing solutions to renewable energy resources. Since most of +the energy from primary sources is lost as waste heat, potential thermoelectric materials +come as a rescue by harvesting this waste heat. Transport properties (Seebeck Coefficient +(S), electrical conductivity (σ) and thermal conductivity (κ) which define the thermoelectric +figure of merit (ZT), are closely interrelated, which makes it quite challenging to find novel +materials with optimal ZT. Though there are several materials reported in the literature,1–9 +the hunt for more efficient novel materials is still ongoing. +Half-Heusler (HH) alloys have emerged as promising thermoelectric materials due to a +variety of interesting properties such as good thermal stability, easily tunable band gaps, good +mechanical properties etc.10–14 They can be classified on the basis of the valence electron +count (VEC). The 18 valence-electron HH alloys are very stable because of fully occupied +bonding and empty anti-bonding states. The 17 and 19 valence-electron HH alloys on the +other hand, are unstable because of partially occupied states. +However, mixing of a 17 +and a 19 VEC HH alloy can form an 18 VEC double half heusler alloy. These double half +2 + +heusler alloys exhibit much lower values of lattice thermal conductivity (κL) as compared to +their ternary counterpart because of smaller phonon group velocity and disorder scattering. +Apart from κL, if the electronic transport properties of these double HH alloys can be made +more superior than the corresponding ternary systems, they can be very promising for TE +applications. This is one of the motivation of the present work. +Anand et al.15 explored a large number of unexplored double half-Heusler alloys and +predicted many of them stable. The double half-heusler compounds have a general formula +unit X2YY +′Z2 where Y and Y +′ are not isovalent, X is transition metal and Z is a main +group element. For example, the nominal net valence (NV)̸=0 systems such as TiNiSb and +TiFeSb are the two ternary components of the quaternary double half-Heusler compound, +Ti2FeNiSb2. +The members, namely TiFeSb, TiCoSb, and TiNiSb all have different net +valence (NV = -1, 0, and 1). Anand et al.15 showed that κL of double half-Heusler Ti2FeNiSb2 +is lower in comparison to that of its corresponding (NV = 0) ternary counterpart TiCoSb +(with the same average atomic mass) by a factor of 3 at room temperature. +There are +some reports on the effect of doping in these double half heuslers as well. For instance, +Liu et al.16 showed that by alloying Ti by Hf and by tuning Fe/Ni ratio, a high figure of +merit can be achieved for both p-type and n-type conduction in TiFe0.5Ni0.5Sb. Wang et al. +showed enhanced thermoelectric performance due to very low value of thermal conductivity +and high power factor in p-type double half heusler Ti2-yHfyFeNiSb2-xSnx compounds.17 +Recently, Hasan et al.18 showed enhanced figure of merit in Ti2FeNiSb1.8Sn0.2 as compared +to the pristine Ti2FeNiSb2. +Since the ternary half-heuslers ZrNiSn10,11,19 and HfNiSn11,20 have been widely studied +for reporting promising thermoelectric performance, we chose to study the corresponding +(NV=0) double half heusler counterparts Zr2Ni2InSb and Hf2Ni2InSb. Anand et al.15 theo- +retically studied the Gruniesen parameter and thermal conductivity of these compounds in +tetragonal (I-42d) structure as reported in the open quantum materials database (OQMD). +Both these compounds, however, are experimentally reported to crystallize in cubic (F¯43m +3 + +(#216)) structure with few impurities, prepared under a specific processing condition.21, 22 +In this letter, we use first-principles calculation to investigate the electronic, phonon and +thermoelectric properties of double half heuslers Zr2Ni2InSb and Hf2Ni2InSb. In contrast +to previous studies, we have simulated these properties in all of three relevant phases i.e., +cubic, tetragonal as well as solid solution, of these compounds. Depending on the synthesis +condition and temperature, there is a possibility to realize all these three phases, ordered +in low temperature (T) while disordered in high T-range. Ordered structures could show +high carrier mobility than solid solution.23 Thus, ordered structures with low lattice thermal +conductivity (κL) can prove to be better thermoelectric material. For an accurate estimate +of band gap, HSE06 functional is used, which yields a band gap of 0.17 eV (0.59 eV) for +cubic (tetragonal) phase of Zr2Ni2InSb, while the same for Hf2Ni2InSb are 0.24 eV (0.4 eV). +The corresponding solid solution phase shows a narrow band gap 0.05 eV and 0.06 eV for +Zr and Hf based double half heusler alloys. As expected the simulated κL for double half +heusler alloys are smaller than the ternary counterpart (ZrNiSn and HfNiSn). However these +values of κL is not small enough to give promising TE properties. Interestingly, these double +HH alloys are found to show extremely high electronic transport (S, σ and power-factor), +which actually makes them promising for TE application. Ordered phases are found to be +almost equally competitive with respect to their TE performance with figure of merit (ZT) +value as high as 2.46, at high T. Such comparative study of different structural phases of a +single compound is extremely essential and useful to understand the nature of electrons and +phonons excitation at different T-ranges. +Computational Details +We use Vienna Ab-initio Simulation Package (VASP)25, 26,,27 within DFT, with a projector +augmented wave basis28 and the generalized gradient approximation exchange-correlation +functional of Perdew−Burke−Ernzerhof (PBE).29 HSE0630 calculations including spin-orbit +4 + +coupling (SOC) were performed for the accurate estimation of band gaps. A plane-wave +energy cutoff of 500 eV was used. The Brillouin zone sampling was done by using a Γ- +centered k-mesh. For all the compounds, K-meshes of 10 × 10 × 10 (ionic relaxations) and +20 × 20 × 20 (self-consistent-field solutions) were used for PBE calculations. Cell volume, +shape, and atomic positions for all the structures were fully relaxed using conjugate gradient +algorithm until the energy (forces) converges to 10−6 eV (0.001 eV/˚A). A tetrahedron method +with Blochl corrections was used for accurate electronic density of states. +Density functional perturbation theory (DFPT) combined with phonopy31 was used to +obtain relevant phonon properties. Alloy Theoretic Automated Toolkit (ATAT)32 was used +to generate special quasi-random structures to simulate disordered phases of these com- +pounds. Ab-initio Scattering and Transport (AMSET)33 code was used to calculate the +electronic transport properties which uses the variable carrier relaxation time to evalu- +ate the transport distribution function while solving the Boltzmann transport equations. +Debye-Callaway (DC) model43 was used to calculate lattice thermal conductivity. +More +details about this model is described in supplementary information. +Results and Discussion +Table 1 shows the optimized lattice constant and the relative energies of cubic (F¯43m), +tetragonal (I¯42d) and SQS structures (P1) of Hf2Ni2InSb and Zr2Ni2InSb. +Six different +configurations with different site occupancies of In and Sb were simulated using the conven- +tional cell of cubic phase and the energetically most stable configuration is presented in the +manuscript. +Tetragonal structure of these compounds is theoretically predicted to be stable in open +quantum materials database (OQMD). Solid solution phase (represented by SQS structure +here) is usually inevitable for this class of compounds at high temperature.23, 35 As such a +comparative study of all these three phases is highly desired to facilitate an in-depth analysis +5 + +Table 1: Theoretically optimized lattice constants and relative energies of cubic, tetragonal +and SQS structures of Hf2Ni2InSb and Zr2Ni2InSb +Compound +Crystal +Optimized lattice +△E (meV/ +structure +constants (˚A) +atom) +cubic +a=b=c=6.11 +8.0 +Hf2Ni2InSb +tetragonal +a=b=6.11, c=12.24 +0.0 +SQS +a=6.09, b=6.12,c=5.94 +62.0 +cubic +a=b=c= 6.15 +6.0 +Zr2Ni2InSb +tetragonal +a=b=6.14, c=12.31 +0.0 +SQS +a=b=6.18, c=6.12 +134.0 +of TE properties of these alloys in different T-range. Interestingly, the energy difference +between the two ordered phases is very small (8 meV for Hf-based) and 6 meV for Zr-based +double HH-alloys), where SQS phase is much higher in energy (60-130 meV) as compared to +the lower energy tetragonal phase. +As evident from table 1, the SQS structure are off-cubic due to the presence of disorder. +Figure 1 shows the theoretically optimized cubic, tetragonal and SQS structure of Hf2Ni2InSb +(top) and Zr2Ni2InSb (bottom). Few bond lengths in both cubic and tetragonal structures +are same i.e., dHf-In = dHf-Sb = 3.05 ˚A, dIn-Ni = 2.68 ˚A, dSb-Ni = 2.61 ˚A(in Hf2Ni2InSb) and +dZr-In = dZr-Sb = 3.07 ˚A, dIn-Ni = 2.69 ˚A, dSb-Ni = 2.63 ˚A(in Zr2Ni2InSb), while dHf-Ni bond +length in cubic and tetragonal structure are 2.61 ˚Aand 2.64 ˚Arespectively. The same for +Zr2Ni2InSb are 2.63 and 2.66 ˚Arespectively. In SQS structure, there is large variation in +bond length due to randomness, ranging from 2.6 to 6.04 ˚A. +Electronic structure +Figures 2(a) and (b) show the atom-projected band structure of cubic Hf2Ni2InSb and +Zr2Ni2InSb respectively. For Hf2Ni2InSb, the conduction bands are mostly contributed by +Hf and Ni atoms whereas the valence band edges are composed of In and Sb atoms. Multi- +ple valleys are favorable for thermoelectric performance of a compound as it leads to large +band degeneracy. The second conduction band minima (CBM-2) and third conduction band +minima (CBM-3) lie at an energy difference of 0.03 and 0.05 eV (also contributed by Hf and +6 + +Figure 1: Theoretically optimized crystal structures of Hf2Ni2InSb (top) and Zr2Ni2InSb +(bottom) in (a) cubic (b) tetragonal and (c) SQS phases. +-1.5 +-1 +-0.5 +0 +0.5 +1 +1.5 +E-EF (eV) +Zr +Ni +In +Sb +Γ +X +M +Γ +Z +R +A +Z +Eg = 0.17 eV +(b) +-1 +-0.5 +0 +0.5 +1 +E-EF (eV) +Zr +Ni +In +Sb +Γ +X|Y +Γ +Z|R2 Γ +T2|U2 +Γ +V2 +Eg = 0.05 eV +(f) +-1.5 +-1 +-0.5 +0 +0.5 +1 +1.5 +E-EF (eV) +Hf +Ni +In +Sb +Γ +X +M +Γ +Z +R +A +Z +Eg = 0.24 eV +(a) +-1.5 +-1 +-0.5 +0 +0.5 +1 +1.5 +E-EF (eV) +Hf +Ni +In +Sb +Γ X Y Σ +Γ +Z Σ1 +N +P +Y1Z +Eg = 0.4 eV +(c) +-1.5 +-1 +-0.5 +0 +0.5 +1 +1.5 +E-EF (eV) +Zr +Ni +In +Sb +Γ X Y Σ +Γ +Z Σ1 +N +P +Y1Z +Eg = 0.59 eV +(d) +-1 +-0.5 +0 +0.5 +1 +E-EF (eV) +Hf +Ni +In +Sb +ΓX|Y Γ +Z|R +Γ +T|U +Γ +V +Eg = 0.059 eV +(e) +Figure 2: Atom/orbital-projected electronic band structures of (a,b) cubic (c,d) tetragonal +and (e,f) disordered SQS structure of Hf- and Zr-based double HH alloys respectively. +Ni atoms). For Zr2Ni2InSb, the conduction bands are mostly contributed by Ni atoms while +valence bands by In atoms. The CBM-2 lies at an energy difference of 0.08 eV. This shows +7 + +(a) +(b) +(c) +Hf +Ni +In +SbZr +Ni +In +Sblarge conduction band degeneracy in the cubic phase of these compounds. The band gap +calculated using HSE-SOC functional for Hf and Zr-based compounds are 0.24 eV and 0.17 +eV respectively. The former (latter) is an indirect (direct) band gap semiconductor. SOC +plays a crucial role due to heavy element Sb, and causes a reduction in the band gap as +compared to non-SOC values. +Figures 2(c) and (d) show the band structures of Hf2Ni2InSb and Zr2Ni2InSb respectively +in their tetragonal phase. Both are direct band gap semiconductors with a value of 0.4 and +0.59 eV respectively. Valence band edges show dominant contribution from Hf and Ni atoms +whereas conduction band edges are dominated by Ni atoms for Hf2Ni2InSb. For Zr2Ni2InSb, +the dominant contribution arises from In and Ni atoms near valence band edges whereas +conduction band edges are mostly composed of Ni. Figures 2 (e) and (f) show the band +structures of disordered phase for Hf and Zr-based compounds respectively. HSE-SOC band +gap for Hf2Ni2InSb and Zr2Ni2InSb are 0.06 eV and 0.05 eV respectively. Hf-atoms have the +dominant contribution near valence band edges while a mixed contribution from Hf and Ni- +atoms near conduction band edges. Zr2Ni2InSb has a similar contribution with Hf replaced +by Zr. +The ordered structures have larger band gaps as compared to the disordered SQS struc- +tures, which can help in suppressing the bipolar effect36 and hence better electronic transport +properties. The ordered tetragonal phase has the largest band gap with flat valence bands +which leads to higher density of states effective mass and hence larger p-type thermopower +Whereas conduction bands for ordered cubic phase are flat and show multiple valleys at very +small energy difference. This indicates better n-type thermopower in ordered cubic phase. +All the thermoelectric properties are calculated using PBE band structure with scissor shifted +band gap evaluated from HSE-SOC functional. +8 + +Electronic Transport +Transport parameters of most thermoelectric materials are strongly dependent on carrier +relaxation time (τ). Most previous simulations are based on constant relaxation time ap- +proximation(CRTA),37, 38 which is a very crude approximation. τ is dictated by different +scattering mechanisms such as acoustic scattering, optical scattering, scattering by impuri- +ties and defects and electric polarization in case of the polar lattice. In the present work, we +have estimated the relaxation time for electron and hole transport, using Ab-initio Scattering +and Transport (AMSET).33 AMSET is a numerical code for calculating carrier relaxation +time and transport properties within first principles framework. We have taken into account +all four types of scattering mechanisms in our TE calculations i.e. acoustic deformation +potential (ADP), ionized impurity scattering (IMP), polar optical phonon scattering (POP) +and piezoelectric scattering (PIE). We have also captured the effect of charge carrier screen- +ing arising out of free carriers at high concentrations. The details of these mechanisms is +given in supplementary information (SI). +The relaxation time (τ) for Hf2Ni2InSb and Zr2Ni2InSb (in cubic phase) is shown in +Figures 8 and 9 of supplementary information respectively. As can be seen, τ is strongly +dependent on both carrier concentration (n) and temperature (T). For Hf2Ni2InSb, at a high +temperature of 900 K, τ varies between 25-21 fs for holes and 35-20 fs for electrons with +increasing carrier concentration. For Zr2Ni2InSb, τ varies between 26-22 fs for holes and +37-22 fs for electrons at 900 K. +The relaxation time (τ) of the two compounds in tetragonal phase are shown in Figure 10 +and 11 respectively. For Hf2Ni2InSb, τ varies between 18-17 fs for holes and 40-19 fs for elec- +trons at 900 K. For Zr2Ni2InSb, τ varies between 18-20 fs for holes and 30-20 fs for electrons +at 900 K. Thus, the order of magnitude of τ do not vary much, as we go from one phase to +the other for a given compound. For SQS structure, we expect a similar or relatively lower +value of τ. Figure 3 shows a comparison of the T-dependent electronic transport properties +of Hf and Zr-based compounds in different phases at a fixed carrier concentration of 1 × 1021 +9 + +300 +400 +500 +600 +700 +800 +900 +-200 +-100 +0 +100 +200 +300 +S(µVK +-1) +Cubic-p +Tetra-p +SQS-p +Cubic-n +Tetra-n +SQS-n +T (K) +(d) Zr2Ni2InSb +300 +400 +500 +600 +700 +800 +900 +0 +5 +10 +15 +20 +25 +30 +35 +S +2σ (mWm +-1K +-2) +T(K) +(b) +300 +400 +500 +600 +700 +800 +900 +0 +5 +10 +15 +20 +κe + κb (Wm-1K-1) +T(K) +(f) +300 +400 +500 +600 +700 +800 +900 +0 +5 +10 +15 +20 +25 +30 +35 +40 +S2σ (mWm-1K-2) +T(K) +(e) +300 +400 +500 +600 +700 +800 +900 +-200 +-100 +0 +100 +200 +300 +S(µVK +-1) +Cubic-p +Tetra-p +SQS-p +Cubic-n +Tetra-n +SQS-n +T (K) +(a)Hf2Ni2InSb +300 +400 +500 +600 +700 +800 +900 +0 +5 +10 +15 +20 +κe + κb (Wm +-1K +-1) +T(K) +(c) +Figure 3: Temperature dependence of (a,d) Seebeck coefficient(S), (b,e) power factor (S2σ), +(c,f) electronic thermal conductivity (κe+κb) for p-type (circle) and n-type (square) conduc- +tion in cubic (black line), tetragonal (red line) and SQS (blue line) structures of Hf2Ni2InSb +and Zr2Ni2InSb respectively at a fixed carrier concentration of 1 × 1021 cm−3. +cm−3. The choice of carrier concentration is guided by a previous experimental report on +Zr-based double HH-alloy.22 The ordered structures show better electronic properties than +the SQS structures. Also, as expected from electronic band structures topology, the p-type +thermopower of tetragonal structures is larger whereas for cubic phase, n-type thermopower +is most promising for both the compounds (see figs. +3(a,d)). +As a result, power-factor +of p-type tetragonal phase and n-type cubic phase is largest in a large T-range. (see figs. +3(b,e)). Figures 3(c,f) shows the T-dependence of thermal conductivity (κe+κb) for both the +compounds, where κb is bipolar thermal conductivity. The (κe+κb) values of SQS structures +show a rise at higher T (around 500-600 K) because of the bipolar component of κ. This is +also the reason for a slight decrease in Seebeck coefficient at higher T for SQS structures. +The ordered phases show larger and comparable values of electronic thermal conductivity +for n-type. +These double half heusler compounds show promising electronic transport properties +10 + +0 +2 +4 +6 +8 +Frequency (THz) +K +Γ +L +U +Γ +W +X +Γ +0 +2 +4 +6 +8 +Frequency (THz) +A +Γ +M +R +Γ +X +Z Γ +0 +2 +4 +6 +8 +Frequency (THz) +N Γ +P S0 Γ S +X +Γ R G +Γ M +0 +1 +2 +3 +4 +5 +6 +7 +8 +ν (THz) +0.6 +0.9 +1.2 +1.5 +1.8 +2.1 +2.4 +γ +TA’ +TA +LA +optical +0 +1 +2 +3 +4 +5 +6 +7 +8 +ν (THz) +1.2 +1.5 +1.8 +2.1 +2.4 +γ +TA’ +TA +LA +optical +0 +1 +2 +3 +4 +5 +6 +7 +8 +ν (THz) +0.9 +1.2 +1.5 +1.8 +2.1 +2.4 +γ +TA’ +TA +LA +optical +(i) +(a) +(b) +(e) +(f) +(j) +0 +2 +4 +6 +8 +Frequency (THz) +K +Γ +L +U +Γ +W +X +Γ +0 +1 +2 +3 +4 +5 +6 +7 +8 +ν (THz) +0.6 +0.9 +1.2 +1.5 +1.8 +2.1 +2.4 +γ +TA’ +TA +LA +optical +0 +1 +2 +3 +4 +5 +6 +7 +8 +ν (THz) +0.6 +0.8 +1 +1.2 +1.4 +1.6 +1.8 +2 +2.2 +2.4 +γ +TA’ +TA +LA +optical +0 +2 +4 +6 +8 +Frequency (THz) +N Γ +P S0 Γ +S +X +Γ +R G +Γ M +0 +1 +2 +3 +4 +5 +6 +7 +8 +ν (THz) +1.2 +1.5 +1.8 +2.1 +2.4 +γ +TA’ +TA +LA +optical +0 +2 +4 +6 +8 +Frequency (THz) +A +Γ +M +R +Γ +X +Z Γ +(k) +(c) +(d) +(g) +(h) +(l) +Figure 4: Phonon dispersion and mode Gruneisen parameter for ternary (a,b) HfNiSn and +(c,d) ZrNiSn; cubic double HH (e,f) Hf2Ni2InSb and (g,h) Zr2Ni2InSb; tetragonal double HH +(i,j) Hf2Ni2InSb and (k,l) Zr2Ni2InSb respectively. +(large power-factor) due to favorable band features such as flat bands in the tetragonal +phase and high band degeneracy in cubic phase. +Thus, the present compounds super- +sede/compete well with previously reported high-thermoelectric performance materials. For +example, ZrCoBi0.65Sb0.15Sn0.20 shows a ZT value of 1.42 at around 970 K.39 The power-factor +for this compound is around 3.8 mWm−1K−2. Another promising p-type HH compound +FeNb0.8Ti0.2Sb shows a ZT value of 1.1 (at around 970 K)40 with the corresponding power fac- +tor of 5.3 mWm−1K−2. For n-type HH compounds, Ti0.5Zr0.25Hf0.25NiSn0.998Sb0.002 shows a +ZT value of 1.5 (at 700 K)41 and a power-factor of 6.2 mWm−1K−2. Hf0.6Zr0.4NiSn0.995Sb0.005 +is yet another system with a promising ZT value of 1.2 (at 900 K)42 and a power-factor of +4.7 mWm−1K−2. In double half heusler family, Ti4Fe2Ni2Sb4(in solid solution) shows a ZT +value of 0.5 and 0.4 for p-type and n-type conduction with power-factor values of around 1.7 +and 1.2 mWm−1K−2 respectively at around 900 K.16 At the same temperature, the ordered +structure for the same compound shows ZT value of 1.5 for p-type and 0.5 for n-type.23 The +11 + +corresponding theoretically calculated power factor values are around 7 mWm−1K−2 and 4 +mWm−1K−2. Hf2Ni2InSb and Zr2Ni2InSb (see Fig 3) show much higher power-factor val- +ues at similar temperature range. Hf2Ni2InSb has a value of 9.2 (10.7) mWm−1K−2 and 32 +(24) mWm−1K−2 for for p-type and n-type cubic (tetragonal) structure respectively, while +Zr2Ni2InSb has a value of 7.4 (12) mWm−1K−2 and 35 (25.4) mWm−1K−2 for p-type and +n-type cubic (tetragonal) structures respectively. The disordered (SQS) phases also show +comparable/better power-factor values (7.3 mWm−1K−2 for Hf-based and 6.4 mWm−1K−2 +for Zr-based alloy). +Phonon Transport +For comparison sake, we have not only calculated the phonon properties (specially the order +of magnitude of lattice thermal conductivity, κL) of the present double HH alloys, but also +their ternary counterparts ZrNiSn and HfNiSn (with same VEC) which are extremely studied +in the literature and reported to crystallize in F¯43m (♯216) structure.10,11,19 Figures 4(a,c), +4(e,g) and 4(i,k) show the phonon dispersion for ternary (HfNiSn, ZrNiSn), cubic double +HH (Hf2Ni2InSb, Zr2Ni2InSb) and tetragonal double HH (Hf2Ni2InSb, Zr2Ni2InSb) alloys +respectively. The ternary, cubic and tetragonal phases have 3-atoms, 6-atoms and 12-atoms +in the primitive unit cell giving rise to 9, 18 and 36 phonon branches respectively. Of them, +the lowest 3 branches are acoustic and the rest corresponds to optical branches respectively. +The acoustic branches are further classified into one longitudinal (LA) and two transverse +(TA, TA +′) modes. The velocity of each acoustic mode (γ) is calculated from the slope of +the band corresponding to the vibrational band at Γ-point. +Debye temperature (θ) can +be estimated from the maximum frequency corresponding to the given vibrational mode, +within a reasonable approximation. Apart from ν and θ, DC model for κL also requires few +other quantities including gruneisen parameter for HfNiSn, cubic Hf2Ni2InSb and tetragonal +Hf2Ni2InSb respectively. The same for Zr-based compounds are shown in Figs. 4(d), 4(h) +and 4(l). The simulated values of various parameters, phonon group velocities (νi), Debye +12 + +300 +400 +500 +600 +700 +800 +900 +Temperature (K) +0 +5 +10 +15 +20 +KL (Wm +1 K +1) +HfNiSn +Hf8In8In4Sb4 (Tetragonal) +Hf4Ni4In2Sb2 (Cubic) +300 +400 +500 +600 +700 +800 +900 +Temperature (K) +0 +4 +8 +12 +16 +20 +KL (Wm +-1K +-1) +ZrNiSn +Zr8Ni8In4Sb4 (Tetragonal) +Zr4Ni4In2Sb2 (Cubic) +Figure 5: Comparison of simulated lattice thermal conductivity (κL) of ternary XNiSn with +those of cubic and tetragonal phases of double HH X2Ni2InSb for (left) Hf-based and (right) +Zr-based compounds. +temperature (θi), cell volume (V), atomic mass (M), Gruneisen parameter (γi for different +vibrational modes (LA, TA, TA +′) for the three Hf and Zr-based compounds are presented +in SI. Larger values of γ indicates high degree of anharmonicity. +The group velocity of +acoustic phonons is reduced in double half heuslers (cubic phase) as compared to their ternary +counterparts due to large mixing of acoustic and optical phonon modes in the former. This +indicates reduced lattice thermal conductivity for double HH as compared to ternary alloys. +Thermal transport properties +Figures 5 show the comparison of κL for ordered phases of double half heuslers and their +corresponding ternary counterparts. The simulated values of κL vary between 13.3 to 3.3 +Wm−1K−1 for Hf2Ni2InSb (in cubic phase) and 12.4 to 2.8 Wm−1K−1 (in tetragonal phase) +whereas for ternary HfNiSn, it ranges between 18.9 to 4.7 Wm−1K−1 in the temperature +range 300-900 K. For Zr2Ni2InSb, κL varies between 17.8 to 4.3 Wm−1K−1 (for cubic) and +12.1 to 2.9 Wm−1K−1 (for tetragonal) in the temperature range 300-900 K whereas it varies +between 18.7 to 4.5 Wm−1K−1 for ZrNiSn for the same temperature range. The previous +theoretically reported room temperature κL values of ZrNiSn and HfNiSn are 19.69 and 18.5 +Wm−1K−1, whereas for Zr2Ni2InSb and Hf2Ni2InSb (in tetragonal phase), the values are +13 + +13.58 and 12.5 Wm−1K−1 respectively at 300 K.15 These values are in fair agreement with +our simulated values obtained from DC model. Clearly, we can see the reduction of lattice +thermal conductivity in double half heuslers as compared to the corresponding ternary alloys +which is definitely useful for enhancing the TE figure of merit (ZT). For disorder SQS phase, +we expect a further reduction of κL due to enhanced disorder induced scatterings. +Thermoelectric performance +As evident from Fig. 3, the power factor for ordered phases is reasonably high (better or +comparable to some of the best TE materials in the literature,39, 42 along with the relatively +low lattice thermal conductivity (see Fig. 5). This indicates their potential for efficient TE +performance. In this section, we shall focus on a comparison of TE figure of merit (ZT) +for n and p-type conduction of ordered phases of these two alloys. Figure 6 and 7 show +the carrier concentration dependence of ZT at different T for cubic and tetragonal phases +of the two alloys respectively. The left (right) panel indicates the result for p-type (n-type) +conduction. The change in carrier concentration (n) can be thought of as mimicking the +effect of doping/alloying the host material, keeping the topology of band structure intact +(the so-called rigid band approximation). +For cubic phase, Hf2Ni2InSb show a peak ZT value of 0.49 for p-type conduction at a +carrier concentration of 1 × 1021 cm−3 and 1.48 for n-type at a carrier concentration of 4 +× 1020 cm−3 respectively at 900 K. At these carrier concentrations, the maximum value +of simulated Seebeck coefficient (Smax) and power factor (PFmax) are 165.5 µVK−1 and +9.39 mWm−1K−2 for p-type and 248.9 µVK−1 and 29.6 mWm−1K−2 for n-type conduction +respectively. The maximum ZT value for Zr2Ni2InSb is 0.44 at a carrier concentration of +1 × 1021 cm−3 for p-type and 1.82 at a carrier concentration of 4 × 1020 cm−3 for n-type +respectively at 900 K. The simulated Smax and PFmax at these carrier concentrations are 134.6 +µVK−1 and 7.41 mWm−1K−2 for p-type and are 250.4 mWm−1K−2 and 29.7 mWm−1K−2 +14 + +for n-type respectively. +1e+19 +1e+20 +1e+21 +n (cm +-3) +0 +0.1 +0.2 +0.3 +0.4 +0.5 +ZT +300 +400 +500 +600 +700 +800 +900 +p-type +Hf2Ni2InSb (Cubic) +1e+19 +1e+20 +1e+21 +n (cm +-3) +0 +0.3 +0.6 +0.9 +1.2 +1.5 +ZT +n-type +1e+19 +1e+20 +1e+21 +n (cm +-3) +0 +0.1 +0.2 +0.3 +0.4 +0.5 +ZT +300 +400 +500 +600 +700 +800 +900 +p-type +Zr2Ni2InSb (Cubic) +1e+19 +1e+20 +1e+21 +n (cm +-3) +0 +0.5 +1 +1.5 +2 +ZT +n-type +Figure 6: Thermoelectric figure of merit (ZT) as a function of carrier concentration (n) at +different temperatures for cubic Hf2Ni2InSb (above) and Zr2Ni2InSb (below) alloy for p-type +(left) and n-type (right) conduction. +1e+19 +1e+20 +1e+21 +n (cm +-3) +0 +0.3 +0.6 +0.9 +1.2 +1.5 +1.8 +2.1 +ZT +n-type +1e+19 +1e+20 +1e+21 +n (cm +-3) +0 +0.5 +1 +1.5 +ZT +300 +400 +500 +600 +700 +800 +900 +p-type +Hf2Ni2InSb (tetragonal) +1e+19 +1e+20 +1e+21 +n (cm +-3) +0 +0.5 +1 +1.5 +2 +2.5 +ZT +300 +400 +500 +600 +700 +800 +900 +p-type +Zr2Ni2InSb (tetragonal) +1e+19 +1e+20 +1e+21 +n (cm +-3) +0 +0.5 +1 +1.5 +2 +2.5 +ZT +n-type +Figure 7: The thermoelectric figure of merit (ZT) as a function of carrier concentration (n) at +different temperatures (T) for tetragonal Hf2Ni2InSb (above) and Zr2Ni2InSb (below) alloy +for p-type (left) and n-type (right) conduction. +The tetragonal phase of both alloys shows the lowest value of lattice thermal conductivity +along with better p-type electronic transport properties. The peak value of ZT for Hf2Ni2InSb +15 + +for p-type and n-type conduction occur at carrier concentrations of 7 × 1020 cm−3 and 2 +× 1020 cm−3 respectively. At these values of carrier concentrations, the Smax and PFmax +for p-type are 210.6 µVK−1 and 10.96 mWm−1K−2, and for n-type are 270.1 µVK−1 and +19.48 mWm−1K−2 respectively. This gives a ZT value of ∼ 1.35 at 800 K for p-type and ∼ +2.0 at 900 K for n-type conduction. For Zr2Ni2InSb, the peak value of ZT for p-type and +n-type conduction is obtained at carrier concentration of 9 × 1020 cm−3 and 2 × 1020 cm−3 +respectively. The Smax and PFmax for p-type are 247.2 µVK−1 and 11.96 mWm−1K−2 and +for n-type are 282.5 µVK−1 and 20.03 mWm−1K−2 respectively. As expected, a high ZT +value of ∼ 2.19 and ∼ 2.46 at 900 K for p-type and n-type conduction were obtained for +tetragonal Zr2Ni2InSb. +Although the power factor of these double half heuslers is quite high in cubic phase for +n-type conduction, the alloys actually show higher ZT value in tetragonal phase due to lower +values of lattice thermal conductivity. Thus, the reduction of lattice thermal conductivity +along with enhanced power factor leads to the improvement of thermoelectric performance +for these double half-heusler compounds as compared to corresponding 18 VEC ternary +half-heusler alloys. +Conclusion +In summary, we report an ab-initio study of two double half heusler alloys, X2Ni2InSb +(X=Hf/Zr), in their three competing structural phases: tetragonal, cubic and solid solution. +Double half-heusler alloys are formed via the transmutation of two single heusler compounds +and hence have higher flexibility for tuning their properties. +The simulated band gaps +(using HSE06 hybrid functional) for Hf2Ni2InSb lie in the range 0.06-0.4 eV while those for +Zr2Ni2InSb in 0.05-0.59 eV depending on the structure. Spin-orbit coupling plays a crucial +role in splitting the bands due to heavy Sb-atom. Thermoelectric performance is mostly +dominated by the promising electronic transport in these alloys, with ZT value as high as 2.46 +16 + +for n-type Zr2Ni2InSb and 2.0 for n-type Hf2Ni2InSb at high temperature. Tetragonal phase +show a relatively lower lattice thermal conductivity, responsible for the best thermoelectric +figure of merit (ZT). The TE performance of p-type conduction is almost equally promising +in tetragonal phase, with highest ZT of ∼ 1.35 and ∼ 2.19 for Hf- and Zr-based compounds. +We expect the present study to receive immediate attention from experimentalists with a +possibility to synthesize and cross validate our findings. +References +(1) Tritt, T. M. Thermoelectrics Run Hot and Cold. Science 1996, 272, 1276-1277. +(2) DiSalvo, F. J. Thermoelectric Cooling and Power Generation. Science 1999, 285, 703- +706. +(3) Chung, D. Y.; Hogan, T. H.; Brazis, P.; Rocci-Lane, M.; Kannewurf, C.; Bastae, M.; +Uher, C.; Kanatzidis, M. G. 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Uncovering +high thermoelectric figure of merit in (Hf,Zr)NiSn half-Heusler alloys Appl. Phys. Lett. +2015, 107, 041902. +(43) Zhang, Y. First-principles Debye–Callaway approach to lattice thermal conductivity. J +Materiomics 2016, 2, 237-247. +21 + +(44) Vikram; Kangsabanik J. ; Enamullah; Alam A. Bismuth based half-Heusler alloys with +giant thermoelectric figures of merit. J. Mater. Chem. A 2017, 5, 6131-6139. +(45) Vikram ; Sahni, B.; Barman, C. K.; Alam A. Accelerated Discovery of New 8-Electron +Half-Heusler Compounds as Promising Energy and Topological Quantum Materials. J. +Phys. Chem. C 2019, 123, 7074-7080 +(46) Sahni, B; Vikram; Kangsabanik, J.; Alam A. ”Reliable Prediction of New Quantum +Materials for Topological and Renewable-Energy Applications: A High-Throughput +Screening.” J. Phys. Chem. Lett. 2020, 15, 6364-6372. +Supporting Information +Here, we present auxiliary information on the formalism of lattice thermal conductivity and +carrier relaxation time calculation. +Lattice Thermal Conductivity +Lattice thermal conductivity (κL) is calculated using the Debye-Callaway (D-C) model.43 +The Debye-Callaway (DC) model has been proven to be useful for estimating the lattice +thermal conductivity for various experimentally synthesized compounds.44, 45, 46 Comparison +of κL for three systems (Cu3SbSe4, Cu3SbSe3 and SnSe) with the previous reported exper- +imental values have been shown in reference.43 Within this model, the major contribution +to the lattice thermal conductivity comes from the 3-phonon scattering process (normal & +Umklapp phonon scattering) and is majorly due to the 3 acoustic modes of vibrations. The +thermal contribution from each of the acoustic vibrational mode in D-C model43,45 is given +as, +κi +L = CiT 3/3 +� +� +� +� +� +θi/T +� +0 +τ i +c(x)x4ex +(ex − 1)2 dx + +�� θi/T +0 +τ i +c(x)x4ex +τ i +N(ex−1)2dx +�2 +� θi/T +0 +τ ic(x)x4ex +τ i +Nτ i +U(ex−1)2dx +� +� +� +� +� +(1) +22 + +where x = (¯hω/kBT) and the index i denotes LA, TA and TA′ modes of vibration. The +constant Ci is given by, +Ci = +k4 +B +2π2¯h3νi +(2) +The scattering rates corresponding to the normal and Umklapp scattering processes are, +1 +τ LA +N (x) = k3 +Bγ2 +LAV +M¯h2ν5 +LA +�kB +¯h +�2 +x2T 5 +(3) +1 +τ TA/TA′ +N +(x) += +k4 +Bγ2 +TA/TA′V +M¯h3ν5 +TA/TA′ +�kB +¯h +� +xT 5 +(4) +1 +τ i +U(x) = +¯hγ2 +Mν2 +i θi +�kB +¯h +�2 +x2T 3e−θi/3T +(5) +where, kB is the Boltzmann’s constant, ¯h is the reduced Plank’s constant, T is the tempera- +ture, θ is the Debye temperature, V and M are the volume and mass per atom respectively, +γ is the mode greenness parameter, ν is phonon group velocity and ω is the angular fre- +quency. The input parameters such as the mode velocities, Debye temperature, Gruneisen +parameter, etc were calculated from the phonon band structure and the mode Gruneisen +parameter from the Phonopy code,31 which was used as post processing tool after doing the +density functional perturbation theory (DFPT) calculations in Vienna Ab-initio Simulation +Package (VASP). +Simulated parameters needed for κL calculations +The numerical values of various parameters used to calculate (κL) for HfNiSn are; νLA = +4746.3246 m/s, νTA = 3120.5535 m/s, νTA′ = 2508.8948 m/s, γLA = 1.55, γTA = 1.38,γTA′ += 1.35, γ= 1.59, θLA = 195.65 K, θTA = 160.06 K, θTA′ = 145.33 K, V = 19.05 × 10−30 m3, +and M = 196.93 × 10−27 kg. The numerical values of various parameters used to calculate +(κL) for Hf2Ni2InSb (cubic phase) are; νLA = 4152.7021 m/s, νTA = 3438.3485 m/s, νTA′ = +23 + +2304.3666 m/s, γLA = 1.43, γTA = 1.26,γTA′ = 1.22, γ= 1.60, θLA = 149.73 K, θTA = 148.64 +K, θTA′ = 148.19 K, V = 19.05 × 10−30 m3, and M = 196.69 × 10−27 kg. The numerical +values of various parameters used to calculate (κL) for Hf2Ni2InSb (tetragonal phase) are; +νLA = 4963.8142, νTA = νTA′ = 2324.1295 m/s, γLA= 1.40, γTA = 1.16,γTA′ = 1.23, γ= 1.6, +θLA = 148.13 K, θTA = 129.40 K, θTA′ = 123.89 K, V = 19.05 × 10−30 m3, and M = 196.69 +× 10−27 kg. +The numerical values of various parameters used to calculate (κL) for ZrNiSn are; νLA = +5322.6442 m/s, νTA = 3600.8698 m/s, νTA′ = 2852.1311 m/s, γLA = 1.69, γTA = 1.36,γTA′ += 1.30, γ= 1.71, θLA = 213.76 K, θTA = 184.12 K, θTA′ = 166.74 K, V = 19.38 × 10−30 m3, +and M = 148.64 × 10−27 kg. The numerical values of various parameters used to calculate +(κL) for Zr2Ni2InSb (cubic phase) are; νLA = 4784.2714 m/s, νTA = 4040.5217 m/s, νTA′ = +2774.3478 m/s, γLA = 1.42, γTA = 1.28,γTA′ = 1.24, γ=1.60, θLA = 170.93 K, θTA = 171.02 +K, θTA′ = 171.32 K, V = 19.37 × 10−30 m3, and M = 148.40 × 10−27 kg. The numerical +values of various parameters used to calculate (κL) for Zr2Ni2InSb (tetragonal phase) are; +νLA = 4696.1338 m/s, νTA = 3907.6455 m/s, νTA′ = 2588.8792 m/s, γLA = 1.43, γTA = +1.26,γTA′ = 1.22, γ= 1.64, θLA = 166.19 K, θTA = 149.18 K, θTA′ = 142.16 K, V = 19.36 × +10−30 m3, and M = 148.45 × 10−27 kg. +Carrier Relaxation Time +1e+19 +1e+20 +1e+21 +20 +30 +40 +50 +60 +70 +80 +τ (fs) +300 +500 +700 +900 +n-type +n (cm +-3) +1e+20 +1e+21 +20 +30 +40 +50 +60 +τ (fs) +300 +500 +700 +900 +p-type +n (cm +-3) +Figure 8: Temperature dependence of p-type (holes) and n-type(electrons) relaxation time +for Cubic Hf2Ni2InSb. +24 + +1e+20 +1e+21 +20 +30 +40 +50 +60 +τ (fs) +300 +500 +700 +900 +p-type +n (cm +-3) +1e+19 +1e+20 +1e+21 +20 +30 +40 +50 +60 +70 +80 +90 +τ (fs) +300 +500 +700 +900 +n-type +n (cm +-3) +Figure 9: Temperature dependence of p-type (holes) and n-type(electrons) relaxation time +for Cubic Zr2Ni2InSb. +1e+20 +1e+21 +10 +20 +30 +40 +50 +τ (fs) +300 +500 +700 +900 +p-type +n (cm +-3) +1e+19 +1e+20 +1e+21 +20 +40 +60 +80 +τ (fs) +300 +500 +700 +900 +n-type +n (cm +-3) +Figure 10: Temperature dependence of p-type (holes) and n-type(electrons) relaxation time +for tetragonal Hf2Ni2InSb. +1e+20 +1e+21 +10 +20 +30 +40 +50 +60 +70 +80 +τ (fs) +300 +500 +700 +900 +p-type +n (cm +-3) +1e+19 +1e+20 +1e+21 +10 +20 +30 +40 +50 +60 +70 +τ (fs) +300 +500 +700 +900 +n-type +n (cm +-3) +Figure 11: Temperature dependence of p-type (holes) and n-type(electrons) relaxation time +for tetragonal Zr2Ni2InSb. +We have estimated the carrier relaxation time for electron and hole transport, using +Ab-initio Scattering and Transport (AMSET).33 AMSET is a numerical code for calculat- +ing carrier relaxation time and transport properties within the ab-initio framework. Four +scattering mechanisms are simulated in AMSET. They are acoustic deformation potential +25 + +scattering, ionized impurity scattering, polar optical phonon scattering and piezoelectric +scattering. The mode dependent scattering rates are calculated using Born approximation. +Based on Fermi’s golden rule, the differential scattering rate from initial state |nk⟩ to final +state |mk+q⟩ is calculated as: +τ −1 +nk→mk+q = 2π +¯h |gnm(k, q)|2δ(ϵnk − ϵmk+q), +(6) +where gnm(k, q) is the matrix element for scattering from state |nk⟩ into state |mk + q⟩ and +ϵnk is the energy of the state |nk⟩. +The acoustic deformation potential (ADP) scattering is calculated by assuming that the +lattice potential perturbation due to the thermal motion has linear dependence on relative +volume change. The constant of proportionality is the deformation potential. A more general +acoustic deformation potential (ADP) matrix element is given by: +gad +nm(k, q) = +� +kBT +� +G̸=−q +� ˜Dnk : ˜Sl +cl√ρ ++ +˜Dnk : ˜St1 +ct1√ρ ++ +˜Dnk : ˜St2 +ct2√ρ +� +⟨mk + q|ei(q+G).r|nk⟩ +(7) +where ˆS = ˆq⊗ ˆu is the unit strain for an acoustic mode, ˜Dnk = Dnk + vnk ⊗ vnk in which Dnk +is the rank 2 deformation potential tensor, ˆu is the unit vector of phonon polarization, and +the subscripts l, t1 and t2 indicate properties belonging to the longitudinal and transverse +modes. +In polar semiconductors, an electric polarization along with the deformation potential is +produced by the lattice oscillations. Since the ions oscillate out-of-phase for optical modes, +an electric dipole moment varying with time is generated. +This leads to an additional +interaction of optical phonons with charge carriers. The electric field due to perturbation of +dipole moment between atoms leads to scattering of carriers. Thus, Polar optical phonon +26 + +(POP) scattering rate is given by: +gpo +nm(k, q) = +�¯hωpo +2 +�1/2 � +G̸=−q +� +1 +ˆn.ϵ∞.ˆn − +1 +ˆn.ϵs.ˆn +�1/2⟨mk + q|ei(q+G).r|nk⟩ +|q + G| +(8) +where ϵs and ϵ∞ are the static and high-frequency dielectric tensors and ωpo is the polar +optical phonon frequency. +The ionized impurity scattering is an important scattering mechanism at lower temper- +atures. The mobile carriers are attracted by ionized impurity which leads to screening of +potential. The matrix element for ionized impurity scattering33 is given by: +gpo +nm(k, q) = +� +G̸=−q +n1/2 +ii Ze +ˆn.ϵs.ˆn +⟨mk + q|ei(q+G).r|nk⟩ +|q + G|2 + β2 +(9) +where Z is defect charge, nii is the concentration of ionized impurities which is equal to +charge compensation × (nholes-nelectrons)/Z and β is the inverse screening length. +The piezoelectric scattering (PIE) is basically polar acoustic phonon scattering. This +scattering mechanism is important at very low temperatures. However we have included +this scattering as well in our calculations. In 300 to 900K, POP and ADP are the dominant +scattering mechanisms. +Acknowledgment +BS acknowledges financial support from the Indian Institute of Technology, Bombay in the +form of teaching assistantship. BS acknowledges Vikram for some initial discussions regard- +ing the project. A.A. acknowledges DST-SERB (Grant No. CRG/2019/002050) for funding +to support this research. +27 + +Author Contributions +BS and AA conceived the initial idea of the project. BS performed all the calculations and +analyzed them with the help of AA. BS wrote the initial draft of the manuscript, which was +further corrected by AA. AA supervised the entire project. +Competing financial interests +The authors declare no competing financial interests. +28 + diff --git a/PtAyT4oBgHgl3EQftfn_/content/tmp_files/load_file.txt b/PtAyT4oBgHgl3EQftfn_/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fa0341fa86b7678db723e71bbba09ccd2df6b92b --- /dev/null +++ b/PtAyT4oBgHgl3EQftfn_/content/tmp_files/load_file.txt @@ -0,0 +1,1414 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf,len=1413 +page_content='Double Half-Heusler Alloys X2Ni2InSb (X= Zr/Hf) with promising Thermoelectric Performance: Role of varying structural phases Bhawna Sahni and Aftab Alam∗ Department of Physics, Indian Institute of Technology, Bombay, Powai, Mumbai 400 076, India E-mail: aftab@phy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='iitb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='in Abstract Double half-heusler alloys are the new class of compounds which can be seen as transmuted version of two single half-heusler with higher flexibility of tuning their properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Here, we report a detailed study of thermoelectric (TE) properties of two double half-heusler (HH) alloys X2Ni2InSb (X=Hf/Zr), using first-principles calcula- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' These alloys exhibit a rich phase diagram with the possibility of tetragonal, cubic and solid solution phase at different temperature range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' As such, a comparative study of TE properties of all these phases is performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' The ordered phases show quite fa- vorable electronic transport as compared to the disordered ones in both compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Lattice thermal conductivity of double HH alloys is lower than their ternary counter- part, making them most promising for TE application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Simulated band gap, obtained using hybrid functional, of ordered phases of Hf2Ni2InSb and Zr2Ni2InSb lie in the range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='24-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='4 eV and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='17-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='59 eV respectively, while for disordered phase, it is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='05- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='06 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Hf2Ni2InSb shows a reasonably high ZT value of ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='19, while Zr2Ni2InSb yields 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='46 at high temperature for n-type conduction in tetragonal phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' The ZT 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='00598v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='mtrl-sci] 2 Jan 2023 value for p-type conduction is also quite promising (∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='35 and ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='19 for Hf- and Zr-based compounds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' In both the compounds, electronic transport (Seebeck and elec- trical conductivity) plays the dominant role for the high ZT-value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Keeping in mind the promising TE performance, we propose immediate attention from experimentalists to synthesize and cross validate our findings for these new candidate materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Keywords Double half-heusler alloys, Thermoelectric, Ab-initio calculation, Carrier relaxation time, Electronic and thermal transport Introduction Thermoelectric technology which enables to convert waste heat into electricity, has proven to be extremely useful in providing solutions to renewable energy resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Since most of the energy from primary sources is lost as waste heat, potential thermoelectric materials come as a rescue by harvesting this waste heat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Transport properties (Seebeck Coefficient (S), electrical conductivity (σ) and thermal conductivity (κ) which define the thermoelectric figure of merit (ZT), are closely interrelated, which makes it quite challenging to find novel materials with optimal ZT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Though there are several materials reported in the literature,1–9 the hunt for more efficient novel materials is still ongoing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Half-Heusler (HH) alloys have emerged as promising thermoelectric materials due to a variety of interesting properties such as good thermal stability, easily tunable band gaps, good mechanical properties etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='10–14 They can be classified on the basis of the valence electron count (VEC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' The 18 valence-electron HH alloys are very stable because of fully occupied bonding and empty anti-bonding states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' The 17 and 19 valence-electron HH alloys on the other hand, are unstable because of partially occupied states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' However, mixing of a 17 and a 19 VEC HH alloy can form an 18 VEC double half heusler alloy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' These double half 2 heusler alloys exhibit much lower values of lattice thermal conductivity (κL) as compared to their ternary counterpart because of smaller phonon group velocity and disorder scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Apart from κL, if the electronic transport properties of these double HH alloys can be made more superior than the corresponding ternary systems, they can be very promising for TE applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' This is one of the motivation of the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Anand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='15 explored a large number of unexplored double half-Heusler alloys and predicted many of them stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' The double half-heusler compounds have a general formula unit X2YY ′Z2 where Y and Y ′ are not isovalent, X is transition metal and Z is a main group element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' For example, the nominal net valence (NV)̸=0 systems such as TiNiSb and TiFeSb are the two ternary components of the quaternary double half-Heusler compound, Ti2FeNiSb2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' The members, namely TiFeSb, TiCoSb, and TiNiSb all have different net valence (NV = -1, 0, and 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Anand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='15 showed that κL of double half-Heusler Ti2FeNiSb2 is lower in comparison to that of its corresponding (NV = 0) ternary counterpart TiCoSb (with the same average atomic mass) by a factor of 3 at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' There are some reports on the effect of doping in these double half heuslers as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' For instance, Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='16 showed that by alloying Ti by Hf and by tuning Fe/Ni ratio, a high figure of merit can be achieved for both p-type and n-type conduction in TiFe0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='5Ni0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='5Sb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' showed enhanced thermoelectric performance due to very low value of thermal conductivity and high power factor in p-type double half heusler Ti2-yHfyFeNiSb2-xSnx compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='17 Recently, Hasan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='18 showed enhanced figure of merit in Ti2FeNiSb1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='8Sn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='2 as compared to the pristine Ti2FeNiSb2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Since the ternary half-heuslers ZrNiSn10,11,19 and HfNiSn11,20 have been widely studied for reporting promising thermoelectric performance, we chose to study the corresponding (NV=0) double half heusler counterparts Zr2Ni2InSb and Hf2Ni2InSb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Anand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='15 theo- retically studied the Gruniesen parameter and thermal conductivity of these compounds in tetragonal (I-42d) structure as reported in the open quantum materials database (OQMD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Both these compounds, however, are experimentally reported to crystallize in cubic (F¯43m 3 (#216)) structure with few impurities, prepared under a specific processing condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='21, 22 In this letter, we use first-principles calculation to investigate the electronic, phonon and thermoelectric properties of double half heuslers Zr2Ni2InSb and Hf2Ni2InSb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' In contrast to previous studies, we have simulated these properties in all of three relevant phases i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=', cubic, tetragonal as well as solid solution, of these compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Depending on the synthesis condition and temperature, there is a possibility to realize all these three phases, ordered in low temperature (T) while disordered in high T-range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Ordered structures could show high carrier mobility than solid solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='23 Thus, ordered structures with low lattice thermal conductivity (κL) can prove to be better thermoelectric material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' For an accurate estimate of band gap, HSE06 functional is used, which yields a band gap of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='17 eV (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='59 eV) for cubic (tetragonal) phase of Zr2Ni2InSb, while the same for Hf2Ni2InSb are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='24 eV (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='4 eV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' The corresponding solid solution phase shows a narrow band gap 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='05 eV and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='06 eV for Zr and Hf based double half heusler alloys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' As expected the simulated κL for double half heusler alloys are smaller than the ternary counterpart (ZrNiSn and HfNiSn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' However these values of κL is not small enough to give promising TE properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Interestingly, these double HH alloys are found to show extremely high electronic transport (S, σ and power-factor), which actually makes them promising for TE application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Ordered phases are found to be almost equally competitive with respect to their TE performance with figure of merit (ZT) value as high as 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='46, at high T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Such comparative study of different structural phases of a single compound is extremely essential and useful to understand the nature of electrons and phonons excitation at different T-ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Computational Details We use Vienna Ab-initio Simulation Package (VASP)25, 26,,27 within DFT, with a projector augmented wave basis28 and the generalized gradient approximation exchange-correlation functional of Perdew−Burke−Ernzerhof (PBE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='29 HSE0630 calculations including spin-orbit 4 coupling (SOC) were performed for the accurate estimation of band gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' A plane-wave energy cutoff of 500 eV was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' The Brillouin zone sampling was done by using a Γ- centered k-mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' For all the compounds, K-meshes of 10 × 10 × 10 (ionic relaxations) and 20 × 20 × 20 (self-consistent-field solutions) were used for PBE calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Cell volume, shape, and atomic positions for all the structures were fully relaxed using conjugate gradient algorithm until the energy (forces) converges to 10−6 eV (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='001 eV/˚A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' A tetrahedron method with Blochl corrections was used for accurate electronic density of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Density functional perturbation theory (DFPT) combined with phonopy31 was used to obtain relevant phonon properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Alloy Theoretic Automated Toolkit (ATAT)32 was used to generate special quasi-random structures to simulate disordered phases of these com- pounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Ab-initio Scattering and Transport (AMSET)33 code was used to calculate the electronic transport properties which uses the variable carrier relaxation time to evalu- ate the transport distribution function while solving the Boltzmann transport equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Debye-Callaway (DC) model43 was used to calculate lattice thermal conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' More details about this model is described in supplementary information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Results and Discussion Table 1 shows the optimized lattice constant and the relative energies of cubic (F¯43m), tetragonal (I¯42d) and SQS structures (P1) of Hf2Ni2InSb and Zr2Ni2InSb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Six different configurations with different site occupancies of In and Sb were simulated using the conven- tional cell of cubic phase and the energetically most stable configuration is presented in the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Tetragonal structure of these compounds is theoretically predicted to be stable in open quantum materials database (OQMD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Solid solution phase (represented by SQS structure here) is usually inevitable for this class of compounds at high temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='23, 35 As such a comparative study of all these three phases is highly desired to facilitate an in-depth analysis 5 Table 1: Theoretically optimized lattice constants and relative energies of cubic, tetragonal and SQS structures of Hf2Ni2InSb and Zr2Ni2InSb Compound Crystal Optimized lattice △E (meV/ structure constants (˚A) atom) cubic a=b=c=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='11 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='0 Hf2Ni2InSb tetragonal a=b=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='11, c=12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='0 SQS a=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='09, b=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='12,c=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='94 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='0 cubic a=b=c= 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='15 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='0 Zr2Ni2InSb tetragonal a=b=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='14, c=12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='0 SQS a=b=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='18, c=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='12 134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='0 of TE properties of these alloys in different T-range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Interestingly, the energy difference between the two ordered phases is very small (8 meV for Hf-based) and 6 meV for Zr-based double HH-alloys), where SQS phase is much higher in energy (60-130 meV) as compared to the lower energy tetragonal phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' As evident from table 1, the SQS structure are off-cubic due to the presence of disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Figure 1 shows the theoretically optimized cubic, tetragonal and SQS structure of Hf2Ni2InSb (top) and Zr2Ni2InSb (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Few bond lengths in both cubic and tetragonal structures are same i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=', dHf-In = dHf-Sb = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='05 ˚A, dIn-Ni = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='68 ˚A, dSb-Ni = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='61 ˚A(in Hf2Ni2InSb) and dZr-In = dZr-Sb = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='07 ˚A, dIn-Ni = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='69 ˚A, dSb-Ni = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='63 ˚A(in Zr2Ni2InSb), while dHf-Ni bond length in cubic and tetragonal structure are 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='61 ˚Aand 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='64 ˚Arespectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' The same for Zr2Ni2InSb are 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='63 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='66 ˚Arespectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' In SQS structure, there is large variation in bond length due to randomness, ranging from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='6 to 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='04 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Electronic structure Figures 2(a) and (b) show the atom-projected band structure of cubic Hf2Ni2InSb and Zr2Ni2InSb respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' For Hf2Ni2InSb, the conduction bands are mostly contributed by Hf and Ni atoms whereas the valence band edges are composed of In and Sb atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Multi- ple valleys are favorable for thermoelectric performance of a compound as it leads to large band degeneracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' The second conduction band minima (CBM-2) and third conduction band minima (CBM-3) lie at an energy difference of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='03 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='05 eV (also contributed by Hf and 6 Figure 1: Theoretically optimized crystal structures of Hf2Ni2InSb (top) and Zr2Ni2InSb (bottom) in (a) cubic (b) tetragonal and (c) SQS phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='5 E-EF (eV) Zr Ni In Sb Γ X M Γ Z R A Z Eg = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='17 eV (b) 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='5 1 E-EF (eV) Zr Ni In Sb Γ X|Y Γ Z|R2 Γ T2|U2 Γ V2 Eg = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='05 eV (f) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='5 E-EF (eV) Hf Ni In Sb Γ X M Γ Z R A Z Eg = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='24 eV (a) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='5 E-EF (eV) Hf Ni In Sb Γ X Y Σ Γ Z Σ1 N P Y1Z Eg = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='4 eV (c) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='5 E-EF (eV) Zr Ni In Sb Γ X Y Σ Γ Z Σ1 N P Y1Z Eg = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='59 eV (d) 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='5 1 E-EF (eV) Hf Ni In Sb ΓX|Y Γ Z|R Γ T|U Γ V Eg = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='059 eV (e) Figure 2: Atom/orbital-projected electronic band structures of (a,b) cubic (c,d) tetragonal and (e,f) disordered SQS structure of Hf- and Zr-based double HH alloys respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Ni atoms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' For Zr2Ni2InSb, the conduction bands are mostly contributed by Ni atoms while valence bands by In atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' The CBM-2 lies at an energy difference of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='08 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' This shows 7 (a) (b) (c) Hf Ni In SbZr Ni In Sblarge conduction band degeneracy in the cubic phase of these compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' The band gap calculated using HSE-SOC functional for Hf and Zr-based compounds are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='24 eV and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='17 eV respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' The former (latter) is an indirect (direct) band gap semiconductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' SOC plays a crucial role due to heavy element Sb, and causes a reduction in the band gap as compared to non-SOC values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Figures 2(c) and (d) show the band structures of Hf2Ni2InSb and Zr2Ni2InSb respectively in their tetragonal phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Both are direct band gap semiconductors with a value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='4 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='59 eV respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Valence band edges show dominant contribution from Hf and Ni atoms whereas conduction band edges are dominated by Ni atoms for Hf2Ni2InSb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' For Zr2Ni2InSb, the dominant contribution arises from In and Ni atoms near valence band edges whereas conduction band edges are mostly composed of Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Figures 2 (e) and (f) show the band structures of disordered phase for Hf and Zr-based compounds respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' HSE-SOC band gap for Hf2Ni2InSb and Zr2Ni2InSb are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='06 eV and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='05 eV respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Hf-atoms have the dominant contribution near valence band edges while a mixed contribution from Hf and Ni- atoms near conduction band edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Zr2Ni2InSb has a similar contribution with Hf replaced by Zr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' The ordered structures have larger band gaps as compared to the disordered SQS struc- tures, which can help in suppressing the bipolar effect36 and hence better electronic transport properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' The ordered tetragonal phase has the largest band gap with flat valence bands which leads to higher density of states effective mass and hence larger p-type thermopower Whereas conduction bands for ordered cubic phase are flat and show multiple valleys at very small energy difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' This indicates better n-type thermopower in ordered cubic phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' All the thermoelectric properties are calculated using PBE band structure with scissor shifted band gap evaluated from HSE-SOC functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' 8 Electronic Transport Transport parameters of most thermoelectric materials are strongly dependent on carrier relaxation time (τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Most previous simulations are based on constant relaxation time ap- proximation(CRTA),37, 38 which is a very crude approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' τ is dictated by different scattering mechanisms such as acoustic scattering, optical scattering, scattering by impuri- ties and defects and electric polarization in case of the polar lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' In the present work, we have estimated the relaxation time for electron and hole transport, using Ab-initio Scattering and Transport (AMSET).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='33 AMSET is a numerical code for calculating carrier relaxation time and transport properties within first principles framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' We have taken into account all four types of scattering mechanisms in our TE calculations i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' acoustic deformation potential (ADP), ionized impurity scattering (IMP), polar optical phonon scattering (POP) and piezoelectric scattering (PIE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' We have also captured the effect of charge carrier screen- ing arising out of free carriers at high concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' The details of these mechanisms is given in supplementary information (SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' The relaxation time (τ) for Hf2Ni2InSb and Zr2Ni2InSb (in cubic phase) is shown in Figures 8 and 9 of supplementary information respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' As can be seen, τ is strongly dependent on both carrier concentration (n) and temperature (T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' For Hf2Ni2InSb, at a high temperature of 900 K, τ varies between 25-21 fs for holes and 35-20 fs for electrons with increasing carrier concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' For Zr2Ni2InSb, τ varies between 26-22 fs for holes and 37-22 fs for electrons at 900 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' The relaxation time (τ) of the two compounds in tetragonal phase are shown in Figure 10 and 11 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' For Hf2Ni2InSb, τ varies between 18-17 fs for holes and 40-19 fs for elec- trons at 900 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' For Zr2Ni2InSb, τ varies between 18-20 fs for holes and 30-20 fs for electrons at 900 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Thus, the order of magnitude of τ do not vary much, as we go from one phase to the other for a given compound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' For SQS structure, we expect a similar or relatively lower value of τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Figure 3 shows a comparison of the T-dependent electronic transport properties ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='of Hf and Zr-based compounds in different phases at a fixed carrier concentration of 1 × 1021 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='700 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='800 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='900 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='S(µVK ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='Cubic-p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='Tetra-p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='SQS-p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='Cubic-n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='Tetra-n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='SQS-n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='T (K) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='(d) Zr2Ni2InSb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='700 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='800 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='κe + κb (Wm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='1K ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='T(K) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='(c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='Figure 3: Temperature dependence of (a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='d) Seebeck coefficient(S),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' (b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='e) power factor (S2σ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' (c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='f) electronic thermal conductivity (κe+κb) for p-type (circle) and n-type (square) conduc- tion in cubic (black line),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' tetragonal (red line) and SQS (blue line) structures of Hf2Ni2InSb and Zr2Ni2InSb respectively at a fixed carrier concentration of 1 × 1021 cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' The choice of carrier concentration is guided by a previous experimental report on Zr-based double HH-alloy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='22 The ordered structures show better electronic properties than the SQS structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Also, as expected from electronic band structures topology, the p-type thermopower of tetragonal structures is larger whereas for cubic phase, n-type thermopower is most promising for both the compounds (see figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' 3(a,d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' As a result, power-factor of p-type tetragonal phase and n-type cubic phase is largest in a large T-range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' (see figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' 3(b,e)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Figures 3(c,f) shows the T-dependence of thermal conductivity (κe+κb) for both the compounds, where κb is bipolar thermal conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' The (κe+κb) values of SQS structures show a rise at higher T (around 500-600 K) because of the bipolar component of κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' This is also the reason for a slight decrease in Seebeck coefficient at higher T for SQS structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' The ordered phases show larger and comparable values of electronic thermal conductivity for n-type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' These double half heusler compounds show promising electronic transport properties 10 0 2 4 6 8 Frequency (THz) K Γ L U Γ W X Γ 0 2 4 6 8 Frequency (THz) A Γ M R Γ X Z Γ 0 2 4 6 8 Frequency (THz) N Γ P S0 Γ S X Γ R G Γ M 0 1 2 3 4 5 6 7 8 ν (THz) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='4 γ TA’ TA LA optical 0 1 2 3 4 5 6 7 8 ν (THz) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='4 γ TA’ TA LA optical 0 1 2 3 4 5 6 7 8 ν (THz) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='4 γ TA’ TA LA optical (i) (a) (b) (e) (f) (j) 0 2 4 6 8 Frequency (THz) K Γ L U Γ W X Γ 0 1 2 3 4 5 6 7 8 ν (THz) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='4 γ TA’ TA LA optical 0 1 2 3 4 5 6 7 8 ν (THz) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='8 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='4 γ TA’ TA LA optical 0 2 4 6 8 Frequency (THz) N Γ P S0 Γ S X Γ R G Γ M 0 1 2 3 4 5 6 7 8 ν (THz) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='4 γ TA’ TA LA optical 0 2 4 6 8 Frequency (THz) A Γ M R Γ X Z Γ (k) (c) (d) (g) (h) (l) Figure 4: Phonon dispersion and mode Gruneisen parameter for ternary (a,b) HfNiSn and (c,d) ZrNiSn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' cubic double HH (e,f) Hf2Ni2InSb and (g,h) Zr2Ni2InSb;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' tetragonal double HH (i,j) Hf2Ni2InSb and (k,l) Zr2Ni2InSb respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' (large power-factor) due to favorable band features such as flat bands in the tetragonal phase and high band degeneracy in cubic phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Thus, the present compounds super- sede/compete well with previously reported high-thermoelectric performance materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' For example, ZrCoBi0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='65Sb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='15Sn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='20 shows a ZT value of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='42 at around 970 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='39 The power-factor for this compound is around 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='8 mWm−1K−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Another promising p-type HH compound FeNb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='8Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='2Sb shows a ZT value of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='1 (at around 970 K)40 with the corresponding power fac- tor of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='3 mWm−1K−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' For n-type HH compounds, Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='5Zr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='25Hf0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='25NiSn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='998Sb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='002 shows a ZT value of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='5 (at 700 K)41 and a power-factor of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='2 mWm−1K−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Hf0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='6Zr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='4NiSn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='995Sb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='005 is yet another system with a promising ZT value of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='2 (at 900 K)42 and a power-factor of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='7 mWm−1K−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' In double half heusler family, Ti4Fe2Ni2Sb4(in solid solution) shows a ZT value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='5 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='4 for p-type and n-type conduction with power-factor values of around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='7 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='2 mWm−1K−2 respectively at around 900 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='16 At the same temperature, the ordered structure for the same compound shows ZT value of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='5 for p-type and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='5 for n-type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='23 The 11 corresponding theoretically calculated power factor values are around 7 mWm−1K−2 and 4 mWm−1K−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Hf2Ni2InSb and Zr2Ni2InSb (see Fig 3) show much higher power-factor val- ues at similar temperature range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Hf2Ni2InSb has a value of 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='2 (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='7) mWm−1K−2 and 32 (24) mWm−1K−2 for for p-type and n-type cubic (tetragonal) structure respectively, while Zr2Ni2InSb has a value of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='4 (12) mWm−1K−2 and 35 (25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='4) mWm−1K−2 for p-type and n-type cubic (tetragonal) structures respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' The disordered (SQS) phases also show comparable/better power-factor values (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='3 mWm−1K−2 for Hf-based and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='4 mWm−1K−2 for Zr-based alloy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Phonon Transport For comparison sake, we have not only calculated the phonon properties (specially the order of magnitude of lattice thermal conductivity, κL) of the present double HH alloys, but also their ternary counterparts ZrNiSn and HfNiSn (with same VEC) which are extremely studied in the literature and reported to crystallize in F¯43m (♯216) structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='10,11,19 Figures 4(a,c), 4(e,g) and 4(i,k) show the phonon dispersion for ternary (HfNiSn, ZrNiSn), cubic double HH (Hf2Ni2InSb, Zr2Ni2InSb) and tetragonal double HH (Hf2Ni2InSb, Zr2Ni2InSb) alloys respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' The ternary, cubic and tetragonal phases have 3-atoms, 6-atoms and 12-atoms in the primitive unit cell giving rise to 9, 18 and 36 phonon branches respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Of them, the lowest 3 branches are acoustic and the rest corresponds to optical branches respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' The acoustic branches are further classified into one longitudinal (LA) and two transverse (TA, TA ′) modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' The velocity of each acoustic mode (γ) is calculated from the slope of the band corresponding to the vibrational band at Γ-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Debye temperature (θ) can be estimated from the maximum frequency corresponding to the given vibrational mode, within a reasonable approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Apart from ν and θ, DC model for κL also requires few other quantities including gruneisen parameter for HfNiSn, cubic Hf2Ni2InSb and tetragonal Hf2Ni2InSb respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' The same for Zr-based compounds are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' 4(d), 4(h) and 4(l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' The simulated values of various parameters,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' phonon group velocities (νi),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Debye ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='700 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='800 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='900 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='Temperature (K) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='KL (Wm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='1 K ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='HfNiSn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='Hf8In8In4Sb4 (Tetragonal) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='Hf4Ni4In2Sb2 (Cubic) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='700 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='800 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='900 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='Temperature (K) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='KL (Wm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='1K ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='ZrNiSn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='Zr8Ni8In4Sb4 (Tetragonal) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='Zr4Ni4In2Sb2 (Cubic) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='Figure 5: Comparison of simulated lattice thermal conductivity (κL) of ternary XNiSn with ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='those of cubic and tetragonal phases of double HH X2Ni2InSb for (left) Hf-based and (right) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='Zr-based compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' temperature (θi), cell volume (V), atomic mass (M), Gruneisen parameter (γi for different vibrational modes (LA, TA, TA ′) for the three Hf and Zr-based compounds are presented in SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Larger values of γ indicates high degree of anharmonicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' The group velocity of acoustic phonons is reduced in double half heuslers (cubic phase) as compared to their ternary counterparts due to large mixing of acoustic and optical phonon modes in the former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' This indicates reduced lattice thermal conductivity for double HH as compared to ternary alloys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Thermal transport properties Figures 5 show the comparison of κL for ordered phases of double half heuslers and their corresponding ternary counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' The simulated values of κL vary between 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='3 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='3 Wm−1K−1 for Hf2Ni2InSb (in cubic phase) and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='4 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='8 Wm−1K−1 (in tetragonal phase) whereas for ternary HfNiSn, it ranges between 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='9 to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='7 Wm−1K−1 in the temperature range 300-900 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' For Zr2Ni2InSb, κL varies between 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='8 to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='3 Wm−1K−1 (for cubic) and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='1 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='9 Wm−1K−1 (for tetragonal) in the temperature range 300-900 K whereas it varies between 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='7 to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='5 Wm−1K−1 for ZrNiSn for the same temperature range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' The previous theoretically reported room temperature κL values of ZrNiSn and HfNiSn are 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='69 and 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='5 Wm−1K−1, whereas for Zr2Ni2InSb and Hf2Ni2InSb (in tetragonal phase), the values are 13 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='58 and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='5 Wm−1K−1 respectively at 300 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='15 These values are in fair agreement with our simulated values obtained from DC model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Clearly, we can see the reduction of lattice thermal conductivity in double half heuslers as compared to the corresponding ternary alloys which is definitely useful for enhancing the TE figure of merit (ZT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' For disorder SQS phase, we expect a further reduction of κL due to enhanced disorder induced scatterings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Thermoelectric performance As evident from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' 3, the power factor for ordered phases is reasonably high (better or comparable to some of the best TE materials in the literature,39, 42 along with the relatively low lattice thermal conductivity (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' This indicates their potential for efficient TE performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' In this section, we shall focus on a comparison of TE figure of merit (ZT) for n and p-type conduction of ordered phases of these two alloys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Figure 6 and 7 show the carrier concentration dependence of ZT at different T for cubic and tetragonal phases of the two alloys respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' The left (right) panel indicates the result for p-type (n-type) conduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' The change in carrier concentration (n) can be thought of as mimicking the effect of doping/alloying the host material, keeping the topology of band structure intact (the so-called rigid band approximation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' For cubic phase, Hf2Ni2InSb show a peak ZT value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='49 for p-type conduction at a carrier concentration of 1 × 1021 cm−3 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='48 for n-type at a carrier concentration of 4 × 1020 cm−3 respectively at 900 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' At these carrier concentrations, the maximum value of simulated Seebeck coefficient (Smax) and power factor (PFmax) are 165.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='5 µVK−1 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='39 mWm−1K−2 for p-type and 248.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='9 µVK−1 and 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='6 mWm−1K−2 for n-type conduction respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' The maximum ZT value for Zr2Ni2InSb is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='44 at a carrier concentration of 1 × 1021 cm−3 for p-type and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='82 at a carrier concentration of 4 × 1020 cm−3 for n-type respectively at 900 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' The simulated Smax and PFmax at these carrier concentrations are 134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='6 µVK−1 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='41 mWm−1K−2 for p-type and are 250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='4 mWm−1K−2 and 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='7 mWm−1K−2 14 for n-type respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' 1e+19 1e+20 1e+21 n (cm 3) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='5 ZT 300 400 500 600 700 800 900 p-type Hf2Ni2InSb (Cubic) 1e+19 1e+20 1e+21 n (cm 3) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='5 ZT n-type 1e+19 1e+20 1e+21 n (cm 3) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='5 ZT 300 400 500 600 700 800 900 p-type Zr2Ni2InSb (Cubic) 1e+19 1e+20 1e+21 n (cm 3) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='5 2 ZT n-type Figure 6: Thermoelectric figure of merit (ZT) as a function of carrier concentration (n) at different temperatures for cubic Hf2Ni2InSb (above) and Zr2Ni2InSb (below) alloy for p-type (left) and n-type (right) conduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' 1e+19 1e+20 1e+21 n (cm 3) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='1 ZT n-type 1e+19 1e+20 1e+21 n (cm 3) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='5 ZT 300 400 500 600 700 800 900 p-type Hf2Ni2InSb (tetragonal) 1e+19 1e+20 1e+21 n (cm 3) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='5 ZT 300 400 500 600 700 800 900 p-type Zr2Ni2InSb (tetragonal) 1e+19 1e+20 1e+21 n (cm 3) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='5 ZT n-type Figure 7: The thermoelectric figure of merit (ZT) as a function of carrier concentration (n) at different temperatures (T) for tetragonal Hf2Ni2InSb (above) and Zr2Ni2InSb (below) alloy for p-type (left) and n-type (right) conduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' The tetragonal phase of both alloys shows the lowest value of lattice thermal conductivity along with better p-type electronic transport properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' The peak value of ZT for Hf2Ni2InSb 15 for p-type and n-type conduction occur at carrier concentrations of 7 × 1020 cm−3 and 2 × 1020 cm−3 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' At these values of carrier concentrations, the Smax and PFmax for p-type are 210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='6 µVK−1 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='96 mWm−1K−2, and for n-type are 270.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='1 µVK−1 and 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='48 mWm−1K−2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' This gives a ZT value of ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='35 at 800 K for p-type and ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='0 at 900 K for n-type conduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' For Zr2Ni2InSb, the peak value of ZT for p-type and n-type conduction is obtained at carrier concentration of 9 × 1020 cm−3 and 2 × 1020 cm−3 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' The Smax and PFmax for p-type are 247.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='2 µVK−1 and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='96 mWm−1K−2 and for n-type are 282.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='5 µVK−1 and 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='03 mWm−1K−2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' As expected, a high ZT value of ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='19 and ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='46 at 900 K for p-type and n-type conduction were obtained for tetragonal Zr2Ni2InSb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Although the power factor of these double half heuslers is quite high in cubic phase for n-type conduction, the alloys actually show higher ZT value in tetragonal phase due to lower values of lattice thermal conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Thus, the reduction of lattice thermal conductivity along with enhanced power factor leads to the improvement of thermoelectric performance for these double half-heusler compounds as compared to corresponding 18 VEC ternary half-heusler alloys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Conclusion In summary, we report an ab-initio study of two double half heusler alloys, X2Ni2InSb (X=Hf/Zr), in their three competing structural phases: tetragonal, cubic and solid solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Double half-heusler alloys are formed via the transmutation of two single heusler compounds and hence have higher flexibility for tuning their properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' The simulated band gaps (using HSE06 hybrid functional) for Hf2Ni2InSb lie in the range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='06-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='4 eV while those for Zr2Ni2InSb in 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='05-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='59 eV depending on the structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Spin-orbit coupling plays a crucial role in splitting the bands due to heavy Sb-atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Thermoelectric performance is mostly dominated by the promising electronic transport in these alloys, with ZT value as high as 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='46 16 for n-type Zr2Ni2InSb and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='0 for n-type Hf2Ni2InSb at high temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Tetragonal phase show a relatively lower lattice thermal conductivity, responsible for the best thermoelectric figure of merit (ZT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' The TE performance of p-type conduction is almost equally promising in tetragonal phase, with highest ZT of ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='35 and ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='19 for Hf- and Zr-based compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' We expect the present study to receive immediate attention from experimentalists with a possibility to synthesize and cross validate our findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' References (1) Tritt, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Tritt, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Kolis, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Ketchum, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Transition-metal pen- tatellurides as potential low-temperature thermoelectric refrigeration materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' B 1999, 60, 13453.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' (5) Nolas, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Kaeser M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Littleton IV, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Tritt, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' High figure of merit in partially filled ytterbium skutterudite materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Appl.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Wood M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Xia Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Wolverton C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Synder G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Double Half-Heuslers, Joule 2019, 3, 1-13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' (16) Liu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Guo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Wu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Mao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Zhu, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Zhu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Pei, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Sui, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Ren Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Design of High-Performance Disordered Half-Heusler Thermoelectric Materials Using 18-Electron Rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' 2019, 29, 1905044.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' 18 (17) Wang, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Li, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Chen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Xue, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Xie, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Cao, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Sui, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Liu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Zhang Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Enhanced Thermoelectric Properties in p-Type DoubleHalf-Heusler Ti2-yHfyFeNiSb2-xSnx Compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Status Solidi A , 2020, 217, 2000096.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' (18) Hasan, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Park, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Kim, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Kim, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Jo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Lee K.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' He J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' High-performance half- Heusler thermoelectric materials Hf1−xZrxNiSn1−ySby prepared by levitation melting and spark plasma sintering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Acta Materialia 2009, 57, 2757–2764.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' (21) Skolozdra, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Romaka, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Aksel ′rud, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Mel ′nik, GA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Tatomir, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' New phases of MgAgAs, LiGaGe and TiNiSi structural types containing d-and p-elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Inorg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Mater.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Giebeler, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Nielsch, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' He R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Improving the thermo- electric performance of ZrNi(In,Sb)-based double half-Heusler compounds.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Olmsted, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Asta, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Dick, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=';' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Liu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Efficient stochastic generation of special quasirandom structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Calphad,2013, 42, 13–18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' (33) Ganose, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' M.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' He, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=';' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='5NiSn (A, B = Ti, Zr, Hf) with a special quasirandom structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' J Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' 2021, 56, 4280–4290.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' 20 (36) Gong, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' J.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' C 2019, 123, 7074-7080 (46) Sahni, B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Vikram;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Kangsabanik, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Alam A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' ”Reliable Prediction of New Quantum Materials for Topological and Renewable-Energy Applications: A High-Throughput Screening.” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' 2020, 15, 6364-6372.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Supporting Information Here, we present auxiliary information on the formalism of lattice thermal conductivity and carrier relaxation time calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Lattice Thermal Conductivity Lattice thermal conductivity (κL) is calculated using the Debye-Callaway (D-C) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='43 The Debye-Callaway (DC) model has been proven to be useful for estimating the lattice thermal conductivity for various experimentally synthesized compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='44, 45, 46 Comparison of κL for three systems (Cu3SbSe4, Cu3SbSe3 and SnSe) with the previous reported exper- imental values have been shown in reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='43 Within this model, the major contribution to the lattice thermal conductivity comes from the 3-phonon scattering process (normal & Umklapp phonon scattering) and is majorly due to the 3 acoustic modes of vibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' The thermal contribution from each of the acoustic vibrational mode in D-C model43,45 is given as, κi L = CiT 3/3 � � � � � θi/T � 0 τ i c(x)x4ex (ex − 1)2 dx + �� θi/T 0 τ i c(x)x4ex τ i N(ex−1)2dx �2 � θi/T 0 τ ic(x)x4ex τ i Nτ i U(ex−1)2dx � � � � � (1) 22 where x = (¯hω/kBT) and the index i denotes LA, TA and TA′ modes of vibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' The constant Ci is given by,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Ci = k4 B 2π2¯h3νi (2) The scattering rates corresponding to the normal and Umklapp scattering processes are,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' 1 τ LA N (x) = k3 Bγ2 LAV M¯h2ν5 LA �kB ¯h �2 x2T 5 (3) 1 τ TA/TA′ N (x) = k4 Bγ2 TA/TA′V M¯h3ν5 TA/TA′ �kB ¯h � xT 5 (4) 1 τ i U(x) = ¯hγ2 Mν2 i θi �kB ¯h �2 x2T 3e−θi/3T (5) where,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' kB is the Boltzmann’s constant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' ¯h is the reduced Plank’s constant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' T is the tempera- ture,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' θ is the Debye temperature,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' V and M are the volume and mass per atom respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' γ is the mode greenness parameter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' ν is phonon group velocity and ω is the angular fre- quency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' The input parameters such as the mode velocities, Debye temperature, Gruneisen parameter, etc were calculated from the phonon band structure and the mode Gruneisen parameter from the Phonopy code,31 which was used as post processing tool after doing the density functional perturbation theory (DFPT) calculations in Vienna Ab-initio Simulation Package (VASP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Simulated parameters needed for κL calculations The numerical values of various parameters used to calculate (κL) for HfNiSn are;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' νLA = 4746.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='3246 m/s, νTA = 3120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='5535 m/s, νTA′ = 2508.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='8948 m/s, γLA = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='55, γTA = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='38,γTA′ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='35, γ= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='59, θLA = 195.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='65 K, θTA = 160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='06 K, θTA′ = 145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='33 K, V = 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='05 × 10−30 m3, and M = 196.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='93 × 10−27 kg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' The numerical values of various parameters used to calculate (κL) for Hf2Ni2InSb (cubic phase) are;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' νLA = 4152.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='7021 m/s, νTA = 3438.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='3485 m/s, νTA′ = 23 2304.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='3666 m/s, γLA = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='43, γTA = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='26,γTA′ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='22, γ= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='60, θLA = 149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='73 K, θTA = 148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='64 K, θTA′ = 148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='19 K, V = 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='05 × 10−30 m3, and M = 196.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='69 × 10−27 kg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' The numerical values of various parameters used to calculate (κL) for Hf2Ni2InSb (tetragonal phase) are;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' νLA = 4963.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='8142, νTA = νTA′ = 2324.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='1295 m/s, γLA= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='40, γTA = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='16,γTA′ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='23, γ= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='6, θLA = 148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='13 K, θTA = 129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='40 K, θTA′ = 123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='89 K, V = 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='05 × 10−30 m3, and M = 196.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='69 × 10−27 kg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' The numerical values of various parameters used to calculate (κL) for ZrNiSn are;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' νLA = 5322.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='6442 m/s, νTA = 3600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='8698 m/s, νTA′ = 2852.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='1311 m/s, γLA = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='69, γTA = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='36,γTA′ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='30, γ= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='71, θLA = 213.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='76 K, θTA = 184.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='12 K, θTA′ = 166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='74 K, V = 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='38 × 10−30 m3, and M = 148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='64 × 10−27 kg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' The numerical values of various parameters used to calculate (κL) for Zr2Ni2InSb (cubic phase) are;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' νLA = 4784.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='2714 m/s, νTA = 4040.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='5217 m/s, νTA′ = 2774.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='3478 m/s, γLA = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='42, γTA = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='28,γTA′ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='24, γ=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='60, θLA = 170.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='93 K, θTA = 171.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='02 K, θTA′ = 171.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='32 K, V = 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='37 × 10−30 m3, and M = 148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='40 × 10−27 kg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' The numerical values of various parameters used to calculate (κL) for Zr2Ni2InSb (tetragonal phase) are;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' νLA = 4696.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='1338 m/s, νTA = 3907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='6455 m/s, νTA′ = 2588.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='8792 m/s, γLA = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='43, γTA = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='26,γTA′ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='22, γ= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='64, θLA = 166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='19 K, θTA = 149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='18 K, θTA′ = 142.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='16 K, V = 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='36 × 10−30 m3, and M = 148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='45 × 10−27 kg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Carrier Relaxation Time 1e+19 1e+20 1e+21 20 30 40 50 60 70 80 τ (fs) 300 500 700 900 n-type n (cm 3) 1e+20 1e+21 20 30 40 50 60 τ (fs) 300 500 700 900 p-type n (cm 3) Figure 8: Temperature dependence of p-type (holes) and n-type(electrons) relaxation time for Cubic Hf2Ni2InSb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' 24 1e+20 1e+21 20 30 40 50 60 τ (fs) 300 500 700 900 p-type n (cm 3) 1e+19 1e+20 1e+21 20 30 40 50 60 70 80 90 τ (fs) 300 500 700 900 n-type n (cm 3) Figure 9: Temperature dependence of p-type (holes) and n-type(electrons) relaxation time for Cubic Zr2Ni2InSb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' 1e+20 1e+21 10 20 30 40 50 τ (fs) 300 500 700 900 p-type n (cm 3) 1e+19 1e+20 1e+21 20 40 60 80 τ (fs) 300 500 700 900 n-type n (cm 3) Figure 10: Temperature dependence of p-type (holes) and n-type(electrons) relaxation time for tetragonal Hf2Ni2InSb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' 1e+20 1e+21 10 20 30 40 50 60 70 80 τ (fs) 300 500 700 900 p-type n (cm 3) 1e+19 1e+20 1e+21 10 20 30 40 50 60 70 τ (fs) 300 500 700 900 n-type n (cm 3) Figure 11: Temperature dependence of p-type (holes) and n-type(electrons) relaxation time for tetragonal Zr2Ni2InSb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' We have estimated the carrier relaxation time for electron and hole transport, using Ab-initio Scattering and Transport (AMSET).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='33 AMSET is a numerical code for calculat- ing carrier relaxation time and transport properties within the ab-initio framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Four scattering mechanisms are simulated in AMSET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' They are acoustic deformation potential 25 scattering, ionized impurity scattering, polar optical phonon scattering and piezoelectric scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' The mode dependent scattering rates are calculated using Born approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Based on Fermi’s golden rule, the differential scattering rate from initial state |nk⟩ to final state |mk+q⟩ is calculated as: τ −1 nk→mk+q = 2π ¯h |gnm(k, q)|2δ(ϵnk − ϵmk+q), (6) where gnm(k, q) is the matrix element for scattering from state |nk⟩ into state |mk + q⟩ and ϵnk is the energy of the state |nk⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' The acoustic deformation potential (ADP) scattering is calculated by assuming that the lattice potential perturbation due to the thermal motion has linear dependence on relative volume change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' The constant of proportionality is the deformation potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' A more general acoustic deformation potential (ADP) matrix element is given by: gad nm(k, q) = � kBT � G̸=−q � ˜Dnk : ˜Sl cl√ρ + ˜Dnk : ˜St1 ct1√ρ + ˜Dnk : ˜St2 ct2√ρ � ⟨mk + q|ei(q+G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='r|nk⟩ (7) where ˆS = ˆq⊗ ˆu is the unit strain for an acoustic mode, ˜Dnk = Dnk + vnk ⊗ vnk in which Dnk is the rank 2 deformation potential tensor, ˆu is the unit vector of phonon polarization, and the subscripts l, t1 and t2 indicate properties belonging to the longitudinal and transverse modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' In polar semiconductors, an electric polarization along with the deformation potential is produced by the lattice oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Since the ions oscillate out-of-phase for optical modes, an electric dipole moment varying with time is generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' This leads to an additional interaction of optical phonons with charge carriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' The electric field due to perturbation of dipole moment between atoms leads to scattering of carriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Thus, Polar optical phonon 26 (POP) scattering rate is given by: gpo nm(k, q) = �¯hωpo 2 �1/2 � G̸=−q � 1 ˆn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='ϵ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='ˆn − 1 ˆn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='ϵs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='ˆn �1/2⟨mk + q|ei(q+G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='r|nk⟩ |q + G| (8) where ϵs and ϵ∞ are the static and high-frequency dielectric tensors and ωpo is the polar optical phonon frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' The ionized impurity scattering is an important scattering mechanism at lower temper- atures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' The mobile carriers are attracted by ionized impurity which leads to screening of potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' The matrix element for ionized impurity scattering33 is given by: gpo nm(k, q) = � G̸=−q n1/2 ii Ze ˆn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='ϵs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='ˆn ⟨mk + q|ei(q+G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='r|nk⟩ |q + G|2 + β2 (9) where Z is defect charge, nii is the concentration of ionized impurities which is equal to charge compensation × (nholes-nelectrons)/Z and β is the inverse screening length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' The piezoelectric scattering (PIE) is basically polar acoustic phonon scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' This scattering mechanism is important at very low temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' However we have included this scattering as well in our calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' In 300 to 900K, POP and ADP are the dominant scattering mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Acknowledgment BS acknowledges financial support from the Indian Institute of Technology, Bombay in the form of teaching assistantship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' BS acknowledges Vikram for some initial discussions regard- ing the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' acknowledges DST-SERB (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' CRG/2019/002050) for funding to support this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' 27 Author Contributions BS and AA conceived the initial idea of the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' BS performed all the calculations and analyzed them with the help of AA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' BS wrote the initial draft of the manuscript, which was further corrected by AA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' AA supervised the entire project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' Competing financial interests The authors declare no competing financial interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} +page_content=' 28' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQftfn_/content/2301.00598v1.pdf'} diff --git a/RtE4T4oBgHgl3EQfKwzR/content/tmp_files/2301.04933v1.pdf.txt b/RtE4T4oBgHgl3EQfKwzR/content/tmp_files/2301.04933v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..042bfea00a6169313bc78c4998f6b9ccdf2cf48b --- /dev/null +++ b/RtE4T4oBgHgl3EQfKwzR/content/tmp_files/2301.04933v1.pdf.txt @@ -0,0 +1,2001 @@ +RATIONALITY AND PARAMETRIZATIONS OF ALGEBRAIC CURVES +UNDER SPECIALIZATIONS +SEBASTIAN FALKENSTEINER AND J.RAFAEL SENDRA +Abstract. Rational algebraic curves have been intensively studied in the last decades, +both from the theoretical and applied point of view. In applications (e.g. level curves, linear +homotopy deformation, geometric constructions in computer aided design, etc.), there often +appear unknown parameters. It is possible to adjoin these parameters to the coefficient field +as transcendental elements. In some particular cases, however, the curve has a different +behavior than in the generic situation treated in this way. In this paper, we show when +the singularities and thus the (geometric) genus of the curves might change. More precisely, +we give a partition of the affine space, where the parameters take values, so that in each +subset of the partition the specialized curve is either reducible or its genus is invariant. In +particular, we give a Zariski-closed set in the space of parameter values where the genus of +the curve under specialization might decrease or the specialized curve gets reducible. For +the genus zero case, and for a given rational parametrization, a better description is possible +such that the set of parameters where Hilbert’s irreducibility theorem does not hold can be +isolated, and such that the specialization of the parametrization parametrizes the specialized +curve. We conclude the paper by illustrating these results by some concrete applications. +Algebraic curves, parameters, rational parametrizations, singularities, geometric genus, +Hilbert’s irreducibility theorem +Acknowledgements +Authors partially supported by the grant PID2020-113192GB-I00 (Mathematical Visu- +alization: Foundations, Algorithms and Applications) from the Spanish MICINN. Part of +this work was developed during a research visit of the first author to CUNEF University in +Madrid. +1. Introduction +The study and analysis of the behavior of algebraic or algebraic-geometric objects under +specializations is of great interest from a theoretical, computational or applied point of view. +For instance, some techniques for computing resultants, gcds, or polynomial factorizations, +rely on Hensel’s lemma or the Chinese remainder theorem (see e.g. [11], [39]). From a more +theoretical point of view, also computational, it is important to control, for instance, when a +resultant, or more generally a Gr¨obner basis with parameters, specializes properly (see e.g. +[6], [17]). The question whether a given irreducible polynomial over K(a1,...,an) remains +irreducible when the parameters are replaced by values in a field K was studied intensively +by Hilbert [8] and Serre [31] and is the defining property of “Hilbertian fields”. The work +of Serre can be seen in a more general context. With respect to applications, there is a vast +Max Planck Institute Leipzig, Germany. +Universidad de Alcal´a, Dpto. F´ısica y Matem´aticas, Alcal´a de Henares, Madrid, Spain +E-mail addresses: sebastian.falkensteiner@mis.mpg.de, rafael.sendra@uah.es. +Date: January 13, 2023. +1 +arXiv:2301.04933v1 [math.AG] 12 Jan 2023 + +2 +RATIONALITY AND PARAMETRIZATIONS OF ALGEBRAIC CURVES UNDER SPECIALIZATIONS +amount of applications of algebraic curves involving parameters: level curves of surfaces [2], +linear homotopy deformation of curves (see Section 7), curve recognition [34], geometric +constructions in computer aided design, like offsets, conchoids, cissoids etc, where the final +object depends on the distance, the focus, etc.; see e.g. [3], [4], [16], [19], [24]. Another type +of applications is the computation with meromorphic functions in linear algebra (see [25]) or +the rational solutions of functional algebraic equations (see Section 7). +In this work we study algebraic curves C(F) given as the zero-set of a polynomial +(1.1) +F(x,y) = 0 with F ∈ K(a1,...,an)[x,y] +where K is a computable field of characteristic zero, a1,...,an are a set of parameters, and +F is irreducible over the algebraic closure of the coefficient field K(a1,...,an), denoted by +K(a1,...,an). +In this paper, we focus on the problem that for certain values of the pa- +rameters a1,...,an the algebraic properties of the resulting curve do not coincide with the +generic properties of C(F). More precisely, we define several Zariski-closed sets in the space +of parameter values where non-generic behavior may appear. Of particular interest are the +singularities, their multiplicities and their character. This leads to a partition of the affine +space, where the parameters take values, so that in each subset of the partition the special- +ized curve is either reducible or its (geometric) genus is invariant. When the generic curve +has genus zero, for a given rational parametrization can be given a better description. In +particular, the set of parameters where Hilbert’s irreducibility theorem does not hold can be +isolated. Moreover, the proper specialization of the rational parametrization is guaranteed. +In [30, 37] and references therein are studied algebraic curves and their rationality. The +problem of finding rational parametrizations of plane curves is a classical problem and has +already been studied by Hilbert [9], and more recently in [13], [21], [27],[28]. In addition, +for evaluating the parameters, it is important to control field extensions which might be +necessary for computing parametrizations. +Optimal fields of parametrizations have been +studied in [13] and [28]. When introducing parameters in the coefficients, new phenomena +have to be considered and lead to Tsen’s study of finding solutions in a minimal field [7]. +The structure of the paper is as follows. In Section 2 we present notations, preliminaries on +algebraic curves and rational parametrizations. Of particular interest is the computation of +the genus and a rational parametrization, if it exists. Some of the details are attached in the +appendix1 A. In Section 3, we introduce the unspecified parameters and their specialization. +The computation of the genus and rational parametrizations is followed to define several +computable Zariski-open subsets Ω where the specialized curve behaves, up to irreducibility, +as in the generic case. The actual computation of the genus is presented in Section 4. In +Theorem 4.2 is shown that the genus of the specialized curve, where the parameters take +values in ΩsingOrd, is less or equal to the generic genus or the defining polynomial is reducible. +A direct corollary of that is that specialized curves of rational curves are also rational or +reducible (Corollary 4.3). For values in a smaller set ΩgenusOrd, it is shown that the genus of +the curve remains exactly the same, again up to irreducibibility, see Theorem 4.8. Section 5 +is devoted to the case where the generic curve is rational; in this frame the irreducibility can +be guaranteed. For some of the parameter values the genus may remain the same but an +evaluation of the parametrization is not possible. In Theorem 5.5, however, is presented an +1There exist different methods to deal computationally with the genus: the adjoint curve based method +(see e.g. [37] and [30]), the method based on the anticanonical divisor (see [13]) or the method based on +Puiseux expansions (see [20]), among others. In this paper we will follow the adjoint curve based method +which is described, for completeness, in the appendix. + +RATIONALITY AND PARAMETRIZATIONS OF ALGEBRAIC CURVES UNDER SPECIALIZATIONS +3 +open set where the specialization is possible and results in a parametrization of the specialized +curve. These open sets can be recursively used for decomposing the whole parameter space as +it is explained in Section 6. Applications as described above are presented by using illustrative +examples in Section 7. +This manuscript is a self-contained work on the computation of the genus and rational +parametrizations of algebraic curves involving parameters. Results from various mathemat- +ical disciplines are combined for this purpose and presented in a coherent way. A rigorous +construction of such computable Zariski-open sets were, up to our knowledge, missing in the +literature. The theorems mentioned in the previous paragraph are novel and can be directly +applied in several interesting problems involving parametric curves. +2. Preliminaries and notation +Throughout this paper, the following notation will be used. K is a computable field of +characteristic zero. We denote by a a tuple of parameters, and we represent by L the field +extension L ∶= K(a). In addition, we consider an algebraic element γ over L. Let F be the +field F ∶= L(γ). Furthermore, K represents any field extension of K. We denote by K the +algebraic closure of K, similarly for any field appearing in the paper. S is the affine space +(2.1) +S = K +#(a) +where a will take values. +For G ∈ K[x,y] ∖ K, we denote by C(G) the plane affine algebraic curve +C(G) = {p ∈ K +2 ∣G(p) = 0}. +We denote by Gh(x,y,z) the homogenization of G, and by Gx,Gy (similarly for Gh +x,Gh +y,Gh +z) +the partial derivative of G w.r.t. +x and y respectively. +For a homogeneous polynomial +M ∈ K[x,y,z] ∖ {0}, C(M) denotes the projective plane curve +C(M) = {p ∈ P2(K)∣M(p) = 0}. +For polynomials f,g in the variable t, and coefficients in an integral domain, we denote by +rest(f,g) the resultant of f and g w.r.t. t. +Let {f1,...,fk} ⊂ K[v], where v is a tuple of variables. We denote by V(f1,...,fk) the +zero set, over K, of the polynomials {f1,...,fk}; similarly for V(I) where I is an ideal in +K[v]. +2.1. Rational Curves. Throughout this section, let G ∈ K[x,y]∖K be irreducible over K. A +rational (affine) parametrization of the irreducible affine plane curve C(G) is a pair of rational +functions P(t) ∈ K(t)2 ∖ K +2 such that G(P(t)) = 0. A rational (projective) parametrization +of C(Gh) is of form Q(h,t) = (p1(h,t) ∶ p2(h,t) ∶ p3(h,t)) where pi are homogeneous co-prime +polynomials of the same degree over K, not all zero, such that Gh(Q) = 0. We observe that +the degree, the irreducibility and the rationality of C(G) and C(Gh) are equivalent. Moreover +the parametrizations of C(G) and C(Gh) relate each other by means of homogenizing and +dehomogenizing. So, in the following we will focus on affine parametrizations. +The parametrization P(t) is called birational or proper if the map K ⇢ C(G);t ↦ P(t) is +injective in a non–empty open Zariski subset of K (see e.g. [30] for further details). Curves +admitting a rational parametrization are called rational, and they correspond to those of genus +zero; note that the genus of C(G) is defined as the genus of C(Gh). There exist algorithmic +methods to compute the genus of an algebraic curves and to determine, when the genus is + +4 +RATIONALITY AND PARAMETRIZATIONS OF ALGEBRAIC CURVES UNDER SPECIALIZATIONS +zero, a rational parametrization of the curve (see e.g. [13], [27], [28], [30]). In Appendix A +we summarize the adjoint curves based method for parametrizing curves. Some of the ideas +in this paper will use those methods. +In general, if one computes a parametrization P(t) of C(G), the ground field K has to be +extended (see e.g. Sections 4.7. and 4.8. in [30]). A subfield E of K is called a parametriz- +ing field or field of parametrization of C(G) if there exists a parametrization of C(G) with +coefficients in E. +2.2. Fields of Parametrization. In this section, we work with the field L ∶= K(a). Let +G ∈ L[x,y] be an irreducible (over L) non-constant polynomial, and let us assume C(G) is a +rational curve. We analyze the fields of parametrization of C(G). +L is always a field of parametrization of C(G). Nevertheless, in [28] (see also Chapter 5 +in [30]), the optimality of the fields of parametrization is analyzed and, as a consequence of +Hilbert-Hurwitz Theorem (see [9]), there always exists a field extension of L, of degree at +most 2, being a field of parametrization of C(G). Indeed, this field extension, of degree at +most two, is the field extension used in Step (4), of the parametrization computation (see +Subsection A.2), to express the simple point utilized in the parametrization of either the +conic or the line. +We observe that if the two degree field extension is L(α), with minimal polynomial t2 + +bt + c ∈ L[t], then L(α) = L(β) where β = α + b/2 which minimal polynomial is t2 + c − b2/4. +Therefore, the following holds. +Theorem 2.1. +(1) If deg(C(G)) is odd then L is a field of parametrization. +(2) If deg(C(G)) is even then either L is a field of parametrization or there exists δ ∈ L +algebraic over L, with minimal polynomial t2 − α ∈ L[t], such that L(δ) is a field of +parametrization of C(G). +Remark 2.2. Observe that the previous result is valid taking L as any field extension of K. +The case where a contains a single element admits a particular treatment because of Tsen’s +Theorem; we refer to [7] for this topic. +Corollary 2.3. If #(a) = 1, then L is a field of parametrization of C(G). +Proof. By Hilbert-Hurwitz Theorem (see e.g. Theorem 5.8. in [30] or Subsection A.2), C(G) +is L–birationally equivalent to either a line or a conic. +So, fields of parametrization are +preserved. In the line case, the result is clear. In the conic case, the result follows from +Tsen’s Theorem (see e.g. Corollary 1.11. in [32]). +□ +Remark 2.4. The proof of Tsen’s Theorem provides a method for computing an L-simple +point on the conic. An alternative approach for computing this point can be found in [12] +and [33]. +Remark 2.5. In the following section we will work with G ∈ K[a,γ][x,y] where γ is algebraic +over K(a). In the case where #(a) = 1, we can view γ as the only parameter and write a in +terms of γ. More precisely, let M(a,c) ∈ K(a)[c] be the minimal polynomial of γ. We can +view M as rational expression in a and consider H(a) ∶= num(M)(a = a,c = γ) ∈ K(γ)[a] +as polynomial in a with the root a. +Thus, K(γ,a) ∶ K(γ) is a field extension of degree +d ≤ degc(M). If d = 1, by Corollary 2.3, K(γ) is a field of parametrization of C(G). + +RATIONALITY AND PARAMETRIZATIONS OF ALGEBRAIC CURVES UNDER SPECIALIZATIONS +5 +Remark 2.6. In Corollary 2.3, we have seen that if #(a) = 1 then L is a field of parametriza- +tion. The following example shows that if #(a) > 1, in general, L is not a field of parametriza- +tion. We consider the conic defined by F ∶= a1x2 +a2y2 −1, and we see that it does not have a +parametrization over C(a1,a2). Let us assume that C(a1,a2) is a field of parametrization of +C(F), then C(F) has infinitely many point in C(a1,a2)2. C(F) can be properly parametrized +by +P ∶= (√a1 +1 − t2 +t2 + 1,√a2 +2t +t2 + 1), +which inverse is +P−1(x,y) = +√a2 (√a1 + x) +√a1 y +. +So, there are infinitely many points in C(F)∩C(a1,a2)2 that are injectively reachable, via P, +for t ∈ C(√a1,√a2). Indeed, note that all points of C(F), with the exception of (−√a1,0), +are reachable by P. Let t0 ∈ C(√a1,√a2) ∖ {0,±i} be one of these parameter values; say +P(t0) = (x0,y0) ∈ C(a1,a2)2. Then t2 +0 = (√a1 + x0)/(√a1 − x0) ∈ C(√a1,a2). For x0 ≠ 0, it +holds that √a1 +x0, √a1 −x0 are coprime (seen as polynomials in √a1) and t0 = ± +√√a1+x0 +√a1−x0 ∉ +C(√a1,√a2), a contradiction. For x0 = 0 we have the curve-points (x0,y0) = (0,±1/√a2) +which are not in C(a1,a2). +3. Specializations +Throughout the paper, we will specialize the tuple of parameters a taking values in S (see +(2.1)). We will write a0 to emphasize that the parameters in a have been substituted by +elements in K. In the following we discuss different aspects on the specializations. +3.1. General statements. The elements in K(a) are assumed to be represented in reduced +form; that is, the numerator and denominator are assumed to be coprime. Then, for f ∶= +p/q ∈ K(a), where by assumption gcd(p,q) = 1, and for a0 ∈ S (see (2.1)) such that q(a0) ≠ 0, +we denote by f(a0) the K–element p(a0)/q(a0). +We may need to work in the finite field extension F = L(γ). Let p(a,t) ∈ K(a)[t], of degree +k in t, be the minimal polynomial of γ. We might simply write p(t) instead of p(a,t) and +express it as +(3.1) +p(t) = tk + Nk−1(a) +Dk−1(a) tk−1 + ⋯ + N0(a) +D0(a), Ni,Di ∈ K[a] +where gcd(Ni,Di) = 1. +Then, for a0 ∈ S such that all Di(a0) ≠ 0, we denote by γ0 the +algebraic element, over K(a0), defined by an irreducible factor of +p(a0,t) = tk + Nk−1(a0) +Dk−1(a0) tk−1 + ⋯ + N0(a0) +D0(a0) ∈ K(a0)[t] ⊂ K[t]. +For an element f ∈ F, specialized at a0 ∈ S, we might simply write f(a0) instead of f(a0,γ0). +Definition 3.1. We define the open subset Ωγ ∶= S ∖ V(D) where D ∶= lcm(D0,...,Dn−1). +Clearly for a0 ∈ Ωγ, γ(a0) is well–defined. The elements in F are assumed to be expressed +in canonical form; that is, f ∈ F is expressed as +(3.2) +f = +k−1 +∑ +i=0 +Ui(a) +W(a)γi + +6 +RATIONALITY AND PARAMETRIZATIONS OF ALGEBRAIC CURVES UNDER SPECIALIZATIONS +where Ui,W ∈ K[a] and gcd(U1,...,Uk−1,W) = 1. In addition, the coefficients of polynomials +in F[v], w.r.t. to the tuple of variables v, are also supposed to be written in canonical form. +Moreover, for f as in (3.2), we denote by Norm(f) the L–field element +Norm(f) ∶= ∏f(a,γi) ∈ L +where the product is taken over all roots γi in L of p(a,t) (see (3.1)). Note that Norm(f) = +rest(f(a,t),p(a,t)). +Lemma 3.2. Let f be as in (3.2). Let a0 ∈ S be such that D(a0)W(a0) ≠ 0 (see Def. 3.1). +If f(a0) = 0, then Norm(f)(a0) = 0. +Proof. Since D(a0)W(a0) ≠ 0, then γ(a0), f(a0).Then Norm(f)(a0) = ∏f(a0,γi(a0)) is +well-defined and, since one of the factors on the right hand side is equal to f(a0,γ(a0)) = 0, +we obtain Norm(f)(a0) = 0. +□ +Definition 3.3. Let H ∈ F[v], where v is a tuple of variables. Let S be the set of all non-zero +coefficients of H w.r.t. v. Let +D(H) ∶= lcm({denom(C)∣C ∈ S}) ∈ K[a], +and let +V(H) ∶= {Norm(numer(C))∣C ∈ S} ⊂ K[a]. +We associate to H the following open subsets +(1) Ωdef(H) ∶= Ωγ ∩ (S ∖ V(D(H))). +(2) ΩnonZ(H) ∶= Ωdef(H) ∩ (S ∖ V(V(H))). +Remark 3.4. Throughout the paper, we will define several open subsets of S. All these open +subsets will be included in Ωγ (for the corresponding algebraic element γ). So, we observe +that γ0 will be always well–defined. +The next lemma justifies the previous definitions. +Lemma 3.5. Let H ∈ F[v], where v is a tuple of variables. It holds that +(1) If a0 ∈ Ωdef(H) then H(a0,γ0,v) is well-defined. +(2) If a0 ∈ ΩnonZ(H) then H(a0,γ0,v) ≠ 0. +Proof. (1) Let a0 ∈ Ωdef(H) ⊂ Ωγ. Then, γ0 = γ(a0) is well–defined, and the result follows +from the definition of D. +(2) Let a0 ∈ ΩnonZ(H) ⊂ Ωdef(H). Then, by (1), H(a0,γ0,v) is well–defined. Furthermore, +there exists a coefficient of H w.r.t. v, say C(a,γ), such that Norm(numer(C))(a0) ≠ 0. +Since D(a0) ≠ 0 and the denominator of C does not vanish at a0, by Lemma 3.2, we get that +C(a0,γ0) ≠ 0. So, H(a0,γ0,v) ≠ 0. +□ +The following lemma is an adaptation of Lemma 3 in [29] to our case, and will be used to +control the birationality of a curve parametrization P(a,t) under specializations of a. +Definition 3.6. Let f1,f2 ∈ F[u][v]∖{0} for i ∈ {1,2}, where u,v are variables. Let fi = f∗ +i g, +for i ∈ {1,2}, where g = gcd(f1,f2). Let Ai ∈ F[u] be the leading coefficient of fi w.r.t. v for +i ∈ {1,2} and B ∈ F[u] the leading coefficient of g w.r.t. v. Let R = resv(f∗ +1 ,f∗ +2 ) ∈ F[u]. Let +Ω1 ∶= Ωdef(f1) ∩ Ωdef(f2) ∩ Ωdef(f∗ +1 ) ∩ Ωdef(f∗ +2 ) ∩ Ωdef(g) ∩ Ωdef(R), +Ω2 ∶= ΩnonZ(A1) ∩ ΩnonZ(A2) ∩ ΩnonZ(B) ∩ ΩnonZ(R). + +RATIONALITY AND PARAMETRIZATIONS OF ALGEBRAIC CURVES UNDER SPECIALIZATIONS +7 +We define the set +Ωgcd(f1,f2) ∶= Ω1 ∩ Ω2. +Lemma 3.7. Let f1,f2,f∗ +1 ,f∗ +2 ,g be as in Def. 3.6. For a0 ∈ Ωgcd(f1,f2), it holds that +g(a0,γ0,u,v) = λ(u)gcd(f1(a0,γ0,u,v),f2(a0,γ0,u,v)), +with λ(u) ∈ K[u] ∖ {0}. Moreover, degv(g(a0,γ0,u,v)) = degv(g(a,γ,u,v)). +Proof. Let Ai,B,R be as in Def. 3.6. +Since a0 ∈ Ωgcd(f1,f2) ⊂ Ω1 (see Def. +3.6), by +Lemma 3.5 (1), the specializations of fi,f∗ +i ,g,R at a0 are well-defined. So, +(3.3) +fi(a0,γ0,u,v) = f∗ +i (a0,γ0,u,v)g(a0,γ0,u,v). +Moreover, since a0 ∈ Ωgcd(f1,f2) ⊂ Ω2 (see Def. 3.6), the specializations of fi,g,R at a0 preserve +the degree in v and, in particular, are non–zero. This implies that f∗ +i (a0,γ0,u,v) are non–zero +too. From (3.3), one has that there exists λ ∈ K[u] ∖ {0} such that +gcd(f1(a0,γ0,u,v),f2(a0,γ0,u,v)) = λ(u) gcd(f∗ +1 (a0,γ0,u,v),f∗ +2 (a0,γ0,u,v))g(a0,γ0,u,v). +Let us assume that gcd(f∗ +1 (a0,γ0,u,v),f∗ +2 (a0,γ0,u,v)) has positive degree in v. Then, if +˜R(u) is the resultant w.r.t. v of f∗ +1 (a0,γ0,u,v) and f∗ +2 (a0,γ0,u,v), we get that ˜R is zero +(see e.g. Corollary page 288 in [11]). However, since a0 ∈ Ωgcd(f1,f2) ⊂ Ω2 (see Def. 3.6), by +Lemma 3.5(2), A1,A2 do not vanish at a0 and, hence, the leading coefficients of f∗ +i do not +vanish either at a0. Therefore, by Lemma 4.3.1 in [39], R(a0,γ0,u) = ˜R(u). Nevertheless, +since a0 ∈ Ω2 by Lemma 3.5 (2), R(a0,γ0,u) ≠ 0 which is a contradiction. So, g(a0,γ0,u,v) +and gcd(f1(a0,γ0,u,v),f2(a0,γ0,u,v)) are associated. +In addition, since a0 ∈ Ω2, by +Lemma 3.5 (2), B(a0,γ0,u) ≠ 0 and, hence, degv(g(a0,γ0,u,v)) = degv(g(a,γ,u,v)). +□ +If in Lemma 3.7 all coefficients are assumed to be in a field, the statement can be simplified +as follows. +Corollary 3.8. Let f1,f2 ∈ F[v] ∖ {0} for i ∈ {1,2}. Let fi = f∗ +i g, for i ∈ {1,2}, where +g = gcd(f1,f2). For a0 ∈ Ωgcd(f1,f2), it holds that +g(a0,γ0,t) = gcd(f1(a0,γ0,v),f2(a0,γ0,v)). +Moreover, degv(g(a0,γ0,v)) = degv(g(a,γ,v)). +Let us now generalize the previous statement to several univariate polynomials with coef- +ficients in F. +Definition 3.9. Let f1,...,fr ∈ F[v] ∖ {0}. +Let fi = f∗ +i g, for i ∈ {1,...,r}, where g = +gcd(f1,...,fr). We consider the polynomial fZ ∶= f2 + f3Z + ⋯ + frZr−2 ∈ F(Z)[v] where Z is +a new variable. We define +Ωgcd(f1,...,fr) = Ωgcd(f1,fZ) ∩ Ωdef(g) ∩ ΩnonZ(A) +where A is the leading coefficient of g w.r.t. v. +Remark 3.10. Observe that if r = 2 in Def. 3.9, then Def. 3.6 and 3.9 coincide. +Theorem 3.11. Let f1,...,fr,f∗ +1 ,...,f∗ +r ,g be as in Def. 3.9. For a0 ∈ Ωgcd(f1,...,fr), it holds +that +g(a0,γ0,v) = gcd(f1(a0,γ0,v),...,fr(a0,γ0,v)). +Moreover, degv(g(a0,γ0,v)) = degv(g(a,γ,v)). + +8 +RATIONALITY AND PARAMETRIZATIONS OF ALGEBRAIC CURVES UNDER SPECIALIZATIONS +Proof. Let g∗ ∶= gcd(f1,fZ) ∈ F(Z)[v]. +Since f1 does not depend on Z, g∗ ∈ F[v]. +This implies that g = λg∗ with λ ∈ F. +By Corollary 3.8, we know that g∗(a0,γ0,t) = +gcd(f1(a0,γ0,v),fZ(a0,γ0,Z,v)) and that degv(g∗(a0,γ0,v)) = degv(g∗(a,γ,v)). Note that +degv(g∗(a,γ,v)) = degv(g(a,γ,v)). Moreover, since a0 ∈ Ωdef(g) ∩ ΩnonZ(A), then g(a0,γ0,v) +is well–defined. Furthermore, since both leading coefficients of g and g∗ w.r.t. v do not +vanish at a0, then λ(a0,γ0) is well–defined and non–zero. +Thus, degv(g∗(a0,γ0,v)) = +degv(g(a0,γ0,v)). Summarizing, degv(g(a0,γ0,v)) = degv(g(a,γ,v)). On the other hand, +g(a0,γ0,v) = λ(a0,γ0)g∗(a0,γ0,v) = λ(a0,γ0) gcd(f1(a0,γ0,v),fZ(a0,γ0,Z,v)) +and, since λ(a0,γ0) ≠ 0, this implies that g(a0,γ0,v) = gcd(f1(a0,γ0,v),...,fr(a0,γ0,v)). +□ +Our next step is to analyze the squarefreeness. +Definition 3.12. Let f ∈ F[v]∖F be squarefree. Let R be the discriminant of f w.r.t. v and +let A be the leading coefficient of f w.r.t. v. We define the open subset +Ωsqfree(f) ∶= Ωdef(f) ∩ ΩnonZ(R) ∩ ΩNonZ(A)). +Lemma 3.13. Let f ∈ F[v] ∖ F be squarefree. +If a0 ∈ Ωsqfree(f), then degv(f(a,γ,v)) = +degv(f(a0,γ0,v)) and f(a0,γ0,v) is squareefree. +Proof. Since a0 ∈ Ωdef(f), by Lemma 3.5, f(a0,γ0,v) is well–defined and, since a ∈ ΩNonZ(A)), +f(a0,γ0,v) ≠ 0. +Furthermore, one has the equality of the degrees. +Moreover, since a0 ∈ +ΩnonZ(R), also by Lemma 3.5, R(a0,γ0,v) is well-defined and non–zero. Since A(a0,γ0,v) ≠ 0, +by [39, Lemma 4.3.1], the discrimininant of f(a0,γ0,v) is not zero. Then, by [39, Theorem +4.4.1], f(a0,γ0,v) is squarefree. +□ +3.2. Specialization of the curve defining polynomial. In this subsection, we deal with +the specialization of defining polynomials of irreducible plane curves. Let G ∈ F[x,y] ∖ F +be irreducible over F of total degree d and let Gh ∈ F[x,y,z] be its homogenization. In the +following, let G be written as +(3.4) +G = gd(x,y) + ⋯ + g0(x,y) +where gi is either the zero polynomial or a form of degree i. +Definition 3.14. We associate to G the open subset (see (3.4)) ΩG ∶= Ωdef(G) ∩ ΩnonZ(gd). +Lemma 3.15. Let G be as above and let a0 ∈ ΩG. Then +(1) G(a0,x,t) is well–defined and deg(G(a0,x,y)) = deg(G); +(2) G(a0,x,y)h = Gh(a0,x,y,z); +(3) the partial derivatives of Gh, of any order, specialize properly. +Proof. Since a0 ∈ Ωdef(G), by Lemma 3.5, G(a0,x,y) is well–defined and, since a0 ∈ ΩnonZ(gd), +the equality on the degree holds. +Since G(a0,x,y) is well–defined, the other statements +directly follow. +□ + +RATIONALITY AND PARAMETRIZATIONS OF ALGEBRAIC CURVES UNDER SPECIALIZATIONS +9 +3.3. Specialization of families of points. Let us now deal with the specialization of conju- +gate families of points associated to a curve. More precisely, let G and Gh be as in Subsection +3.2. We will study the specialization of families in the standard decomposition of the singular +locus of C(Gh). For this purpose, we observe that, for each a0 ∈ S such that G(a0,x,y) /∈ K, +G(a0,x,y) defines an affine plane curve over K. Let us denote by C(G) the first curve and +by C(G,a0) the second. +The conjugate families of C(Gh) will be over F. When we specialize a we need to have a +reference field where the conjugation of the points is defined. This motivates the following +definition. +Definition 3.16. For a0 ∈ S, we define Ka0 as the smallest subfield of K containing the +coefficients of G(a0,γ0,x,y). Moreover, if F = {(f1 ∶ f2 ∶ f3)}m(t) is an F–conjugate family, +and a0 ∈ S is such that γ0,f1(a0,t),f2(a0,t),f3(a0,t),m(a0,t) are well–defined, we denote +by F(a0) the specialization of F at a0 (and γ0). +Let F(Gh) be an F–standard decomposition of the singular locus of C(Gh) obtained using +the process described in Subsection A.1. Let F(Gh) decompose as +(3.5) +F(Gh) = +⋃ +m(t)∈Aa +{(f1,m(t) ∶ f2,m(t) ∶ 1)}m(t) ∪ +⋃ +m(t)∈A∞ +{(L1,m(t) ∶ L2,m(t) ∶ 0)}m(t) +where fi,m,m(t) ∈ F[t], Li,m ∈ K[t] (recall that the transformation L, in the standard +decomposition process in Subsection A.1, can be taken over K) with deg(Li,m) ≤ 1, +gcd(L1,m,L2,m) = 1, and m(t) irreducible over F, and where Aa and A∞ are finite sets +of irreducible polynomials in F[t]. By abuse of notation, we will write F ∈ F(Gh) for such a +component F of F(Gh). +Definition 3.17. Let F ∶= {(f1,m(t) ∶ f2,m(t) ∶ 1)}m(t) ∈ F(Gh) be an irreducible F-conjugate +family of affine singularities of C(Gh) (see (3.5)). +Let A be the product of the leading +coefficient of m w.r.t. t and the leading coefficients w.r.t. t of f1,m,f2,m. We associate to F +the open set +Ωdef(F) ∶= Ωdef(f1,m) ∩ Ωdef(f2,m) ∩ Ωsqfree(m) ∩ ΩNonZ(A)) ∩ ΩG. +Let F ∶= {(L1,m(t) ∶ L2,m(t) ∶ 0)}m(t) ∈ F(Gh) be an irreducible F-conjugate family of +singularities of C(Gh) at infinity (see (3.5)). Let A be the leading coefficient of m w.r.t. t. +We associate to F the open set +Ωdef(F) ∶= Ωsqfree(m) ∩ ΩNonZ(A)) ∩ ΩG. +We start our analysis with a technical lemma. +Lemma 3.18. Let H,m ∈ F[t]. Let H = R mod m, and let A be the leading coefficient of m +w.r.t. t. If H(a0,t),m(a0,t) are well–defined and A(a0) ≠ 0, then H(a0,t) = R(a0,t) mod +m(a0,t). +Proof. Let Q be the quotient of H by m w.r.t. t. So, H = Q ⋅ m + R with degt(R) < degt(m). +Since H(a0),m(a0,t) is well–defined, γ0 is well–defined or all polynomials are independent +of γ. Since A(a0) ≠ 0, then Q(a0,t),R(a0,t) are well–defined. Moreover, degt(m(a,t)) = +degt(m(a0,t)). Then H(a0,t) = Q(a0,t)m(a0,t) + R(a0,t) with +degt(R(a0,t)) ≤ degt(R) < degt(m) = degt(m(a0,t)). +This concludes the proof. +□ + +10 +RATIONALITY AND PARAMETRIZATIONS OF ALGEBRAIC CURVES UNDER SPECIALIZATIONS +Lemma 3.19. Let F ∈ F(Gh) be an irreducible F–family of C(Gh) (see (3.5)). +If a0 ∈ +Ωdef(F), then F(a0) is a Ka0-conjugate family of points of C(Gh,a0) and #(F) = #(F(a0)). +Proof. Let F = {(f1,m(t) ∶ f2,m(t) ∶ 1)}m(t). Let us first show that F(a0) is a Ka0-conjugate +family of points of C(Gh,a0). Since a0 ∈ Ωdef(fi,m) and a0 ∈ Ωsqrfree(m) ⊂ Ωdef(m), we have that +γ0, fi,m(a0,t) and m(a0,t) are well-defined. Furthermore, since a0 ∈ ΩNonZ(A)), the degree of +all non-constant polynomials fi,m and m is preserved under the specialization. In addition, +since a0 ∈ Ωsqfree(m), it holds that m(a0,t) is squarefree (see Lemma 3.13) and condition (3) +in Def. A.1 holds; note that conditions (1) and (2) in Def. A.1 hold trivially. Furthermore, +note that, after specialization, all polynomials are over Ka0. So, F(a0) is a family over Ka0. +It remains to prove that the points in F(a0) are in the specialized curve. Since a0 ∈ ΩG, +by Lemma 3.15, it holds that G(a0,x,y)h = Gh(a0,x,y,z). Let T(a,t) = Gh(a,f1,m,f2,m,1). +Since F is a family of points in C(Gh), it holds that T = 0 mod m. Since Gh(a0,x,y,z) +and fi,m(a0,t) are well–defined, then T(a0,t) is well–defined too. We know that m(a0,t) is +well–defined and that the leading coefficient of m in t does not vanish after specialization. +Therefore, by Lemma 3.18, Gh(a0,f1,m(a0,t),f2,m(a0,t),1) = 0 modulo m(a0,t). +Hence, +F(a0) is a Ka0-conjugate family of points of C(Gh,a0). +If F ∶= {(L1,m(t) ∶ L2,m(t) ∶ 0)}m(t) all the arguments above apply and conditions (1) and +(2) in Def. A.1 also hold because Li,m do not depend on a or γ. +We already know that F(a0) is a Ka0-conjugate family of points of C(Gh,a0), and +degt(m) = degt(m(a0,t)), where m is the defining polynomial of F. It remains to prove +that #(F(a0)) = #(F). Let L be the K–linear change of coordinates transforming C(G) in +regular position; see Step (1) in the standard decomposition process described in Subsection +A.1. Then, #(F) = #(L−1(F)). Furthermore, L−1(F) is in the form appearing either in +(A.2) or in (A.3). Therefore, #(F) = degt(m). Since L is over K, we may apply it to F(a0) +and L−1(F(a0)) will be of the form either {(t ∶ B(a0,t) ∶ 1)}m(a0,t) or {(1 ∶ t ∶ 0)}m(a0,t). In +both cases, #(F(a0)) = #(L−1(F(a0))) = degt(m(a0,t)). Now, the result follows using that +degt(m) = degt(m(a0,t)). +□ +Remark 3.20. Given an F–conjugate family F ∈ F(Gh) (see equation (3.5)), and a0 ∈ Ωdef(F) +(see Def. 3.17), we observe that, even though F is irreducible, F(a0) may be reducible. We +will be interested in working with irreducible specialized families. So, factoring over Ka0 the +defining polynomial of F(a0), the family will be decomposed as +F(a0) = ⋃ +i∈I +Fi +where Fi is an irreducible Ka0–family. We will refer to Fi as the irreducible subfamilies of +F(a0). +In the sequel we analyze the multiplicity of families of singularities under specializations. +Definition 3.21. Let F ∈ F(Gh) (see (3.5)) be an irreducible F-conjugate family of r-fold +points of C(Gh) with defining polynomial m(t), and let H∗ be one of the order r derivatives +of Gh such that H∗(F) ≠ 0 modulo m(t). Let H(a,t) be the reduction of H∗(F) modulo +m(t). Let R(a) ∶= rest(H(t),m(t)). We define the open subset +Ωmult(F) ∶= Ωdef(F) ∩ ΩnonZ(H) ∩ ΩnonZ(R). + +RATIONALITY AND PARAMETRIZATIONS OF ALGEBRAIC CURVES UNDER SPECIALIZATIONS +11 +Remark 3.22. We observe that in Def. +3.21, m is irreducible (note that F belongs to a +standard decomposition of the singular locus) over F, H ∈ F[t] ∖{0} and degt(H) < degt(m). +Therefore, gcd(m,H) = 1 and hence R ≠ 0. +Lemma 3.23. Let F ∈ F(Gh) (see (3.5)) be an irreducible F-conjugate family of r-fold points +of C(Gh). If a0 ∈ Ωmult(F), then every irreducible subfamily of F(a0) (see Remark 3.20) is a +Ka0–conjugate family of r-fold points of C(Gh,a0). +Proof. Let F be expressed as F = {(f1 ∶ f2 ∶ λ)}m(t) where f1,f2,m ∈ F[t], λ ∈ {0,1}, m +irreducible over F, and such that, for the case λ = 0, degt(fi) ≤ 1 and gcd(f1,f2) = 1 (see +(3.5)). +Let H∗ and H be as in Def. 3.21, and let Fi, with defining polynomial mi, be an irreducible +subfamily of F(a0). Since a0 ∈ Ωdef(F), by Lemma 3.19, F(a0) is a Ka0–conjugate family +of points of C(Gh,a0); in particular fi(a0,t) and m(a0,t) are well–defined and the leading +coefficient of m w.r.t. t does not vanish at a0. Moreover, by Lemma 3.15, since a0 ∈ ΩG, all +partial derivatives specialize properly; note that Ωdef(F) ⊂ ΩG. Therefore, by Lemma 3.18, the +multiplicity of Fi is at least r. Moreover, since a0 ∈ ΩG, H∗(a0,f1(a0,t),f2(a0,t),λ) is well– +defined. So, again by Lemma 3.18, H∗(a0,f1(a0,t),f2(a0,t),λ) = H(a0,t) modulo m(a0,t). +Furthermore, since a0 ∈ ΩnonZ(H), then H(a0,t) ≠ 0. Moreover, since the leading coefficient of +m w.r.t. t does not vanish at a0, we have that rest(H(a0,t),m(a0,t)) = µR(a0) for some non- +zero constant µ (see Lemma 4.3.1 in [39]). So, since a0 ∈ ΩnonZ(R), rest(H(a0,t),m(a0,t)) ≠ 0. +Therefore, gcd(H(a0,t),m(a0,t)) = 1 and hence H(a0,t) ≠ 0 mod mi. Summarizing, the +multiplicity of Fi is r. +□ +In the last part of this subsection, we deal with the tangents to C(Gh) at an irreducible +F-conjugate family; since the family F is assumed to be irreducible, one may think on the +tangents at F to the curve C(Gh +F) (see Remark A.2(3) in Subsection A.1). +Definition 3.24. Let F, with defining polynomial m(t), be an irreducible F-conjugate family +of r-fold points of C(Gh). +Let Fm be the quotient field of F[t]/ < m(t) >. +The defining +tangent polynomial of C(Gh) at F is defined as the homogenous polynomial T ∈ Fm[x,y,z] of +degree r which factors over the algebraic closure of Fm into the tangents, with the according +multiplicities, of C(Gh) at F. Similarly, we introduce the defining tangent polynomial to an +specialized curve. +Remark 3.25. +(1) Let us assume that F = {(f1 ∶ f2 ∶ 1)}m(t) ∈ F(Gh) (see equation (3.5)) is a family of +r–fold points with irreducible m; similarly if the family is at infinity. Then T is the +reduction of +(3.6) +T ∗(a,t,x,y,z) = +r +∑ +i=0 +(r +i) +∂rG +∂ix∂r−iy(f1,f2)(x − f1z)i(y − f2z)r−i +modulo m(t). +(2) Let F be as in Def. 3.24. We observe that F is a family of ordinary r–fold points +if and only if T is squarefree over Fm. +In the sequel, we assume w.l.o.g. +that +there is no tangent of C(Gh +F) at F independent of x and for two different tangents +T1(x,y,z),T2(x,y,z) it holds that T1(x,1,1) ≠ T2(x,1,1). Note that, if this is not the +case, one can apply a linear change over K (and thus invariant under specializations + +12 +RATIONALITY AND PARAMETRIZATIONS OF ALGEBRAIC CURVES UNDER SPECIALIZATIONS +of the parameters a). Then, the ordinary character of the family is readable from the +squarefreeness of T(a,t,x,1,1) over Fm. +Definition 3.26. Let F, with defining polynomial m(t), be an irreducible F-conjugate family +of ordinary r-fold points of C(Gh). Let T be the defining tangent polynomial of F, where +we assume w.l.o.g. that the hypotheses in Remark 3.25 (2) are satisfied. Let D(a,t) be the +reduction modulo m(t) of the discriminant w.r.t. x of T(a,t,x,1,1). Let N(a) = rest(D,m). +Let A(a,t) be the leading coefficient of T(a,t,x,1,1) w.r.t. x and let R(a) = rest(A,m). Let +S(a,x) = rest(T(a,t,x,0,0),m). We define the set +Ωord(F) = Ωmult(F) ∩ ΩnonZ(R) ∩ ΩnonZ(N) ∩ ΩnonZ(S). +Remark 3.27. In relation to Def. 3.26, we observe the following. +(1) By construction, degt(A) < degt(m) and clearly A is not zero. Since m is irreducible, +gcd(A,m) = 1. Hence, R ≠ 0. +(2) Since F is ordinary and the two hypotheses in Remark 3.25 (2) are satisfied, D ≠ 0. +Because degt(D) < degt(m) and m is irreducible, gcd(m,H) = 1 and hence, N ≠ 0. +(3) T(a,t,x,y,z) has a factor in Fm[y,z] if and only if T(a,t,x,0,0) = 0. +This fol- +lows from the fact that the tangents are of degree one and thus, T(a,t,x,y,z) = +∏(Ai(a,t)x+Bi(a,t)y+Ci(a,t)z) for some Ai,Bi,Ci ∈ Fm. Thus, under our assump- +tion that T(a,t,x,y,z) does not have a factor independent of x, T(a,t,x,0,0) ≠ 0. +Since m is irreducible, and degt(T(a,t,x,0,0)) < degt(m), it follows that S ≠ 0. +Lemma 3.28. Let F ∈ F(Gh) (see equation (3.5)) be an irreducible F-conjugate family of +ordinary r-fold points of C(Gh). If a0 ∈ Ωord(F), then every irreducible subfamily of F(a0) +(see Remark 3.20) is a Ka0–conjugate family of ordinary r-fold points of C(Gh,a0). +Proof. Since a0 ∈ Ωord(F) ⊂ Ωmult(F) ⊂ Ωdef(F), then degt(m(a,t)) = degt(m(a0,t)) and, by +Lemma 3.23, every irreducible subfamily of F(a0) is a Ka0–conjugate family of r-fold points +of C(Gh,a0). Let us prove that all points in F(a0) are ordinary. +Let T ∗ be as in (3.6), or similarly if the family is at infinity, and let T be the reduction of +T ∗ modulo m. Since a0 ∈ Ωord(F) ⊂ Ωmult(F) ⊂ Ωdef(F) ⊂ ΩG, by Lemma 3.15, T ∗ specializes +properly at a0, and since degt(m(a,t)) = degt(m(a0,t)), by Lemma 3.18, T also specializes +properly at a0. +Now, let P be a point in F(a0). +Then, there exists a root t0 of m(a0,t) such that +P is obtained by specializing F at a0 and t0. +Since P belongs to one of the irreducible +subfamilies, P is an r–fold point of the curve C(Gh,a0). Because of the discussion above, +E(x,y,z) ∶= T(a0,t0,x,y,z) is the defining tangent polynomial of C(Gh,a0) at P. It remains +to prove that E is squareefree. First, let us see that there is no factor of E independent of +x. Assume that e(y,z) is a factor of E. Then E(x,0,0) = 0, and (a0,t0,x,0,0) is a common +zero of T(a,t,x,y,z) and m(a,t). Therefore, see e.g. Theorem 4.3.3 in [39], S(a0,x) = 0 +which contradicts that a0 ∈ ΩnonZ(S). Thus, it is sufficient to prove the squarefreeness of +E(x,1,1). Let us assume that it is not squarefree, then its discriminant is zero. That is, +the discriminant of T(a0,t0,x,1,1) is zero. On the other hand, a0 ∈ ΩnonZ(R), R(a0) ≠ 0 and +thus, A(a0,t0) ≠ 0. By [39, Lemma 4.1.3] and the fact that m(a0,t0) = 0, it follows that +D(a0,t0) = 0 and consequently, N(a0) = 0, in contradiction to a0 ∈ ΩnonZ(N). +□ +As a consequence of Lemmas 3.19, 3.23, 3.28, and taking into account that Ωord(F) ⊂ +Ωmult(F) ⊂ Ωdef(F), we get the following corollary. + +RATIONALITY AND PARAMETRIZATIONS OF ALGEBRAIC CURVES UNDER SPECIALIZATIONS +13 +Corollary 3.29. Let F ∈ F(Gh) (see equation (3.5)) be an irreducible F-conjugate family of +ordinary r-fold points of C(Gh). If a0 ∈ Ωord(F), then all points in F(a0) are ordinary r-fold +points of C(Gh,a0) and #(F) = #(F(a0)). +4. Preservation of the Genus +We consider a polynomial +(4.1) +F(a,γ,x,y) ∈ K[a,γ][x,y] ∖ K[a,γ]. +F, as a non–constant polymomial in F[x,y], defines an affine plane curve over F that we +assume irreducible. +As introduced in Subsection 3.3, for each a0 ∈ S such that F(a0,γ0,x,y) /∈ K, we denote +by C(F,a0) the curve C(F(a0,γ0,x,y)); similarly for C(F h,a0). Also, we denote by Ka0 the +ground field of C(F,a0) (see Def. 3.16). Our goal is to analyze the relation between the genus +of C(F) and the genus of C(F,a0) under the assumption that C(F,a0) is irreducible. +4.1. Ordinary singular locus case. We start our analysis assuming that C(F h) has only +ordinary singularities. Let F(F h) be an F–standard decomposition of the singular locus of +C(F h) obtained by using the process described in Subsection A.1. Let F(F h) decompose as +in (3.5) +(4.2) +F(F h) = +⋃ +m(t)∈Aa +{(f1,m(t) ∶ f2,m(t) ∶ 1)}m(t) ∪ +⋃ +m(t)∈A∞ +{(L1,m(t) ∶ L2,m(t) ∶ 0)}m(t) +where fi,m ∈ F[t], Li,m ∈ K[t] with gcd(L1,m,L2,m) = 1 and deg(Li,m) ≤ 1, and Aa, A∞ are +finite sets of irreducible polynomials in F[t]. We start with the following definition, where +sing(C(F h)) denotes the singular locus of C(F h). +Definition 4.1. We define the open subset (see (4.2) and Def. 3.14 and Def. 3.26) +ΩsingOrd(F h) ∶= +⎧⎪⎪⎪⎨⎪⎪⎪⎩ +⋂ +F∈F(F h) +Ωord(F) +if sing(C(F h)) ≠ ∅ +ΩF +if sing(C(F h)) = ∅ +Then, the following result holds. +Theorem 4.2. Let a0 ∈ ΩsingOrd(F h). If C(F,a0) is irreducible, then +genus(C(F)) ≥ genus(C(F,a0)). +Proof. Let d be the degree of C(F). If sing(C(F h)) = ∅, then a0 ∈ ΩF . So, by Lemma 3.15, +deg(F(a0,γ0,x,y)) = d. Since C(F,a0) is irreducible, by (A.4), one has that +genus(C(F)) = (d − 1)(d − 2) +2 +≥ genus(C(F,a0)). +If sing(C(F h)) ≠ ∅, then a0 ∈ ΩsingOrd(F h) ⊂ ΩF and deg(F(a0,γ0,x,y)) = d. By Corollary +3.29, all elements in sing(C(F h)) have the same multiplicity and character as their corre- +sponding elements in sing(C(F h,a0)) after specialization. New singularities, however, may +appear in sing(C(F h,a0)). So, reasoning as above with the genus formula in (A.4), or (A.8), +we get the result. +□ +The next result is a direct consequence of the previous theorem. + +14 +RATIONALITY AND PARAMETRIZATIONS OF ALGEBRAIC CURVES UNDER SPECIALIZATIONS +Corollary 4.3. Let C(F) be a rational curve. Let a0 ∈ ΩSingOrd(F h). If C(F,a0) is irreducible, +then C(F,a0) is rational. +The inequality in Theorem 4.2 comes from the fact that, using ΩSingOrd(F h), we cannot +ensure that sing(C(F h,a0)) does not include new singularities apart from those coming from +the specialization of the singular locus of C(F h). To control this phenomenon, we will ensure +that certain Gr¨obner bases behave properly under specializations. By, exercises 7, 8, pages +315–316 in [6], or by Proposition 1, page 308 in [6], we know that there exists an open Zariski +set such the Gr¨obner basis specializes properly; in fact, a description of this open subset is +also available. For a more general analysis of Gr¨obner bases with parametric coefficients we +refer to [17] and [38]. On the other hand, since we are working with bivariate polynomials +in F[x,y], the open subset above can be determined by using resultants. This motivates the +next definition. +Definition 4.4. Let I be an ideal in F[v], where v is tuple of variables, generated by G ⊂ F[v]. +Let G be a Gr¨obner basis of G w.r.t. some order. We define ΩspGB(G) ⊂ S as a non-empty +open subset such that for every a0 ∈ ΩspGB(G) it holds that {g(a0,γ0,v)∣g ∈ G} is a Gr¨obner +basis, w.r.t. the same order, of the ideal generated by {g(a0,γ0,v)∣g ∈ G } in Ka0[v]. +Now, we focus our attention on the standard decomposition of the singular locus of C(F h) +described in Subsection A.1. In the first step, if necessary, we apply a K linear change of +coordinates to ensure that the curve is in regular position. Hence, this linear transformation +it is not affected by the specializations of a. Therefore, for our reasonings, we may assume +w.l.o.g. that F is already in regular position. Next, let G1 be a Gr¨obner basis of +w.r.t. the lexicographic order with x < y, and let G2 be a Gr¨obner basis of the same ideal +w.r.t. the lexicographic order with y < x. Let {f(a,γ,x)} = G1 ∩ F[x], {g(a,γ,y)} = G2 ∩ +F[y], ˜f = f/gcd(f, ∂f +∂x) and ˜g = g/gcd(g,gcd(g, ∂g +∂x). Finally, let G3 ∶= {A(a,x),y − B(a,x)}, +with A square-free and deg(B) < deg(A), be the normed reduced Gr¨obner basis w.r.t. the +lexicographic order with x < y of . Then, we introduce the following definition. +Definition 4.5. With the notation introduced above, let (see also Def. 3.3, 3.6, and 3.12) +Ω1 = ΩnonZ(U) ∩ Ωgcd(f, ∂f +∂x ) ∩ Ωsqfree( ˜f), Ω2 = ΩnonZ(V ) ∩ Ωgcd(g, ∂g +∂y ) ∩ Ωsqfree(˜g) +where U and V are the leading coefficients of f and g w.r.t. x and y, respectively. We define +the open subset +Ωsinga(F) ∶= +3 +⋂ +i=1 +ΩspGB(Gi) +⋂ +q∈G1∖{f} +ΩnonZ(Wq,y) +⋂ +q∈G2∖{g} +ΩnonZ(Wq,x) +2 +⋂ +i=1 +Ωi ∩ Ωsqfree(A) +where Wq,y denotes the leading coefficient of q w.r.t. y; similarly with Wq,x. In addition, +let G(a,γ,y,z) ∶= F h(a,γ,1,y,z), let U(a,γ,t) = gcd(G(a,γ,t,0),Gy(a,γ,t,0),Gz(a,γ,t,0)) +and ˜U(a,γ,t) ∶= U/gcd(U,U′), where U ′ is the derivative of U w.r.t. t. Let Ω(0∶1∶0) ∶= S if +(0 ∶ 1 ∶ 0) ∈ sing(C(F h)) and else Ω(0∶1∶0) ∶= ΩnonZ(J(a,γ,0,1,0)) where J is one the first derivatives +of F h not vanishing at (0 ∶ 1 ∶ 0). +We define the open subset (see Definitions 3.9 and 3.12) +Ωsing∞(F) ∶= Ωgcd(G(a,γ,t,0),Gy(a,γ,t,0),Gz(a,γ,t,0)) ∩ Ωgcd(U,U′) ∩ Ω(0∶1∶0). +Then, we define (see Def. 4.1) +ΩgenusOrd(F h) ∶= ΩsingOrd(F h) ∩ Ωsinga(F) ∩ Ωsing∞(F). + +RATIONALITY AND PARAMETRIZATIONS OF ALGEBRAIC CURVES UNDER SPECIALIZATIONS +15 +Remark 4.6. Note that G1 ∩ F[x] = {f}. So all polynomials in G1 ∖ {f} do depend on y; +similarly for G2 ∖ {g}. The idea of controlling the coefficients Wq,x and Wq,y in Def. 4.5 is to +ensure that the elimination ideal of the specialized Gr¨obner basis does not include additional +generators. +In the following lemma, we see that the cardinality of the singular locus, as a set, is +preserved under specializations. +Lemma 4.7. Let a0 ∈ Ωsinga(F) ∩ Ωsing∞(F). Then +#(sing(C(F h)) = #(sing(C(F h,a0)). +Proof. Let F 0(x,y) ∶= F(a0,γ0,x,y). +Let G0 +1 be a Gr¨obner basis of < F 0,F 0 +x,F 0 +y > w.r.t. +the lexicographic order x < y, and let G0 +2 be a Gr¨obner basis of the same ideal w.r.t. the +lexicographic order y < x. Since +a0 ∈ ΩspGB(G1) ∩ +⋂ +q∈G1∖{f} +ΩnonZ(Wq,y) ∩ Ω1, +then {f(a0,γ0,x)} = G0 +1 ∩ Ka0[x] and degx(f(a,γ,x)) = degx(f(a0,γ0,x)). +Similarly, +{g(a0,γ0,x)} = G0 +2 ∩Ka0[y] and degy(f(a,γ,y)) = degy(f(a0,γ0,y)). Since a0 ∈ Ω1, by Corol- +lary 3.8, gcd(f, ∂f +∂x)(a0,γ0,x) = gcd(f(a0,γ0,x), ∂f(a0,γ0,x) +∂x +) and degx(gcd(f, ∂f +∂x)(a0,γ0,x)) = +degx(gcd(f, ∂f +∂x)(a,γ,x)). Thus, +˜f(a0,γ0,x) = +f(a0,γ0,x) +gcd(f(a0,γ0,x), ∂f(a0,γ0,x) +∂x +) +. +By Lemma 3.13, it holds that degx( ˜f(a,γ,x)) = degx( ˜f(a0,γ0,x)) and ˜f(a0,γ0,x) is square- +free. +Similarly for g and ˜g since a0 ∈ Ω2. +In addition, by Lemma 3.15, F 0 +x(x,y) = +Fx(a0,γ0,x,y) and F 0 +y (x,y) = Fy(a0,γ0,x,y). Thus, +√ + = . +Since a0 ∈ ΩspGB(G3), {A(a0,γ0,x),y − B(a0,γ0,x)} is a Gr¨obner basis of +√ +. +Since a0 ∈ Ωsqfree(A), by Lemma 3.13, A(a0,γ0,x) is squarefree. Therefore, the number +of affine singularities of C(F,a0) is degx(A(a0,γ0,x)) and degx(A(a,γ,x)) is the number of +affine singularities of C(F). By Lemma 3.13, we get that degx(A(a0,γ0,x)) = degx(A(a,γ,x)) +and, hence, C(F) and C(F,a0) have the same number of affine singularities. +It remains to prove that the number of singularities at infinity is also the same. First +we observe that, if (0 ∶ 1 ∶ 0) ∈ sing(C(F h)), then (0 ∶ 1 ∶ 0) ∈ sing(C(F h,a0)). Moreover, +since a0 ∈ Ω(0∶1∶0), if (0 ∶ 1 ∶ 0) /∈ sing(C(F h)), then (0 ∶ 1 ∶ 0) /∈ sing(C(F h,a0)). For the +remaining singularities at infinity, denote by Σ the set of the singularities of the form (1 ∶ µ ∶ +0) ∈ sing(C(F h)); similarly let Σ0 be the set of singularities of this type in sing(C(F h,a0)). +Let G0(y,z) ∶= G(a0,γ0,y,z) (see Def. +4.5), U 0(t) ∶= gcd(G0(t,0),G0 +y(t,0),G0 +z(t,0)) and +˜U 0(t) ∶= U 0/gcd(U 0,(U 0)′). Then, +(4.3) +#(Σ) = degt( ˜U) and #(Σ0) = degt( ˜U 0). +Since a0 ∈ Ωgcd(G(a,γ,t,0),Gy(a,γ,t,0),Gz(a,γ,t,0)), by Theorem 3.11, it holds that +(4.4) +U 0(t) = U(a0,γ0,t) and degt(U 0(t)) = degt(U(a,γ,t)). + +16 +RATIONALITY AND PARAMETRIZATIONS OF ALGEBRAIC CURVES UNDER SPECIALIZATIONS +Let D(a,γ,t) = gcd(U,U′) and D0(t) = gcd(U 0,(U 0)′). +Since a0 ∈ Ωgcd(U,U′), by Corol- +lary 3.8, one has that +(4.5) +D0(t) = D(a0,γ0,t) and degt(D0(t)) = degt(D(a,γ,t)). +Now, by (4.3), (4.4) and (4.5), we get that #(Σ) = #(Σ0) and this concludes the proof. +□ +Theorem 4.8. Let a0 ∈ ΩgenusOrd(F h). If C(F,a0) is irreducible, then +genus(C(F)) = genus(C(F,a0)). +Proof. Since a0 ∈ Ωsinga(F) ∩ Ωsing∞(F), by Lemma 4.7, it holds that #(sing(C(F h)) = +#(sing(C(F h,a0)). On the other hand, since a0 ∈ ΩSingOrd(F h), by Corollary 3.29, we know +that each r–fold in sing(C(F h)) generates an ordinary r-fold in sing(C(F h,a0)). Therefore, +applying (A.4), we conclude the proof. +□ +4.2. General case. Let F be as in (4.1). But now, differently to the case of Subsection 4.1, +we do not introduce any assumption on the singular locus of the irreducible curve C(F h). +The key of our analysis is to reduce the general case to the case studied in Subsection 4.1. +For this purpose, we recall that any irreducible curve is birationally equivalent to a curve +having only ordinary singularities; see e.g. [37, Theorem 7.4.] or [30, Section 3.2.] for a +more computational description. This transformation, say ϕ, can be seen as a finite sequence +of blowups of the irreducible families of non-ordinary singularities and, hence, as a finite +sequence of compositions of quadratic Cremona transformations and linear transformations. +Now, our goal is to find an open subset Ωblowup of S such that, when a is specialized in +Ωblowup, the birationality of ϕ is preserved. +For this purpose, let F0(x,y) = F(x,y) and +let F(F h +0 ) be a F-conjugate irreducible family of non-ordinary singularities of C(F h +0 ) with +defining polynomial m1(t1) ∈ F[t1]. Let Fm1 be the quotient field of F[t1]/, that is, +Fm1 = F(t1) with m1(t1) as minimal polynomial of t1. Then we apply a linear transformation +L1, given by a matrix M1 ∈ M3×3(Fm1), and the Cremona transformation Q1 = (yz ∶ xz ∶ xy) +as described in the blow up basic step in Subsection A.1 of the appendix. Denote by ∆1 +the determinant det(M1) and let C(F h +1 ) be the curve over Fm1 obtained after the quadratic +transformation Q1 ○ L1. Note that F h +1 , the quadratic transformation of F h +0 , is the cofactor +of F h +0 (Q1(L1)) not being divisible by neither x, y nor z. We repeat the above process for +F h +1 (t1,x,y,z), F h +2 (t1,t2,x,y,z),...,F h +r (t1,...,tr,x,y,z) until all singularities of C(F h +r ) are +ordinary. Then +ϕ(a,γ,t1,...,tr,x,y,z) = (Qr ○ Lr) ○ ⋯ ○ (Q1 ○ L1). +Note that F h +r is defined over F(t1,...,tr) and C(F h +r ) over the algebraic closure of F(t1,...,tr). +In addition, F(t1,...,tr) = K(a,γ,t1,...,tr) = L(γ,t1,...,tr). So, we consider a primitive +element of the extension over L, say γ∗, and we work over L(γ∗) = L(γ,t1,...,tr); note that +results in Section 3 apply to this new frame. In this situation, let us denote by ∆ = ∆1⋯∆r ∈ +L(γ∗) the product of the determinants of the linear transformations L1,...,Lm. In addition, +let +M ∶= {all entries of Mi ∈ M3×3(L(γ∗))}i∈{1,...,r}. +and let +B ∶= +{F h +i (Qi+1(Li+1))(a,γ∗,0,y,z),F h +i (Qi+1(Li+1))(a,γ∗,x,0,z), +F h +i (Qi+1(Li+1))(a,γ∗,x,y,0)}i∈{0,...,r−1}. + +RATIONALITY AND PARAMETRIZATIONS OF ALGEBRAIC CURVES UNDER SPECIALIZATIONS +17 +Definition 4.9. With the notation introduced above, we define the set +Ωblowup(F) ∶= ΩF ∩ ⋂ +h∈M +Ωdef(h) ∩ ΩnonZ(∆) ∩ ⋂ +h∈B +ΩnonZ(h). +The previous observations lead to the following result. +Lemma 4.10. Let a0 ∈ Ωblowup(F). Then ϕ(a0,(γ∗)0,x,y,z) is birational. +Proof. Since a0 ∈ Ωblowup(F) ⊂ ⋂h∈M Ωdef(h) ∩ ΩnonZ(∆), all Qi ○ Li are well–defined, and +birational, when a is specialized as a0. So ϕ(a0,(γ∗)0,x,y,z) is birational. +□ +Lemma 4.11. Let a0 ∈ Ωblowup(F) and let ϕ0 ∶= ϕ(a0,(γ∗)0,x,y,z). If F0(a0,γ0,x,y,z) is +irreducible, then Fr(a0,(γ∗)0,x,y,z) is the quadratic transformation of F0(a0,γ0,x,y,z). +Proof. First we observe that because of (the proof of) Lemma 4.10 each ϕi ∶= Qi ○ Li is +well defined at a0 and it is birational. +Let us prove the result by induction. +By hy- +pothesis F h +0 (a0,γ0,x,y,z) is irreducible. +We have the equality F h +0 (ϕi) = xn1yn2zn3F h +1 +for some ni ∈ N and that neither x, y nor z divides F h +1 . +So, F0(ϕi)(a0,(γ∗)0,x,y,z) = +xn1yn2zn3F h +1 (a0,(γ∗)0,x,y,z). Moreover, since a0 ∈ ⋂h∈B ΩnonZ(h), we know that neither x, +y nor z divides F1(a0,(γ∗)0,x,y,z). Furthermore, since ϕi is birational when specialized at +a0 and F h +0 (a0,γ0,x,y,z) is irreducible, we have that F h +1 (a0,(γ∗)0,x,y,z) is also irreducible. +Thus, F h +1 (a0,(γ∗)0,x,y,z) is the quadratic transformation of F h +0 . Now, the i-induction step +is reasoned analogously using that, by induction, F h +i (a0,(γ∗)0,x,y,z) is irreducible. +□ +With this, we can now give an open set where the genus is preserved. +Definition 4.12. Let F ∈ F[x,y] be as in (4.1), and let G ∈ F(γ∗)[x,y] be the polynomial +obtained after the blowup process of F h. We define the set +Ωgenus(F) ∶= Ωblowup(F) ∩ ΩgenusOrd(Gh). +Theorem 4.13. Let F ∈ F[x,y] be as in (4.1), and let a0 ∈ Ωgenus(F). If C(F,a0) is irre- +ducible, then +genus(C(F)) = genus(C(F,a0)). +Proof. Since Gh is the quadratic transformation of F h, we have that +(4.6) +genus(C(F h(a,x,y,z))) = genus(C(Gh(a,γ∗,x,y,z))). +Let ϕ0 denote the map ϕ(a0,(γ∗)0,x,y,z). +Since a0 ∈ Ωblowup(F), by Lemma 4.10, ϕ0 +is birational and, by Lemma 4.11, Gh(a0,(γ∗)0,x,y,z) is the quadratic transformation of +F h(a0,γ0,x,y,z) via ϕ0. Therefore, +(4.7) +genus(C(F h(a0,γ0,x,y,z))) = genus(C(Gh(a0,(γ∗)0,x,y,z))). +Moreover, since a0 ∈ ΩgenusOrd(Gh), by Theorem 4.8, it holds that +(4.8) +genus(C(Gh(a,γ∗x,y,z))) = genus(C(Gh(a0,(γ∗)0,x,y,z))). +Now, the proof follows from (4.6), (4.7) and (4.8). +□ + +18 +RATIONALITY AND PARAMETRIZATIONS OF ALGEBRAIC CURVES UNDER SPECIALIZATIONS +5. Birational Parametrization of Parametric Rational Curves +In Section 4, and more precisely in Theorem 4.8 and 4.13, we have described open subsets +of S where the genus of the curve is preserved under specializations; even in Corollary 4.3 +the particular case of genus zero was treated. Nevertheless, in all these results the additional +condition that the specialized polynomial is irreducible over K was required. Avoiding the +irreducibility is in general a difficult problem related to the Hilbert irreducibility problem +(see e.g. [31]). More precisely, there is no algorithm known that finds for given irreducible +F ∈ F[x,y] the specializations a0 ∈ S such that F(a0,x,y) is reducible. Nevertheless, in this +section, we see how in the case of genus zero the problem can be solved. Furthermore, we +described an open subset where the specialized parametrization parametrizes the specialized +curve. +For this purpose, throughout this section, let assume that F ∈ F[x,y] is as in (4.1) and +additionally assume that C(F) is rational. Moreover, let us assume that P is a proper (i.e. +birational) parametrization of C(F) which can be computed, for instance, by the algorithm +described in Subsection A.2 in the appendix. Note that, in general, one may need to extend +F with an algebraic element δ of degree two. If #(a) = 1 and deg(γ) = 1, or degx,y(F) is odd, +then no extension of F is required (see Remark 2.5 and Theorem 2.1). +So, we may consider a primitive element of L(γ,δ), say γ∗, and express our parametrization +in L(γ∗). Throughout this section, by abuse of notation, let F denote the field L(γ∗). Let +us write the proper parametrization P of C(F) as +(5.1) +P(a,γ,t) = (p1 +q1 +, p2 +q2 +) ∈ F(t)2 ∖ F2 +where we assume that P is in reduced form, that is gcd(p1,q1) = gcd(p2,q2) = 1. +Let us start with the simple case of degree one. +Definition 5.1. Let F = A2(a,γ)x+A1(a,γ)y+A0(a,γ) ∈ F[x,y]∖F, and let P be expressed +as P(a,γ,t) = (λ1t + λ0,µ1t + µ0) ∈ F(t)2 ∖ F2 be a proper polynomial parametrization of +C(F). We define the set (see Def. 3.3 and 3.14) +Ωproper(P) ∶= ΩF ∩ Ωdef(λ1t+λ0) ∩ Ωdef(µ1t+µ0) ∩ (ΩnonZ(λ1) ∪ ΩnonZ(µ1)). +Proposition 5.2. Let F and P be as in Def. 5.1. Then, for every a0 ∈ Ωproper(P), it holds +that P(a0,t) is a proper polynomial parametrization of C(F,a0). +Proof. Since a0 ∈ ΩF , by Lemma 3.15, F(a0,γ0,x,y) is well defined and C(F,a0) is an +affine line, obviously irreducible. +Since a0 ∈ Ωdef(λ1t+λ0) ∩ Ωdef(µ1t+µ0) then P(a0,γ0,t) +is well–defined. +Moreover, since a0 ∈ (ΩnonZ(λ1) ∪ ΩnonZ(µ1)), P(a0,γ0,t) is a polynomial +parametrization, clearly proper. +Furthermore, F(a0,γ0,P(a0,γ0,t)) = 0. +Thus, since +F(a0,γ0,x,y) is irreducible, we conclude that P(a0,γ0,t) is a proper parametrization of +C(F,a0). +□ +Remark 5.3. Let us use the notation in Proposition 5.2. If a0 ∈ S ∖ Ωproper(P) then: +(1) If a0 /∈ ΩF ∖ Ωdef(F), it holds that F(a0,γ0,x,y) = A0(a0,γ0) ∈ K, and hence C(F,a0) +does not define an affine curve. +(2) If a0 /∈ Ωdef(λ1t+λ0) but a0 ∈ ΩF (similarly if a0 /∈ Ωdef(µ1t+µ0)), the specialization +P(a0,γ0,t) is not well–defined even though C(F,a0) is a line. Clearly, in this case, +one has that C(F,a0) is rational and a proper parametrization of the specialized line +can be provided. + +RATIONALITY AND PARAMETRIZATIONS OF ALGEBRAIC CURVES UNDER SPECIALIZATIONS +19 +(3) If a0 /∈ (ΩnonZ(λ1) ∪ ΩnonZ(µ1)) but a0 ∈ ΩF ∩ Ωdef(λ1t+λ0) ∩ Ωdef(µ1t+µ0), then +P(a0,γ0,t) ∈ K +2 and hence it is not a parametrization although C(F,a0) is a line. +Again, as in the previous case, one can easily provide a polynomial parametrization +of the specialized line. +In the sequel, we assume that C(F) is not a line. Then, we generalize the open subset in +Def. 5.1 as follows. +Definition 5.4. +(1) Let Ω1 ∶= ΩF (see Def. 3.14). +(2) Ω2 ∶= Ωdef(p1) ∩ Ωdef(p2) ∩ ΩnonZ(q1) ∩ ΩnonZ(q2). +(3) We consider the polynomials Gi = pi(h)qi(t) − pi(t)qi(h) ∈ F[h][t] ∖ {0} for i ∈ {1,2}. +Let Ω3 ∶= Ωgcd(G1,G2) (see Def. 3.6). +(4) Let Ω4 ∶= Ωgcd(p1,q1) ∩Ωgcd(p2,q2); note that pi,qj ∈ F[t] ⊂ F[h,t] and, since C(F) is not +a line, the pi and gi are are non–zero (see Def. 3.6). +We define Ωproper(P) as +Ωproper(P) = +4 +⋂ +i=1 +Ωi. +The following theorem generalizes Prop. 5.2. +Theorem 5.5. Let a0 ∈ Ωproper(P). Then C(F,a0) is a rational affine curve in K +2 properly +parametrized by P(a0,δ0,t). +Proof. If deg(F) = 1, the result follows from Prop. 5.2. Let deg(F) > 1. Since a0 ∈ Ω1, then +F(a0,γ0,x,y) is well–defined and deg(C(F)) = deg(C(F,a0)). In particular C(F,a0) is an +affine curve. On the other hand, since a0 ∈ Ω2, by Lemma 3.5 (1), we have that P(a0,γ0,t) +is well–defined. +In addition, since a0 ∈ Ω4, the leading coefficients of p1,p2,q1,q2 do not +vanish at a0 (see Def. 3.6). Consequently the degree of all numerators and denominators of +P after specialization are preserved. Furthermore, by Corollary 3.8, and using that pi/qi are +in reduced form, we get that pi(a0,γ0,t)/qi(a0,γ0,t) are also in reduced form. Therefore, +(5.2) +degt (pi(a0,γ0,t) +qi(a0,γ0,t)) = degt (pi(a,γ,t) +qi(a,γ,t)) for i ∈ {1,2}. +In particular, P(a0,γ0,t) /∈ K +2, and hence P(a0,γ0,t) is a parametrization. +Moreover, +F(a0,γ0,P(a0,δ0,t)) = 0. Thus, P(a0,γ0,t) parametrizes the curve defined by one of the +factors, say H(x,y), of F(a0,γ0,x,y). Let us see that indeed C(F,a0) = C(H). +Let Gi(a,γ,h,t) as in Def. 5.4 (3), and let G ∶= gcd(G1(a,γ,h,t),G2(a,γ,h,t)). +Let +˜Gi(h,t) be the corresponding polynomials associated, as in Def. 5.4 (3), to P(a0,γ0,t). Let +˜G ∶= gcd( ˜G1, ˜G2). Since pi(a0,γ0,t)/qi(a0,γ0,t) are in reduced form, no simplification of +the rational functions have been required, and therefore ˜Gi(h,t) = Gi(a0,γ0,h,t). Moreover, +since a0 ∈ Ω3, by Lemma 3.7, it holds that +(5.3) +degt(G(a0,γ0,h,t)) = degt( ˜G(h,t)), +and degt(G(a0,γ0,h,t)) = degt(G(a,γ,h,t)). +By [29, Theorem 3], since P(a,γ,t) is +proper, we have that degt(G(a,γ,h,t)) = 1. +Therefore, it holds that degt( ˜G(h,t)) = +degt(G(a0,γ0,h,t)) = 1. Again by [29, Theorem 3], P(a0,γ0,t) is proper. On the other + +20 +RATIONALITY AND PARAMETRIZATIONS OF ALGEBRAIC CURVES UNDER SPECIALIZATIONS +hand, by (5.2), degt(P(a,γ,t)) = degt(P(a0,γ0,t)). Therefore, by Theorem 4.21 in [30], we +have that +max{degx(F(a,γ,x,y)),degy(F(a,γ,x,y)} = +degt(P(a,γ,t)) += +degt(P(a0,γ0,t)) += +max{degx(H),degy(H)}. +Moreover, since F is not linear, by (5.2), no component of P(a0,γ0,t) is constant. Applying +again [30, Theorem 4.21.], we have that +degx(H) = degt (p2(a0,γ0,t) +q2(a0,γ0,t)) = degt (p2(a,γ,t) +q2(a,γ,t)) = degx(F) +degy(H) = degt (p1(a0,γ0,t) +q1(a0,γ0,t)) = degt (p1(a,γ,t) +q1(a,γ,t)) = degy(F). +Finally, since H(x,y) divides F(a0,γ0,x,y), one has that C(F,a0) = C(H), which concludes +the proof. +□ +Remark 5.6. Let us analyze the behavior of F and/or P when specializing in S ∖ Ωproper(P). +(1) If a0 ∈ S ∖ Ω1, since F ∈ K[a][x,y] (see (4.1)), then F(a0,x,y) is always well–defined, +and hence, deg{x,y}(F(a0,x,y)) < deg{x,y}(F(a,x,y)). So, it can happen that either +0 < deg{x,y}(F(a0,x,y)) < deg{x,y}(F(a,x,y)), in which case C(F,a0) is an affine +curve; or 0 = deg{x,y}(F(a0,x,y)), which implies that C(F,a0) is the empty set or +K +2. +(2) If a0 ∈ S ∖ Ω2, then P(a0,δ0,t) is not defined, and hence the specialization fails. +(3) If a0 ∈ (S ∖ (Ω3 ∩ Ω4)) ∩ Ω1 ∩ Ω2, at least one of the following assertions hold. +(a) P(a0,t) is not proper. +(b) P(a0,t) ∈ K +2 and hence, P(a0,t) is not a parametrization. +(c) P(a0,t) parametrizes a proper factor of F(a0,x,y), that is, C(F,a0) decomposes +and one of its components is rational and parametrized by P(a0,t). +The next result follows from Theorem 5.5 and emphasizes the polynomiality of the +parametrizaion. +Corollary 5.7. If P is proper and polynomial and a0 ∈ Ωproper(P), then P(a0,γ0,t) +parametrizes properly and polynomially C(F,a0). +We now analyze the normality (i.e. the surjectivity, see [30]) of the parametrization. We +recall that any parametrization can be reparametrized surjectively (see [30, Theorem 6.26]). +This reparametrization requires, in our case, a new algebraic extension of F via a new algebraic +element. Alternatively, one may reparametrize normally the specialized parametrizations. In +the following we deal with the case where P is already normal and we want to preserve this +property through the specializations. For this purpose, we first introduce a new definition. +Definition 5.8. Let P be as in 5.1. If P is normal, we define the set +Ωnormal(P) ∶= { +S +if degt(p1) > degt(q1) or degt(p2) > degt(q2), +Ωgcd(N1,N2) +if degt(p1) ≤ degt(q1) and degt(p2) ≤ degt(q2) +where (α1/β1,α2/β2) ∈ F2 is the critical point of P (see [30, Def. 6.24]) and, for i ∈ {1,2}, +Ni = αiqi − βipi, + +RATIONALITY AND PARAMETRIZATIONS OF ALGEBRAIC CURVES UNDER SPECIALIZATIONS +21 +Corollary 5.9. Let P be proper and normal. For a0 ∈ Ωproper(P) ∩ Ωnormal(P), P(a0,γ0,t) +parametrizes properly and normally C(F,a0). +Proof. In the proof of Theorem 5.5 we have seen that, for a0 ∈ Ωproper(P), degt(pi(a,γ,t)) = +degt(pi(a0,γ0,t)), similarly for qi, and that the rational functions in P(a0,γ0,t) are in re- +duced form. +Now, if degt(p1) > degt(q1) or degt(p2) > degt(q2), the result follows from +Theorem 5.5 and [30, Theorem 6.22]. If degt(p1) ≤ degt(q1) and degt(p2) ≤ degt(q2), since +a0 ∈ Ω2 in Def. 5.4, we get that +C ∶= (α1(a0,γ0) +β1(a0,γ0), α2(a0,γ0) +β2(a0,γ0)) +is well defined and, by the above remark on the degrees, C is the critical point of P(a0,γ0,t). +Let +˜Ni ∶= αi(a0,γ0)qi(a0,γ0,t) − βi(a0,γ0)pi(a0,γ0) = Ni(a0,γ0,t). +Now, +since a0 +∈ +Ωgcd(N1,N2), +by Corollary 3.8, +one has that degt(gcd( ˜N1, ˜N2)) += +degt(gcd(N1,N2)) > 0; recall that P is normal. Now, the result follows from Theorem 5.5 +and [30, Theorem 6.22]. +□ +Let us illustrate these ideas in an example. +Example 5.10. Let us consider K = Q and F = L ∶= Q(a1,a2). Let +F(a, x, y) =((a5 +1 + a5 +2 + 3a2 +1a2 +2 − a1a2)y2 + (2a3 +1a2 + (−9a2 − 1)a2 +1 + 3a1 − 6a4 +2 + a3 +2)y + a2 +2(a1 + 9a2 − 3))x3 ++ ((−3a3 +1a2 +2 − 6a4 +1 + 3a2 +1a2 − 6a1a2 +2)y2 + 9((a2 + 2/9)a2 +1 + (−(8a2)/9 − 1)a1 − a3 +2/9 + 2a2 + 2/9)a1y ++ 3(a1 − 2/3)a2 +2)x2 − 3(((a2 − 4)a1 − 2a2 +2)a1y + a3 +1/3 − 3a2 +1 + (6a2 + 4/3)a1 − (8a2)/3)a1xy ++ ((a2 +1a2 − 8)y − 3a2 +1 + 2a1)a2 +1y +C(F) is a rational quintic that can be properly parametrized as +(5.4) +P(a,t) = ( +ta1 + 2 +t2a2 + t + a1 +, +t + 3 +t3a1 + a2 +). +The field of parametrization is L. We determine the open subset Ωproper(P) (see Def. (5.4)). +Let us deal with Ω1. Clearly Ωdef(F) = C2. The homogeneous component of F of maximum +degree is +(a5 +1 + a5 +2 + 3a2 +1a2 +2 − a1a2)x3y2 +So, +Ω1 ∶= C2 ∖ V(a5 +1 + a5 +2 + 3a2 +1a2 +2 − a1a2). +One has that +Ω2 ∶= C2 ∖ {(0,0)} + +22 +RATIONALITY AND PARAMETRIZATIONS OF ALGEBRAIC CURVES UNDER SPECIALIZATIONS +Note that, if a0 /∈ Ω2, the second component of P is not well-defined. Let us deal with Ω3. +The polynomials Gi, G∗ +i , R are +G1 += +h2ta1a2 − ht2a1a2 + 2h2a2 − ha2 +1 − 2t2a2 + ta2 +1 + 2h − 2t +G2 += +h3ta1 − ht3a1 + 3h3a1 − 3t3a1 − ha2 + ta2 +G += +h − t +G∗ +1 += +(hta1 + 2h + 2t)a2 − a2 +1 + 2 +G∗ +2 += +((t + 3)h2 + (t2 + 3t)h + 3t2)a1 − a2 +R += +3h4a3 +1a2 +2 − 2h4a2 +1a2 +2 + h3a4 +1a2 + 6h3a2 +1a2 +2 + 3h2a4 +1a2 − h2a2 +1a3 +2 − 2h3a2 +1a2 − 2h2a3 +1a2 ++ha5 +1 − 6h2a2 +1a2 + 12h2a1a2 +2 − 6ha3 +1a2 − 4ha1a3 +2 + 3a5 +1 + 4h2a1a2 − 4ha3 +1 + 12ha1a2 +−12a3 +1 − 4a3 +2 + 4ha1 + 12a1 +Moreover, A1 = −ha1a2 − 2a2,A2 = −ha1 − 3a1,B = −1 (see Def. (3.6)). Hence, Ω1 = C2. +Moreover, ΩnonZ(A1) = C2 ∖ V(a2), ΩnonZ(A2) = C2 ∖ V(a1) and ΩnonZ(B) = C2. On the other +hand, ΩnonZ(R) can be expressed as +C2 ∖ {(0,0),(± +√ +2,0)}. +Therefore, +Ω3 = Ωgcd(G1,G2) = C2∩(C2 ∖ V(a1))∩(C2 ∖ V(a2))∩(C2 ∖ {(0,0),(± +√ +2,0)}) = C2∖V(a1a2). +Finally, we deal with Ω4. We have +p1 = ta1 + 2 +p2 = t + 3 +q1 = t2a2 + t + a1 +q2 = t3a1 + a2 +gcd(p1,q1) = 1 +gcd(p2,q2) = 1 +rest(p1,q1) = a3 +1 − 2a1 + 4a2 +rest(p2,q2) = a2 − 27a1 +Therefore, +Ω4 = C2 ∖ V(a1a2(a3 +1 − 2a1 + 4a2)(a2 − 27a1)). +Summarizing (see Fig. 1, left) +(5.5) +Ωproper(P) = C2 ∖ V(a1a2(a5 +1 + a5 +2 + 3a2 +1a2 +2 − a1a2)(a3 +1 − 2a1 + 4a2)(a2 − 27a1)). +6. Decomposition of S +The goal in this section is to provide an algorithm decomposing the space S so that in +each subset of the decomposition we can give information on the genus of the corresponding +specialized curve. +Let F be as in (4.1), irreducible over F. We first compute the genus of C(F h). Let g ∶= +genus(C(F h)). Furthermore, if g = 0, let P(a,γ,t) be, as in (5.1), a proper parametrization +of C(F h). We consider the open subset +(6.1) +Σ ∶= { Ωgenus(F) +if g > 0 (see Def. (4.12)) +Ωproper(P) +if g = 0 (see Def. (5.4)) +At this level of the process we know that (see Theorems (4.13) and (5.5)) +(1) If g > 0, then for a0 ∈ Σ it holds that C(F,a0) is either reducible or its genus is g. +(2) If g = 0, then for a0 ∈ Σ it holds that C(F,a0) is rational and P(a0,γ0,t) parametrizes +properly C(F,a0). + +RATIONALITY AND PARAMETRIZATIONS OF ALGEBRAIC CURVES UNDER SPECIALIZATIONS +23 +Figure 1. Left: Plot of the real part of the closed set defining Ωproper(P) in +Example 5.10. Right: Plot of the real part of the closed set defining Ωgenus(F) +in Example 6.3 +In the following, we analyze the specializations when working in the closed set +(6.2) +Z ∶= S ∖ Σ. +First, let us discuss the computational issues that may appear. Let A ⊂ K[a] be a set of +generators of Z, and let I be the ideal generated by A in K[a]. +We consider the prime +decomposition of I +I = +ℓ +⋃ +j=1 +Ij. +Now, for each prime ideal J ∈ {I1,...,Iℓ} we consider the quotient field of K[a]/J; we denote +it by LJ. Elements in LJ are quotients of equivalence classes of K[a]/J. We will assume that +elements in K[a]/J are always expressed by means of a canonical representative of the class +in the following sense. We fix a Gr¨obner basis G of J w.r.t. some fixed order. Then, the +elements in K[a]/J are uniquely represented by their normal form w.r.t. G (see e.g. Prop. +1 and Ex. +13, Chap. +2, Sect. +6. +in [6]) and, hence, elements in LJ are represented as +the quotient of the canonical representatives of their numerators and denominators. So, by +abuse of notation, we will identify, via the canonical representation, the elements in LJ with +elements in K(a). In addition, we consider an algebraic element γJ over LJ and we denote +by FJ the field FJ ∶= LJ(γJ). +We observe that FJ is a computable field with a polynomial factorization algorithm avail- +able; zero test and basic arithmetic (addition, multiplication and inverse computation) can +be carried out e.g. by taking the normal forms w.r.t. a Gr¨obner basis of J. For the polyno- +mial factorization we refer to (see Section 10.2 and Appendix B in [36], see also [35]). As a +particular case, as in Example 5.10 and 6.2, if V(J) is a rational variety, one may work over +K(Q(λ1,...,λm)) instead of FJ, where Q(λ1,...,λm) is a parametrization of V(J). +Concerning specializations, instead of working in S (see (2.1)), we take the parameter +values in the irreducible variety V(J). Then, for a0 ∈ V(J) ⊂ S, and f ∈ K[a]/J, we denote +by f(a0) the specialization at a0 of the equivalence class of f; note that since a0 ∈ V(J) the + +4 +2. +2 +3 +-224 +RATIONALITY AND PARAMETRIZATIONS OF ALGEBRAIC CURVES UNDER SPECIALIZATIONS +specialization does not depend on the representative. Similarly, if f ∶= p/q ∈ FJ and q(a0) ≠ 0, +then f(a0) ∶= p(a0)/q(a0) ∈ K. +In this situation, for each prime ideal J ∈ {I1,...,Iℓ} we consider the polynomial F in +(4.1) as a polynomial in FJ[x,y]. To emphasize this fact, we write FJ. First we check the +irreducibility of FJ over the algebraic closure of FJ. If FJ is reducible we can either stop +the decomposition over this closed subset, and claim that the specialization over V(J) is +reducible, or continue the process with each irreducible factor of FJ. For irreducible FJ, the +process continues, as in the initial step, by computing the genus of C(FJ). Since in each +iteration of the process the dimension of the variety V(J) decreases, we, at the end, reach +the zero–dimensional case, and the decomposition ends. +Let us say that a specialization degenerates if either F(a0,γ0,x,y) is not well–defined or +F(a0,γ0,x,y) ∈ K. As a result of the process described above, we find a disjoint decomposition +(6.3) +S = ˙⋃i∈ISi +such that, for every specialization a0 ∈ Si, one of the following holds +(1) the specialization degenerates; +(2) the genus is positive and preserved, or the specialized curve is reducible; +(3) the genus is zero and a proper parametrization of C(F,a0) is provided. +Remark 6.1. Let us remark that in the decomposition (6.3) we can take the union of those Si +corresponding to each of the three items above; say S1,S2,S3 representing the corresponding +item. In this way, we can achieve a unique decomposition of the parameter space S. The Si +obtained in this way are constructible sets of S, and S2,S3 are a finite union of subsets Σ as +in (6.1) and S1 is a closed subset directly defined from the implicit equation F. +Moreover, S3 can further be decomposed into a finite union of subsets S3,j such that for +every j, there is a proper parametrization Pj which is well–defined for every a0 ∈ S3,j and +specialized properly. +Finally, since Σ of F as in (6.1) is open on non-empty, depending on the genus of F, either +S2 or S3 is a dense subset of S. +We illustrate the previous ideas by continuing the analysis of Example 5.10. +Example 6.2. (Continuation of Example 5.10) Taking into account (5.5), the closed set Z +(see (6.2)) decomposes as +Z = V(a1) ∪ V(a2) ∪ V(a2 − 27a1) ∪ V(a3 +1 − 2a1 + 4a2) ∪ V(a5 +1 + a5 +2 + 3a2 +1a2 +2 − a1a2) ⊂ Q +2. +We start with J1 ∶= and V1 ∶= V(J1). Since V1 is rational, surjectively parametrized by +Q1 ∶= (0,λ), we work over the field Q(λ)[x,y]. We have that +FJ1 ∶= λ2x2 (λ3xy2 − 6λ2xy + λxy + 9λx − 3x − 2) +and therefore all specializations in V(a1) lead to a reducible curve. Additionally, one may +distinguish the cases λ = 0, that corresponds to the point (0,0), where the specialization +degenerates, and λ ≠ 0 where C(F,a0) decomposes to the union of a double line and a +rational cubic. +The analysis for J2 ∶=, and V2 ∶= V(J2) looks similar. Since V2 is rational, parametrized +by Q2 ∶= (λ,0), we work over the field Q(λ)[x,y]. We have that +FJ2 ∶=λy(λ4x3y − 6λ3x2y − λ3x + 2λ2x2 + 12λ2xy − λx3 − 3λ3 + 9λ2x − 9λx2 + 3x3 + 2λ2 − 4λx − 8λy + 2x2). + +RATIONALITY AND PARAMETRIZATIONS OF ALGEBRAIC CURVES UNDER SPECIALIZATIONS +25 +Thus, all specializations in V2 lead to a reducible curve; note that Q2 is surjective. The case +λ = 0 is covered above, and for λ ≠ 0, the specialization C(F,a0) decomposes to the union of +a line and a rational quartic. +Let us study J3 ∶= and V3 ∶= V(J3). Again, V3 is rational, parametrized by +Q3 ∶= (λ,27λ), and we work over the field Q(λ)[x,y]. We have that +FJ3 ∶=λ(244λx + 3λ − 3x − 2)(58807λ3x2y2 − 732λ3xy2 + 732λ2x2y2 + 9λ3y2 − 13068λ2x2y ++ 464λ2xy2 + 9λx2y2 + 81λ2xy + 6λ2y2 − 81λx2y − 6λxy2 − λ2y + 729λx2 − 106λxy + 4λy2 − x2y). +The analysis of V3 is identical to V2. +Let us study J4 ∶= and V4 ∶= V(J4) that is again rational and it is properly +and surjectively parametrized by Q4 ∶= (λ,−1 +4λ3 + 1 +2λ). We work over the field Q(λ)[x,y]. +We have that +FJ4 ∶=λ(λ5y − 2λ3y + 12λ2 − 8λ + 32y)(λ9x3y − 8λ7x3y + 12λ6x3 + 24λ5x3y + 8λ5x3 − 32λ4x3y +− 72λ4x3 − 32λ3x3y − 16λ4x2 − 32λ3x3 + 192λ3x2y + 144λ2x3 + 16λx3y + 64λ2x2 − 384λ2xy ++ 32λx3 − 96x3 + 256λy − 64x2). +Again V4 behaves as V2 and V3. Finally, let us analyze J5 ∶= and +V5 ∶= V(J5) which is a rational quintic and properly and surjectively parametrized as +Q5(λ) ∶= ( +(5λ − 1)(5λ − 2)4 +3125λ4 − 3750λ3 + 1750λ2 − 375λ + 31,− +(5λ − 2)(5λ − 1)4 +3125λ4 − 3750λ3 + 1750λ2 − 375λ + 31). +Those values for which the parametrization is not defined, i.e. the poles, play no role in +this analysis. The polynomial FJ5 is FJ5(λ,x,y) = F(Q5(λ),x,y) ∈ Q(λ)[x,y]. It holds that +deg(C(FJ5)) = 4, genus(C(FJ5)) = 0 and a proper surjective parametrization is PJ5(λ,t) ∶= +P(Q(λ),t) (see (5.4)). So, we get (see Def. 5.4) +Ωproper(PJ5) ∶= V5 ∖ {PJ5(λ0)∣f(λ0) = 0} +where f ∶= p1 p2 p3 p4 and +p1 ∶= +5λ − 1, p2 ∶= 5λ − 2, p3 ∶= 25λ2 − 15λ + 3, +p4 ∶= +390625λ8 − 937500λ7 + 1343750λ6 − 1237500λ5 + 711875λ4 − 253500λ3 ++54475λ2 − 6495λ + 331. +By Theorem 5.5, for every a0 ∈ Ωproper(PJ5) it holds that C(F,a0) is rationally parametrized +by P(a0,t) (see (5.4)) or, equivalently, by PJ5(Q−1(a0),t). Now, let us analyze the curve in +Z5 ∶= V5 ∖ Ωproper(PJ5) = {PJ5(λ0)∣f(λ0) = 0} (see (6.2)). We define a0 +1 ∶= (0,0) = PJ5(λ0), +where λ0 is a root of p1 p2; a0 +2 ∶= (−1,−1) = PJ5(λ0), where λ0 is a root of p3; and observe that +p4 generates 8 points on the curve that we denote by a0 +i ,i ∈ {3,...,10} and which correspond +to PJ5(λ0) where λ0 is one of the roots of p4. Thus, Z = {a0 +1,...,a0 +10}, C(F,a0 +1) = C2, and, +for i ∈ {2,...,10}, C(F,a0 +i ) are rational cubics parametrized by PJ5(a0 +i ,t). +Summarizing, S decomposes as (see (6.3) and Remark 6.1) +S =(S1 ∶= {(0,0)}) ∪ (S2 ∶= ∪4 +i=1Vi ∖ {(0,0)}) + +26 +RATIONALITY AND PARAMETRIZATIONS OF ALGEBRAIC CURVES UNDER SPECIALIZATIONS +∪ (S3,1 ∶= Ωproper(P)) ∪ (S3,2 ∶= Ωproper(PJ5)) ∪ (S3,3 ∶= {a0 +2,...,a0 +10}). +For a0 ∈ S1, C(F,a0) degenerates. For a0 ∈ S2, C(F,a0) is reducible (note that a0 +1 ∈ S2). +For a0 ∈ S3,1, the specialized curve C(F,a0) is a quintic parametrized by P(a0,t); for a0 ∈ +S3,2, C(F,a0) is a quartic parametrized by PJ5(a0,t); and for a0 ∈ S3,3, C(F,a0) is a cubic +parametrized by PJ5(a0,t). +Example 6.3. Let K = Q and F = L = Q(a1,a2,a3). We consider +F = x3 + x2a1 + y3 + a2a3 ∈ F[x,y]. +One has that genus(C(F)) = 1. Using the ideas in Section 4, we compute Ωgenus(F) (see Def. +4.12). We observe that since C(F) is an elliptic cubic, no blowup is required and, hence, +Ωgenus(F) = ΩgenusOrd(F). Indeed, one gets that (see Fig. 1, right) +Ωgenus(F) ∶= C3 ∖ V(−a2a3 (−4a13 − 27a2a3)). +Thus, by Theorem 4.8, for every a0 ∈ Ωgenus(F), C(F,a0) is either reducible or it is +a genus 1 cubic curve. +Let us analyze the specializations in Z ∶= C3 ∖ Ωgenus(F) = +V(−a2a3 (−4a13 − 27a2a3)). +Let J1 ∶= and V1 ∶= V(J1). Then, FJ1 ∶= x3+a1x2+y3. We observe that genus(C(FJ1)) = +0 and +PJ1 ∶= (− t3a1 +t3 + 1,− t2a1 +t3 + 1) +is a proper parametrization of C(FJ1). +Applying Def. +5.4 to FJ1 and PJ1 we get that +Ωproper(PJ1) ∶= V1 ∖ V(a1). +Therefore, by Theorem 5.5, for all a0 ∈ Ωproper(PJ1) it holds +that C(F,a0) is a rational curve parametrized by PJ1(a0,t). However, for the remaining case, +namely the points (0,0,µ) for µ ∈ C, C(F,(0,0,µ)) decomposes as the product of three lines. +Let J2 ∶= and V2 ∶= V(J2). Now, the situation is identical to the previous case. Let +J3 ∶=<−4a13 − 27a2a3> and V3 ∶= V(J3). The surface V3 can be properly parametrized as +Q(λ1,λ2) = (λ1,λ2,−4λ13 +27λ2 +). +We observe that Q is not surjective. Indeed, Q(C2) = V3 ∖{(0,0,µ)∣µ ∈ C} (see e.g. Remark +3 in [26]). However, the specializations in {(0,0,µ)∣µ ∈ C} have already been analyzed. So, +we treat the case Q(C2). Then, FJ3 can be taken as +FJ3 ∶= F(Q(λ1,λ2),t) = x3 + x2λ1 + y3 − 4λ13 +27 , +where (λ1,λ2) ∈ C2 ∖ {(0,0)}. It holds that genus(C(FJ3)) = 0 and +PJ3 ∶= (t3λ1 − 2λ1 +3t3 + 3 +, t2λ1 +t3 + 1) +is a proper parametrization. Applying Def. 5.4 to FJ3 and PJ3, we get that Ωproper(PJ3) = +V3 ∖ {(0,0,µ)∣µ ∈ C}. +Therefore, by Theorem 5.5, for all a0 ∈ Ωproper(PJ3) it holds that +C(F,a0) is a rational curve parametrized by PJ3(a0,t). +Summarizing, S decomposes as (see (6.3)) +S = (S2 ∶= Ωgenus(F) ∪ {(0,0,µ)∣µ ∈ C}) ∪ (S3 ∶= Ωproper(PJ1)). + +RATIONALITY AND PARAMETRIZATIONS OF ALGEBRAIC CURVES UNDER SPECIALIZATIONS +27 +For a0 ∈ S2,C(F,a0) is either reducible or an elliptic curve; and for a0 ∈ S3,C(F,a0) is a +rational cubic parametrized by PJ3(a0,t). +7. Some Illustrating Applications +In this section, we illustrate by examples some possible applications of the theory developed +in the paper. In the first example, given a surface, we consider the problem of determining +its rational level curves, if any. +Example 7.1. Level curves of a surface. +Let S be the surface defined over C by the polynomial +F = x6 − 5x4y + 3x4z − y5 + 2y4z − y3z2 − x3z + 5x2y2 − 7x2yz + 3x2z2 + y2z − 2yz2 + z3 − x2 +where F ∈ Q(z)[x,y]. So, with the terminology of the paper, a = z, K = Q and L = F = +Q(z). +With this interpretation, the idea is to analyze the genus of C(F) ⊂ Q(z) +2 under +specializations in S ∶= C. For this purpose, we first compute a standard decomposition of the +singular locus as in (4.2). Following the steps described in Subsection A.1, we get +F(F h) = {(0 ∶ z ∶ 1)}m1=t . +Moreover, one can easily check that the family consists in one ordinary double point. There- +fore, one gets that genus(C(F)) = 9. Since the singularities are all ordinary, we get that (see +Def. 4.12) Ωgenus(F) = ΩgenusOrd(F). Moreover (see Def. 4.5), +ΩgenusOrd(F) = +C ∖ {−1,0,1}. +Using Theorem 4.13, for z0 ∈ ΩgenusOrd(F), C(F,(x,y,z0)) is either reducible or its genus is +9. In any case, no rational level curve appears. For the elements in Z ∶= C ∖ ΩgenusOrd(F), we +get that C(F,(x,y,±1)) are irreducible of genus 7 and C(F,(x,y,0)) is irreducible of genus +0. Indeed, C(F,(x,y,0)) can be parametrized by (t5,t6 − t2). +In the second example, we consider the linear homotopy deformation of two curves and we +analyze the genus of each instance curve. +Example 7.2. Linear homotopy deformation of curves. +Let us consider the linear homotopy between the Fermat cubic curve and the unit circle. +That is we consider the polynomial +F = (1 − λ)(x3 + y3 − 1) + λ(x2 + y2 − 1). +We consider F as a polynomial in Q(λ)[x,y] and we analyze the genus behavior through the +deformation. So, with the terminology of the paper, a = λ, K = Q and L = F = Q(λ). Now, +the idea is to study the genus of C(F) under specializations in S ∶= C; or, in particular, in the +real interval [0,1]. For this purpose, we first observe that genus(C(F)) = 1 and hence (see +Def. 4.12), Ωgenus(F) = ΩgenusOrd(F). We get that ΩgenusOrd(F) = C ∖ V(g) where +g(λ) = − (2λ3 − 5λ2 + 7λ − 3)(λ6 − 4λ5 + 15λ4 − 29λ3 + 43λ2 − 33λ + 9)(4λ − 3)(−1 + λ)(λ − 3) +(2λ9 + 27λ8 + 5049λ7 − 40068λ6 + 148716λ5 − 315657λ4 + 398763λ3 − 295245λ2 + 118098λ − 19683) +(8λ3 − 27λ2 + 54λ − 27). +Using Theorem 4.13, for λ0 ∈ ΩgenusOrd(F), C(F,(x,y,λ0)) is either reducible or its genus +is 1. In any case, no rational deformation instance appears. For the elements in Z = C ∖ + +28 +RATIONALITY AND PARAMETRIZATIONS OF ALGEBRAIC CURVES UNDER SPECIALIZATIONS +ΩgenusOrd(F), we get that C(F,(x,y,1)) is rational, the specialized cubics C(F,(x,y,3/4)) and +C(F,(x,y,3)) factor as a union of a line and a conic, and for all the other cases the genus +remains one (see Fig 2). +Figure 2. Left: Plot of the real part of different instances of the deforma- +tion in Example 7.2. +Right: Plot of the real part of C(F,(x,y,3/4)) and +C(F,(x,y,3)) in Example 7.2. +Let O be a connected open subset of C and let Mer(O) be the field of meromorphic +functions in O (see [14]). We consider a polynomial equation of the form +(7.1) +∑ +i,j∈I +fi,j(t)xiyj = 0 +where I is a finite subset of N2, and where fi,j ∈ Mer(O). Let f be the tuple with all the +functions fi,j appearing in (7.1). The question now is to decide, and indeed compute, whether +there exists rational solutions of the equation; that is p,q ∈ C(f) such that ∑i,j∈I fi,j(t)piqj = 0. +We may proceed as follows. We consider the polynomial F(a,x,y) resulting from the formal +replacement in (7.1) of each function fi,j by a parameter ak. Now, with the terminology of +the paper, we take K = C(f) and L = F = K(a). +Then, we decompose S (see (2.1)) as described in Section 6; note that the computations +can be carried out over C(a) instead of over F. Then, if any subset in the decomposition has +genus zero, and the functions f belong to it, we obtain a (family of) rational solutions. Let +us see a particular example. +Example 7.3. Rational solution of functional algebraic equations. +We consider the functional algebraic equation +(7.2) +x2y3f4 +1 − x2y2f4 +2 f3 − 2xy3f2 +1 f2 + 2xyf1f2 +2 f2 +3 + y3f2 +2 − f2 +1 f3 +3 = 0, +where f = (f1,f2,f3) ∶= (sin(t),cos(t),et). We associate to (7.2) the curve C(F) where +F = x2y3a4 +1 − x2y2a4 +2a3 − 2xy3a2 +1a2 + 2xya1a2 +2a2 +3 + y3a2 +2 − a2 +1a3 +3. +It holds that genus(C(F)) = 0 and that a proper parametrization is +(7.3) +P(a,W) = (W 3a1 + a2 +Wa2 +2 + a2 +1 +, a3 +W 2 ). + +3 +N +-3 +2 +3 +-33 +2 +-3 +0 +2 +-1 +-3RATIONALITY AND PARAMETRIZATIONS OF ALGEBRAIC CURVES UNDER SPECIALIZATIONS +29 +The open subset in Def. 5.4 turns to be +Ωproper(P) = C(f) +3 ∖ VC(f) (a1a2 (a1 − a2)(a6 +1 + a5 +1a2 + a4 +1a2 +2 + a3 +1a3 +2 + a2 +1a4 +2 + a1a5 +2 + a6 +2)a3). +Since f ∈ Ωproper(P), by Theorem 5.5, we have that +(7.4) +{x = +W 3 sin(t) + cos(t) +W (cos2 (t)) + sin2 (t),y = et +W 2 }, +for every W ∈ C(f) such that (7.4) is well–defined, is a rational solution of (7.2). In fact, the +last statement holds more generally for every W ∈ Mer(C). +References +[1] Abhyankar, S.S., Bajaj, C.L.: Automatic Parametrization of Rational Curves and Surfaces III: Algebraic +Plane Curves. 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Vieweg+Teubner Verlag, Wiesbaden, 1997. +[32] Shafarevich, I.R. Basic Algebraic Geometry I and II. Springer–Verlag, Berlin New York, 1994. +[33] N. Thieu Vo. Rational and Algebraic Solutions of First-Order Algebraic ODEs. Research Institute for +Symbolic Computation. PhD Thesis. 2016. +[34] Torrente, M., Beltrametti, M.C., Sendra, J.R. Perturbation of polynomials and applications to the Hough +transform. Journal of Algebra, 486:328–359, 2017. +[35] Wang, D. Elimination Methods. Texts and Monographs in Symbolic Computation. Springer-Verlag, Vi- +enna, 2001. +[36] Wang, D. Elimination Practice: Software Tools and Applications. Imperial College Press, 2004. +[37] Walker, R.J. Algebraic Curves. Princeton Univ. Press (1950). +[38] Weispfenning, V. Comprehensive Gr¨obner Bases. Symbolic Computation, 14:1–29, 1992. +[39] Winkler, F. Polynomial Algorithms in Computer Algebra. Springer–Verlag, Wien New York, 1996. +Appendix A. Rational Curves +In this appendix we recall the main steps to compute the genus of an irreducible plane +curve and, in the affirmative case, how to parametrize a proper rational curve. There exist +different methods to that goal: the adjoint curve based method (see e.g. [37] and [30]), +the method based on the anticanonical divisor (see [13]) or the method based on Puiseux +expansions (see [20]), among others. In this paper we will follow the adjoint curve based +method (see [37]); more precisely, we will follow the symbolic approaches in Chapters 3,4,5 +in [30]. A similar treatment of the problem can be performed using the other algorithmic +approaches. +Throughout this appendix, let G ∈ K[x,y] ∖ K be irreducible over K. +A.1. Genus computation. We denote the genus as either genus(C(G)) or genus(C(Gh)). +The key tool for our symbolic computation of the genus is the notion of conjugate family of +points (see Definition 3.15 in [30]). We adapt the definition to our purposes. +Definition A.1. The set F = {(f1(α) ∶ f2(α) ∶ f3(α))∣m(α) = 0} ⊂ P2(K) is called a +K–conjugate family of points if +(1) at least one of the polynomials fi is not zero, +(2) f1,f2,f3,m ∈ K[t] and gcd(f1,f2,f3,m) = 1, + +RATIONALITY AND PARAMETRIZATIONS OF ALGEBRAIC CURVES UNDER SPECIALIZATIONS +31 +(3) degt(m) > 0 and m is squarefree. +The family is represented as +F ∶= {(f1(t) ∶ f2(t) ∶ f3(t))}m(t) +and m(t) is called the defining polynomial of F. We say that F is a family of points of C(Gh) +if Gh(f1,f2,f3) = 0 modulo m. If m(t) is irreducible over K, we say that F is irreducible. +Remark A.2. +(1) We will assume that families are always represented in canonical form. That is, if +F ∶= {(f1(t) ∶ f2(t) ∶ f3(t))}m(t) then f1,f2,f3 are reduced modulo m(t). +(2) Note that a single point (a ∶ b ∶ c) ∈ P2(K) can be seen as a K-conjugate family, for +instance as {(a ∶ b ∶ c)}t. +(3) Note that, factoring the defining polynomial m(t) over K, every family decomposes +as a finite union of irreducible families. So, in general, we may assume that families +are irreducible. Observe that, for H ∈ K[x,y,z] and F ∶= {(f1(t) ∶ f2(t) ∶ f3(t))}m(t) +irreducible, H(f1,f2,f3) ≠ 0 mod m(t) iff H does not vanish at any point in the +family F. Moreover, if F is irreducible, then Km ∶= K[t]/ is an integral domain, +and we can see (f1(t) ∶ f2(t) ∶ f3(t)) as a single point in the curve defined by Gh over +Km. Let us denote this curve by C(Gh +F). In that situation, by abuse of notation, we +will say that F is a point of C(Gh +F) meaning that (f1(t) ∶ f2(t) ∶ f3(t)) ∈ C(Gh +F). +In [30] it is proved that the singular locus of C(Gh) can be decomposed as a finite union of +irreducible families of K-conjugate points such that in each family the multiplicity and the +singularity character (i.e. whether the singularity is ordinary or non–ordinary) is preserved. +The set of all conjugate families appearing in the union above is called a K–standard decom- +position of the singular locus of C(Gh), or of C(G). In the following, we describe a process +to determine a K–standard decomposition of the singular locus; see also [30] page 83. First, +we introduce the notion of regular position: +Definition A.3. We say that the affine plane curve C(G) is in regular position w.r.t. x if +(1) the coefficient of ydeg(G) in G is a non-zero constant; and +(2) G(a,bi) = Gx(a,bi) = 0 with i ∈ {1,2} implies that b1 = b2. +If C(G) is not in regular position, we may apply a linear change of coordinates over K such +that G is transformed into regular position (see e.g. Lemma 2 in [10]). More precisely, for +this purpose, one may perform a change of coordinates of the form {x = ¯x + q¯y,y = ¯y} where +q is taken in an open subset defined by means of subresultants and by the homogeneous +component of maximum degree in G (see e.g. Remark 3 in [10]). +Condition (1) in Definition (A.3) is easy to check. +Let us now deal with the second +condition. Let us consider the ideal J generated by {G,Gx} in K[x,y]. Taking into account +that G is irreducible over K, one has that J is zero–dimensional. Therefore, using the Shape +lemma (see [39] page 194), the normed reduced Gr¨obner basis of +√ +J, w.r.t. the lexicographic +order x < y, is of the form +{u(x),y − v(x)}, +with u square-free and deg(v) < deg(u), if and only if +√ +J, or equivalently J, satisfies condition +(2) in Definition (A.3). It remains to have a computational approach to determine +√ +J. This + +32 +RATIONALITY AND PARAMETRIZATIONS OF ALGEBRAIC CURVES UNDER SPECIALIZATIONS +can be achieved, for instance, using Seidenberg lemma (see e.g. [22] or [15]). More precisely, +if J ∩ K[x] = and J ∩ K[y] =, then +(A.1) +√ +J =, +where ˜f = f/gcd(f,f′) and ˜g = g/gcd(g,g′) and where f′ and g′ denotes the derivatives of f +and g w.r.t. x and y, respectively. +Then, the process for computing a standard decomposition of the singular locus of C(Gh) +is as follows. +Standard decomposition of the singular locus +Step 1: Regular position. If C(G) is not in regular position, apply an affine linear change of +coordinates over K, say L, such that C(G) is transformed into regular position (see comments +above). +Step 2: Families of singularities at infinity. Factor Gh(x,y,0) over K, +Gh(x,y,0) = ∏ +i∈I +gi(x,y)ei. +Note that none of the polynomials gi is x because C(G) is in regular position w.r.t. x. Then, +the set of points at infinity decomposes in irreducible K–conjugate families as +⋃ +i∈I +{(1 ∶ t ∶ 0)}gi(1,t). +Now, a family {(1 ∶ t ∶ 0)}gi(1,t) is singular if and only if Gh +x(1,t,0) = Gh +y(1,t,0) = Gh +z(1,t,0) = 0 +modulo gi(1,t). +Let I be the set containing all gi(1,t) defining singularities. +Then, the +irreducible families of K–conjugate singularities at infinity are +(A.2) +⋃ +m∈I +{(1 ∶ t ∶ 0)}m(t). +Step 3: Families of affine singularities. Let I be the ideal generated by {G,Gx,Gy}. Since +√ +I is +regular w.r.t. x because of condition (2) in Def. A.3, and zero–dimensional, by the discussion +above on the Shape lemma, the normed reduced Gr¨obner basis w.r.t. the lexicographic order +with x < y of +√ +I is of the form {A(x),y − B(x)} with A square-free and deg(B) < deg(A); +recall that if I ∩ K[x] = and I ∩ K[y] =, then +√ +I =, +where ˜f = f/gcd(f,f′), ˜g = g/gcd(g,g′), and f′ and g′ denote the derivatives of f and g +w.r.t. x and y, respectively. So, each irreducible factor m(x) of A(x) over K generates the +irreducible family {(t ∶ B(t) ∶ 1)}m(t). Therefore, if A denotes the set of all irreducible factors +of A, the affine singularities decompose in irreducible K–families as +(A.3) +⋃ +a∈A +{(t ∶ B(t) ∶ 1)}m(t). +Step 4: Standard decomposition. Applying L−1 to (A.2) and to (A.3) we get an K–standard +decomposition of the singular locus of C(Gh). We observe that the irreducible families of +affine singularities will be of the form {(f1(t) ∶ f2(t) ∶ 1)}m(t) and the singularities at infinity +of the form {(L1(t) ∶ L2(t) ∶ 0)}m(t) with degt(Li) ≤ 1 and gcd(L1,L2) = 1. +Let F be a K–standard decomposition of the singular locus of C(Gh) and let F ∈ F; note +that by construction, F is irreducible. Then, we denote by mult(F) the multiplicity of the + +RATIONALITY AND PARAMETRIZATIONS OF ALGEBRAIC CURVES UNDER SPECIALIZATIONS +33 +points in F, as points of C(Gh). In addition, since the character of the singularity is invariant +within the family we will speak about ordinary and non-ordinary families. The multiplicity +of an irreducible family F = {(f1 ∶ f2 ∶ f3)}m(t) can be computed by determining the greatest +non-negative integer r such that all partial derivatives of Gh of order less than r vanish at +(f1 ∶ f2 ∶ f3) modulo m(t). The character of F can be decided by analyzing the squarefreeness, +modulo m(t), of the polynomial defining the tangents to C(Gh) at (f1 ∶ f2 ∶ f3). Alternatively, +one can work with the curve C(Gh +F) (see Remark A.2 (3)). Now, the family F in C(Gh) turns +to be one point in C(Gh +F), namely (f1 ∶ f2 ∶ f3). Then, the character and multiplicity of F is +the character and multiplicity of (f1 ∶ f2 ∶ f3) as a point in C(Gh +F). +Once the singularities have been detected, one proceeds to recursively, and separately, blow +up each non–ordinary singularity via the neighboring singularities (see [37] or [30] for more +details). Let us briefly recall how the first iteration step works, first for a single point, and +afterwards for an irreducible family of conjugate points. In the following, let p ∈ C(Gh) be a +non–ordinary singular point. +Blow up basic step +Step 1. Apply a linear change of coordinates L such that: p = L(0 ∶ 0 ∶ 1), none of the tangent +to C(Gh(L)) at (0 ∶ 0 ∶ 1) is one of the lines x = 0, and y = 0, and for v ∈ {x,y,z} no point in +(C(Gh(L)) ∖ C(v)) ∖ {(0 ∶ 0 ∶ 1)} is a singularity of C(Gh). +Step 2. Apply the Cremona transformation Q ∶= (yz ∶ xz ∶ xy) to C(Gh(L)). Then, Gh(L(Q)) +factors as Gh(Q(L)) = xn1yn2zn3G∗ for some natural numbers n1,n2,n3. We call G∗ the +quadratic transform of Gh w.r.t. p. +The first neighboring of p is defined as +(C(G∗) ∩ C(z)) ∖ {(1 ∶ 0 ∶ 0),(0 ∶ 1 ∶ 0)}. +The points, and their multiplicities, in the neighborhood, are in one to one correspondence +with the tangents, and their multiplicities, to C(Gh) at p. The (first) neighboring singularities +of p are the neighboring points being singularities of C(G∗). The process continues till no +non-ordinary neighboring point appears in the neighborhoods and until all non-ordinary +singularities of C(Gh) have been blowed up. We will refer to the set of all singularities and +neighboring singularities as the neighboring graph of C(Gh). In this situation, the genus can +be computed as (recall that d = deg(C(G))) +(A.4) +genus(C(G)) = (d − 1)(d − 2) +2 +− 1 +2 ∑mult(p)(mult(p) − 1) +where the sum is taking over all the points in the neighboring graph of C(Gh). +If we work with families, say F = {(f1 ∶ f2 ∶ f3)}m0(t0) is a non-ordinary irreducible family +of singularities, we may introduce the curve C(Gh +F) (see above) and repeat the process with +the single point pF ∶= (f1 ∶ f2 ∶ f3). In this way the first neighboring of pF is decomposed into +irreducible Km0–conjugate families; recall that Km0 is the quotient field of K[t0]/). +These irreducible Km0–conjugate families are of the form +(A.5) +{(t1 ∶ 1 ∶ 0)}m1(t0,t1) +where t1 is a new variable and m1(t0,t1) ∈ Km0[t1] ∖ {t1} is irreducible over Km0. In this +situation, the method continues by applying the same process to the irreducible families of +non-ordinary singularities in the first neighborhood. The irreducible families in the second + +34 +RATIONALITY AND PARAMETRIZATIONS OF ALGEBRAIC CURVES UNDER SPECIALIZATIONS +neighborhood will be of the form +(A.6) +{(t2 ∶ 1 ∶ 0)}m2(t0,t1,t2) +where t2 is a new variable and m2 ∈ Km0,m1[t2]∖{t2} is irreducible over Km0,m1, where Km0,m1 +denotes the quotient field of Km0[t1]/. In general, the irreducible families in the +i-th neighborhood will be of the form +(A.7) +{(ti ∶ 1 ∶ 0)}mi(t0,...,ti). +By abuse of notation, we will say that the families of the form in (A.5), (A.6), (A.7) are K– +conjugate families. Applying this process till no non-ordinary neighboring conjugate family +appears in the neighborhoods and until all non-ordinary families of C(Gh) have been blown +up, we get a decomposition that we call a K–standard decomposition of the neighboring graph +of singularities. In this situation, the genus can be computed as +(A.8) +genus(C(G)) = (d − 1)(d − 2) +2 +− 1 +2 ∑ +F∈N +#(F)mult(F)(mult(F) − 1) +where N is a standard decomposition of the neighboring graph of C(Gh). +A.2. Parametrization algorithm. Once the genus of C(Gh) has been computed, if it is +zero, we may derive a rational parametrization of the curve. There are different approaches to +achieve a rational parametrization, see e.g. [13], [27], [28], [30]. Here we will use a simplified +version of the algebraically optimal algorithm in [28] where Hilbert–Hurwitz theorem (see +e.g. Theorem 5.8. in [30]) is applied directly and recursively. +Let N be a K–standard decomposition of the neighboring graph of singularities of C(Gh). +We consider the linear system Ad−2(C(Gh)) of adjoint curves to C(Gh) of degree d −2 (recall +that d = deg(C(G))); that is, the linear system of all (d −2)–degree curves having each r-fold +point of C(Gh), including the neighboring ones, as a point of multiplicity at least r − 1. In +other words, Ad−2(C(Gh)) is the linear system of curves of degree d−2 defined by the effective +divisor +∑ +F∈N +∑ +p∈F +(mult(p) − 1)p, +where the multiplicity is w.r.t. the curve where F belongs to. In practice, the linear con- +ditions to construct the linear system can be derived by working modulo the corresponding +defining polynomials of the families. In the following, we outline the process for computing +Ad−2(C(Gh)). +Adjoints computation +Step 1. We identify the set of all projective curves, including multiple component curves, of +fixed degree d − 2, with the projective space +Vd−2 ∶= P +(d−2)(d+1) +2 +(K) +via their coefficients, after fixing an order of the monomials. By abuse of notation, we will refer +to the elements in Vd−2 by either their tuple of coefficients, or by the associated {x,y,z}–form, +or by the corresponding curve. Let H(Λ,x,y,z) denote the {x,y,z}–homogeneous polyno- +mial of degree d − 2 defining a generic element in Vd−2, where Λ is a tuple of undetermined +coefficients. +Step 2. For each irreducible K–family F ∶= {(f1 ∶ f2 ∶ f3)}m0(t0), with r ∶= mult(F), in the +standard decomposition of the singular locus of C(Gh), and for each partial derivative M + +RATIONALITY AND PARAMETRIZATIONS OF ALGEBRAIC CURVES UNDER SPECIALIZATIONS +35 +of H of order r − 2 w.r.t. the variables {x,y,z}, compute M(Λ,f1,f2,f3) mod m0(t0) and +collect in a set S of its non–zero coefficients w.r.t. t. +Step 3. For each neighborhood and for each irreducible K–family of neighboring singularities +F ∶= {(ti ∶ 1 ∶ 0)}mi(t0,...,ti), with r ∶= mult(F), compute +(A.9) +(⋯(N(Λ,t1,1,0) mod mi)⋯) mod m1 +where N is every partial derivative of order r−2 of N∗, being N∗ the form obtained applying +to H the same blow up process (linear changes of coordinates, Cremona transformation and +quadratic transformation, see Subsection A.1) as the one to reach the curve where the neigh- +boring family F belongs to; in [27, Theorem 6] appears an alternative method to derive (A.9). +Include in S all non–zero coefficients w.r.t. t of the polynomials in (A.9). +Step 4. Solve the homogeneous linear system of equations {L(Λ) = 0}L∈S and substitute the +result in H. The form resulting from this substitution defines Ad−2(C(Gh)). +Remark A.4. We observe that the defining polynomial of Ad−2(C(Gh)) has coefficients in +K(Λ) and hence the ground field is not extended. +Hilbert-Hurwitz theorem (see [9] or [30, Theorem 5.8]) ensures that for almost all +(φ1,φ2,φ3) ∈ Ad−2(C(Gh))3 the mapping +(A.10) +H ≡ (y1 ∶ y2 ∶ y3) = (φ1(x,y,z),φ2(x,y,z),φ3(x,y,z)) +transforms birationally C(Gh) to an irreducible curve of degree d − 2. Note that, because of +Remark A.4, the map in (A.10) is defined over K. Furthermore, since the map is birational, +the genus is preserved. Thus, applying successively Hilbert–Hurwitz theorem one gets either +a conic (if d is even) or a line (if d is odd) K–birationally equivalent to C(Gh). +After these considerations, we can outline the parametrization algorithm. +Parametrization computation +Step 1. Let G be as above so that the genus of C(Gh) is zero. +Step 2. Set M ∶= G, ρ ∶= deg(M), and T as the identity map in P2(K). +Step 3. While ρ > 2 do +(1) Compute Aρ−2(C(M)). +(2) Take (φ1,φ2,φ3) ∈ Ad−2(C(Gh))3, so that the mapping H in (A.10) is birational over +C(M). +(3) Replace M by M(H −1), ρ by deg(M), and T by H ○ T . +Step 4. Parametrize birationally C(M). Let Q(t) be the output parametrization. +Step 5. Return T −1(Q(t)). +Remark A.5. +(1) If d is odd one may stop the loop in Step 3 when ρ = 3 since cubics can be easily +parametrize over the ground field. For parametrizing a conic or a cubic see [30]. +(2) The direct classical parametrization algorithm, see [37], [30], for C(Gh), in addition +to the computation of the adjoints, needs d−2 simple points of C(Gh) (indeed, a more +sophisticated method in [27] shows that only one simple point is enough). Therefore, +and alternative to Step 5 is: compute d − 2 simple points either on the conic, or on +the line, for instance, using Q(t); apply T to these simple points; and now use the +direct parametrization algorithm for C(Gh). + diff --git a/RtE4T4oBgHgl3EQfKwzR/content/tmp_files/load_file.txt b/RtE4T4oBgHgl3EQfKwzR/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0f357256d7cb6bff71d9547510576ab5008b07a0 --- /dev/null +++ b/RtE4T4oBgHgl3EQfKwzR/content/tmp_files/load_file.txt @@ -0,0 +1,2016 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf,len=2015 +page_content='RATIONALITY AND PARAMETRIZATIONS OF ALGEBRAIC CURVES UNDER SPECIALIZATIONS SEBASTIAN FALKENSTEINER AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='RAFAEL SENDRA Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Rational algebraic curves have been intensively studied in the last decades, both from the theoretical and applied point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' In applications (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' level curves, linear homotopy deformation, geometric constructions in computer aided design, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' ), there often appear unknown parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' It is possible to adjoin these parameters to the coefficient field as transcendental elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' In some particular cases, however, the curve has a different behavior than in the generic situation treated in this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' In this paper, we show when the singularities and thus the (geometric) genus of the curves might change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' More precisely, we give a partition of the affine space, where the parameters take values, so that in each subset of the partition the specialized curve is either reducible or its genus is invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' In particular, we give a Zariski-closed set in the space of parameter values where the genus of the curve under specialization might decrease or the specialized curve gets reducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' For the genus zero case, and for a given rational parametrization, a better description is possible such that the set of parameters where Hilbert’s irreducibility theorem does not hold can be isolated, and such that the specialization of the parametrization parametrizes the specialized curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We conclude the paper by illustrating these results by some concrete applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Algebraic curves, parameters, rational parametrizations, singularities, geometric genus, Hilbert’s irreducibility theorem Acknowledgements Authors partially supported by the grant PID2020-113192GB-I00 (Mathematical Visu- alization: Foundations, Algorithms and Applications) from the Spanish MICINN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Part of this work was developed during a research visit of the first author to CUNEF University in Madrid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Introduction The study and analysis of the behavior of algebraic or algebraic-geometric objects under specializations is of great interest from a theoretical, computational or applied point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' For instance, some techniques for computing resultants, gcds, or polynomial factorizations, rely on Hensel’s lemma or the Chinese remainder theorem (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' [11], [39]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' From a more theoretical point of view, also computational, it is important to control, for instance, when a resultant, or more generally a Gr¨obner basis with parameters, specializes properly (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' [6], [17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' The question whether a given irreducible polynomial over K(a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=',an) remains irreducible when the parameters are replaced by values in a field K was studied intensively by Hilbert [8] and Serre [31] and is the defining property of “Hilbertian fields”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' The work of Serre can be seen in a more general context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' With respect to applications, there is a vast Max Planck Institute Leipzig, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Universidad de Alcal´a, Dpto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' F´ısica y Matem´aticas, Alcal´a de Henares, Madrid, Spain E-mail addresses: sebastian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='falkensteiner@mis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='mpg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='de, rafael.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='sendra@uah.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='es.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Date: January 13, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='04933v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='AG] 12 Jan 2023 2 RATIONALITY AND PARAMETRIZATIONS OF ALGEBRAIC CURVES UNDER SPECIALIZATIONS amount of applications of algebraic curves involving parameters: level curves of surfaces [2], linear homotopy deformation of curves (see Section 7), curve recognition [34], geometric constructions in computer aided design, like offsets, conchoids, cissoids etc, where the final object depends on the distance, the focus, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' [3], [4], [16], [19], [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Another type of applications is the computation with meromorphic functions in linear algebra (see [25]) or the rational solutions of functional algebraic equations (see Section 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' In this work we study algebraic curves C(F) given as the zero-set of a polynomial (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='1) F(x,y) = 0 with F ∈ K(a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=',an)[x,y] where K is a computable field of characteristic zero, a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=',an are a set of parameters, and F is irreducible over the algebraic closure of the coefficient field K(a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=',an), denoted by K(a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=',an).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' In this paper, we focus on the problem that for certain values of the pa- rameters a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=',an the algebraic properties of the resulting curve do not coincide with the generic properties of C(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' More precisely, we define several Zariski-closed sets in the space of parameter values where non-generic behavior may appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Of particular interest are the singularities, their multiplicities and their character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' This leads to a partition of the affine space, where the parameters take values, so that in each subset of the partition the special- ized curve is either reducible or its (geometric) genus is invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' When the generic curve has genus zero, for a given rational parametrization can be given a better description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' In particular, the set of parameters where Hilbert’s irreducibility theorem does not hold can be isolated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Moreover, the proper specialization of the rational parametrization is guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' In [30, 37] and references therein are studied algebraic curves and their rationality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' The problem of finding rational parametrizations of plane curves is a classical problem and has already been studied by Hilbert [9], and more recently in [13], [21], [27],[28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' In addition, for evaluating the parameters, it is important to control field extensions which might be necessary for computing parametrizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Optimal fields of parametrizations have been studied in [13] and [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' When introducing parameters in the coefficients, new phenomena have to be considered and lead to Tsen’s study of finding solutions in a minimal field [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' The structure of the paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' In Section 2 we present notations, preliminaries on algebraic curves and rational parametrizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Of particular interest is the computation of the genus and a rational parametrization, if it exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Some of the details are attached in the appendix1 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' In Section 3, we introduce the unspecified parameters and their specialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' The computation of the genus and rational parametrizations is followed to define several computable Zariski-open subsets Ω where the specialized curve behaves, up to irreducibility, as in the generic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' The actual computation of the genus is presented in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' In Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='2 is shown that the genus of the specialized curve, where the parameters take values in ΩsingOrd, is less or equal to the generic genus or the defining polynomial is reducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' A direct corollary of that is that specialized curves of rational curves are also rational or reducible (Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' For values in a smaller set ΩgenusOrd, it is shown that the genus of the curve remains exactly the same, again up to irreducibibility, see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Section 5 is devoted to the case where the generic curve is rational;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' in this frame the irreducibility can be guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' For some of the parameter values the genus may remain the same but an evaluation of the parametrization is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' In Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='5, however, is presented an 1There exist different methods to deal computationally with the genus: the adjoint curve based method (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' [37] and [30]), the method based on the anticanonical divisor (see [13]) or the method based on Puiseux expansions (see [20]), among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' In this paper we will follow the adjoint curve based method which is described, for completeness, in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' RATIONALITY AND PARAMETRIZATIONS OF ALGEBRAIC CURVES UNDER SPECIALIZATIONS 3 open set where the specialization is possible and results in a parametrization of the specialized curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' These open sets can be recursively used for decomposing the whole parameter space as it is explained in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Applications as described above are presented by using illustrative examples in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' This manuscript is a self-contained work on the computation of the genus and rational parametrizations of algebraic curves involving parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Results from various mathemat- ical disciplines are combined for this purpose and presented in a coherent way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' A rigorous construction of such computable Zariski-open sets were, up to our knowledge, missing in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' The theorems mentioned in the previous paragraph are novel and can be directly applied in several interesting problems involving parametric curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Preliminaries and notation Throughout this paper, the following notation will be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' K is a computable field of characteristic zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We denote by a a tuple of parameters, and we represent by L the field extension L ∶= K(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' In addition, we consider an algebraic element γ over L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let F be the field F ∶= L(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Furthermore, K represents any field extension of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We denote by K the algebraic closure of K, similarly for any field appearing in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' S is the affine space (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='1) S = K #(a) where a will take values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' For G ∈ K[x,y] ∖ K, we denote by C(G) the plane affine algebraic curve C(G) = {p ∈ K 2 ∣G(p) = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We denote by Gh(x,y,z) the homogenization of G, and by Gx,Gy (similarly for Gh x,Gh y,Gh z) the partial derivative of G w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' x and y respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' For a homogeneous polynomial M ∈ K[x,y,z] ∖ {0}, C(M) denotes the projective plane curve C(M) = {p ∈ P2(K)∣M(p) = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' For polynomials f,g in the variable t, and coefficients in an integral domain, we denote by rest(f,g) the resultant of f and g w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let {f1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=',fk} ⊂ K[v], where v is a tuple of variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We denote by V(f1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=',fk) the zero set, over K, of the polynomials {f1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=',fk};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' similarly for V(I) where I is an ideal in K[v].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Rational Curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Throughout this section, let G ∈ K[x,y]∖K be irreducible over K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' A rational (affine) parametrization of the irreducible affine plane curve C(G) is a pair of rational functions P(t) ∈ K(t)2 ∖ K 2 such that G(P(t)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' A rational (projective) parametrization of C(Gh) is of form Q(h,t) = (p1(h,t) ∶ p2(h,t) ∶ p3(h,t)) where pi are homogeneous co-prime polynomials of the same degree over K, not all zero, such that Gh(Q) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We observe that the degree, the irreducibility and the rationality of C(G) and C(Gh) are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Moreover the parametrizations of C(G) and C(Gh) relate each other by means of homogenizing and dehomogenizing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' So, in the following we will focus on affine parametrizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' The parametrization P(t) is called birational or proper if the map K ⇢ C(G);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='t ↦ P(t) is injective in a non–empty open Zariski subset of K (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' [30] for further details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Curves admitting a rational parametrization are called rational, and they correspond to those of genus zero;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' note that the genus of C(G) is defined as the genus of C(Gh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' There exist algorithmic methods to compute the genus of an algebraic curves and to determine, when the genus is 4 RATIONALITY AND PARAMETRIZATIONS OF ALGEBRAIC CURVES UNDER SPECIALIZATIONS zero, a rational parametrization of the curve (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' [13], [27], [28], [30]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' In Appendix A we summarize the adjoint curves based method for parametrizing curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Some of the ideas in this paper will use those methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' In general, if one computes a parametrization P(t) of C(G), the ground field K has to be extended (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' in [30]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' A subfield E of K is called a parametriz- ing field or field of parametrization of C(G) if there exists a parametrization of C(G) with coefficients in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Fields of Parametrization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' In this section, we work with the field L ∶= K(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let G ∈ L[x,y] be an irreducible (over L) non-constant polynomial, and let us assume C(G) is a rational curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We analyze the fields of parametrization of C(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' L is always a field of parametrization of C(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Nevertheless, in [28] (see also Chapter 5 in [30]), the optimality of the fields of parametrization is analyzed and, as a consequence of Hilbert-Hurwitz Theorem (see [9]), there always exists a field extension of L, of degree at most 2, being a field of parametrization of C(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Indeed, this field extension, of degree at most two, is the field extension used in Step (4), of the parametrization computation (see Subsection A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='2), to express the simple point utilized in the parametrization of either the conic or the line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We observe that if the two degree field extension is L(α), with minimal polynomial t2 + bt + c ∈ L[t], then L(α) = L(β) where β = α + b/2 which minimal polynomial is t2 + c − b2/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Therefore, the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' (1) If deg(C(G)) is odd then L is a field of parametrization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' (2) If deg(C(G)) is even then either L is a field of parametrization or there exists δ ∈ L algebraic over L, with minimal polynomial t2 − α ∈ L[t], such that L(δ) is a field of parametrization of C(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Observe that the previous result is valid taking L as any field extension of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' The case where a contains a single element admits a particular treatment because of Tsen’s Theorem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' we refer to [7] for this topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' If #(a) = 1, then L is a field of parametrization of C(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' By Hilbert-Hurwitz Theorem (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' in [30] or Subsection A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='2), C(G) is L–birationally equivalent to either a line or a conic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' So, fields of parametrization are preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' In the line case, the result is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' In the conic case, the result follows from Tsen’s Theorem (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' in [32]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' The proof of Tsen’s Theorem provides a method for computing an L-simple point on the conic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' An alternative approach for computing this point can be found in [12] and [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' In the following section we will work with G ∈ K[a,γ][x,y] where γ is algebraic over K(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' In the case where #(a) = 1, we can view γ as the only parameter and write a in terms of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' More precisely, let M(a,c) ∈ K(a)[c] be the minimal polynomial of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We can view M as rational expression in a and consider H(a) ∶= num(M)(a = a,c = γ) ∈ K(γ)[a] as polynomial in a with the root a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Thus, K(γ,a) ∶ K(γ) is a field extension of degree d ≤ degc(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' If d = 1, by Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='3, K(γ) is a field of parametrization of C(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' RATIONALITY AND PARAMETRIZATIONS OF ALGEBRAIC CURVES UNDER SPECIALIZATIONS 5 Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' In Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='3, we have seen that if #(a) = 1 then L is a field of parametriza- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' The following example shows that if #(a) > 1, in general, L is not a field of parametriza- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We consider the conic defined by F ∶= a1x2 +a2y2 −1, and we see that it does not have a parametrization over C(a1,a2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let us assume that C(a1,a2) is a field of parametrization of C(F), then C(F) has infinitely many point in C(a1,a2)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' C(F) can be properly parametrized by P ∶= (√a1 1 − t2 t2 + 1,√a2 2t t2 + 1), which inverse is P−1(x,y) = √a2 (√a1 + x) √a1 y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' So, there are infinitely many points in C(F)∩C(a1,a2)2 that are injectively reachable, via P, for t ∈ C(√a1,√a2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Indeed, note that all points of C(F), with the exception of (−√a1,0), are reachable by P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let t0 ∈ C(√a1,√a2) ∖ {0,±i} be one of these parameter values;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' say P(t0) = (x0,y0) ∈ C(a1,a2)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Then t2 0 = (√a1 + x0)/(√a1 − x0) ∈ C(√a1,a2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' For x0 ≠ 0, it holds that √a1 +x0, √a1 −x0 are coprime (seen as polynomials in √a1) and t0 = ± √√a1+x0 √a1−x0 ∉ C(√a1,√a2), a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' For x0 = 0 we have the curve-points (x0,y0) = (0,±1/√a2) which are not in C(a1,a2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Specializations Throughout the paper, we will specialize the tuple of parameters a taking values in S (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We will write a0 to emphasize that the parameters in a have been substituted by elements in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' In the following we discuss different aspects on the specializations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' General statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' The elements in K(a) are assumed to be represented in reduced form;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' that is, the numerator and denominator are assumed to be coprime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Then, for f ∶= p/q ∈ K(a), where by assumption gcd(p,q) = 1, and for a0 ∈ S (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='1)) such that q(a0) ≠ 0, we denote by f(a0) the K–element p(a0)/q(a0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We may need to work in the finite field extension F = L(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let p(a,t) ∈ K(a)[t], of degree k in t, be the minimal polynomial of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We might simply write p(t) instead of p(a,t) and express it as (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='1) p(t) = tk + Nk−1(a) Dk−1(a) tk−1 + ⋯ + N0(a) D0(a), Ni,Di ∈ K[a] where gcd(Ni,Di) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Then, for a0 ∈ S such that all Di(a0) ≠ 0, we denote by γ0 the algebraic element, over K(a0), defined by an irreducible factor of p(a0,t) = tk + Nk−1(a0) Dk−1(a0) tk−1 + ⋯ + N0(a0) D0(a0) ∈ K(a0)[t] ⊂ K[t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' For an element f ∈ F, specialized at a0 ∈ S, we might simply write f(a0) instead of f(a0,γ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We define the open subset Ωγ ∶= S ∖ V(D) where D ∶= lcm(D0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=',Dn−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Clearly for a0 ∈ Ωγ, γ(a0) is well–defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' The elements in F are assumed to be expressed in canonical form;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' that is, f ∈ F is expressed as (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='2) f = k−1 ∑ i=0 Ui(a) W(a)γi 6 RATIONALITY AND PARAMETRIZATIONS OF ALGEBRAIC CURVES UNDER SPECIALIZATIONS where Ui,W ∈ K[a] and gcd(U1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=',Uk−1,W) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' In addition, the coefficients of polynomials in F[v], w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' to the tuple of variables v, are also supposed to be written in canonical form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Moreover, for f as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='2), we denote by Norm(f) the L–field element Norm(f) ∶= ∏f(a,γi) ∈ L where the product is taken over all roots γi in L of p(a,t) (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Note that Norm(f) = rest(f(a,t),p(a,t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let f be as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let a0 ∈ S be such that D(a0)W(a0) ≠ 0 (see Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' If f(a0) = 0, then Norm(f)(a0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Since D(a0)W(a0) ≠ 0, then γ(a0), f(a0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='Then Norm(f)(a0) = ∏f(a0,γi(a0)) is well-defined and, since one of the factors on the right hand side is equal to f(a0,γ(a0)) = 0, we obtain Norm(f)(a0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' □ Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let H ∈ F[v], where v is a tuple of variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let S be the set of all non-zero coefficients of H w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let D(H) ∶= lcm({denom(C)∣C ∈ S}) ∈ K[a], and let V(H) ∶= {Norm(numer(C))∣C ∈ S} ⊂ K[a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We associate to H the following open subsets (1) Ωdef(H) ∶= Ωγ ∩ (S ∖ V(D(H))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' (2) ΩnonZ(H) ∶= Ωdef(H) ∩ (S ∖ V(V(H))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Throughout the paper, we will define several open subsets of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' All these open subsets will be included in Ωγ (for the corresponding algebraic element γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' So, we observe that γ0 will be always well–defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' The next lemma justifies the previous definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let H ∈ F[v], where v is a tuple of variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' It holds that (1) If a0 ∈ Ωdef(H) then H(a0,γ0,v) is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' (2) If a0 ∈ ΩnonZ(H) then H(a0,γ0,v) ≠ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' (1) Let a0 ∈ Ωdef(H) ⊂ Ωγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Then, γ0 = γ(a0) is well–defined, and the result follows from the definition of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' (2) Let a0 ∈ ΩnonZ(H) ⊂ Ωdef(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Then, by (1), H(a0,γ0,v) is well–defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Furthermore, there exists a coefficient of H w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' v, say C(a,γ), such that Norm(numer(C))(a0) ≠ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Since D(a0) ≠ 0 and the denominator of C does not vanish at a0, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='2, we get that C(a0,γ0) ≠ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' So, H(a0,γ0,v) ≠ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' □ The following lemma is an adaptation of Lemma 3 in [29] to our case, and will be used to control the birationality of a curve parametrization P(a,t) under specializations of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let f1,f2 ∈ F[u][v]∖{0} for i ∈ {1,2}, where u,v are variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let fi = f∗ i g, for i ∈ {1,2}, where g = gcd(f1,f2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let Ai ∈ F[u] be the leading coefficient of fi w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' v for i ∈ {1,2} and B ∈ F[u] the leading coefficient of g w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let R = resv(f∗ 1 ,f∗ 2 ) ∈ F[u].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let Ω1 ∶= Ωdef(f1) ∩ Ωdef(f2) ∩ Ωdef(f∗ 1 ) ∩ Ωdef(f∗ 2 ) ∩ Ωdef(g) ∩ Ωdef(R), Ω2 ∶= ΩnonZ(A1) ∩ ΩnonZ(A2) ∩ ΩnonZ(B) ∩ ΩnonZ(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' RATIONALITY AND PARAMETRIZATIONS OF ALGEBRAIC CURVES UNDER SPECIALIZATIONS 7 We define the set Ωgcd(f1,f2) ∶= Ω1 ∩ Ω2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let f1,f2,f∗ 1 ,f∗ 2 ,g be as in Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' For a0 ∈ Ωgcd(f1,f2), it holds that g(a0,γ0,u,v) = λ(u)gcd(f1(a0,γ0,u,v),f2(a0,γ0,u,v)), with λ(u) ∈ K[u] ∖ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Moreover, degv(g(a0,γ0,u,v)) = degv(g(a,γ,u,v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let Ai,B,R be as in Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Since a0 ∈ Ωgcd(f1,f2) ⊂ Ω1 (see Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='6), by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='5 (1), the specializations of fi,f∗ i ,g,R at a0 are well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' So, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='3) fi(a0,γ0,u,v) = f∗ i (a0,γ0,u,v)g(a0,γ0,u,v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Moreover, since a0 ∈ Ωgcd(f1,f2) ⊂ Ω2 (see Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='6), the specializations of fi,g,R at a0 preserve the degree in v and, in particular, are non–zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' This implies that f∗ i (a0,γ0,u,v) are non–zero too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='3), one has that there exists λ ∈ K[u] ∖ {0} such that gcd(f1(a0,γ0,u,v),f2(a0,γ0,u,v)) = λ(u) gcd(f∗ 1 (a0,γ0,u,v),f∗ 2 (a0,γ0,u,v))g(a0,γ0,u,v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let us assume that gcd(f∗ 1 (a0,γ0,u,v),f∗ 2 (a0,γ0,u,v)) has positive degree in v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Then, if ˜R(u) is the resultant w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' v of f∗ 1 (a0,γ0,u,v) and f∗ 2 (a0,γ0,u,v), we get that ˜R is zero (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Corollary page 288 in [11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' However, since a0 ∈ Ωgcd(f1,f2) ⊂ Ω2 (see Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='6), by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='5(2), A1,A2 do not vanish at a0 and, hence, the leading coefficients of f∗ i do not vanish either at a0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Therefore, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='1 in [39], R(a0,γ0,u) = ˜R(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Nevertheless, since a0 ∈ Ω2 by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='5 (2), R(a0,γ0,u) ≠ 0 which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' So, g(a0,γ0,u,v) and gcd(f1(a0,γ0,u,v),f2(a0,γ0,u,v)) are associated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' In addition, since a0 ∈ Ω2, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='5 (2), B(a0,γ0,u) ≠ 0 and, hence, degv(g(a0,γ0,u,v)) = degv(g(a,γ,u,v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' □ If in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='7 all coefficients are assumed to be in a field, the statement can be simplified as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let f1,f2 ∈ F[v] ∖ {0} for i ∈ {1,2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let fi = f∗ i g, for i ∈ {1,2}, where g = gcd(f1,f2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' For a0 ∈ Ωgcd(f1,f2), it holds that g(a0,γ0,t) = gcd(f1(a0,γ0,v),f2(a0,γ0,v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Moreover, degv(g(a0,γ0,v)) = degv(g(a,γ,v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let us now generalize the previous statement to several univariate polynomials with coef- ficients in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let f1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=',fr ∈ F[v] ∖ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let fi = f∗ i g, for i ∈ {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=',r}, where g = gcd(f1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=',fr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We consider the polynomial fZ ∶= f2 + f3Z + ⋯ + frZr−2 ∈ F(Z)[v] where Z is a new variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We define Ωgcd(f1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=',fr) = Ωgcd(f1,fZ) ∩ Ωdef(g) ∩ ΩnonZ(A) where A is the leading coefficient of g w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Observe that if r = 2 in Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='9, then Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='6 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='9 coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let f1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=',fr,f∗ 1 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=',f∗ r ,g be as in Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' For a0 ∈ Ωgcd(f1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=',fr), it holds that g(a0,γ0,v) = gcd(f1(a0,γ0,v),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=',fr(a0,γ0,v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Moreover, degv(g(a0,γ0,v)) = degv(g(a,γ,v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' 8 RATIONALITY AND PARAMETRIZATIONS OF ALGEBRAIC CURVES UNDER SPECIALIZATIONS Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let g∗ ∶= gcd(f1,fZ) ∈ F(Z)[v].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Since f1 does not depend on Z, g∗ ∈ F[v].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' This implies that g = λg∗ with λ ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' By Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='8, we know that g∗(a0,γ0,t) = gcd(f1(a0,γ0,v),fZ(a0,γ0,Z,v)) and that degv(g∗(a0,γ0,v)) = degv(g∗(a,γ,v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Note that degv(g∗(a,γ,v)) = degv(g(a,γ,v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Moreover, since a0 ∈ Ωdef(g) ∩ ΩnonZ(A), then g(a0,γ0,v) is well–defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Furthermore, since both leading coefficients of g and g∗ w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' v do not vanish at a0, then λ(a0,γ0) is well–defined and non–zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Thus, degv(g∗(a0,γ0,v)) = degv(g(a0,γ0,v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Summarizing, degv(g(a0,γ0,v)) = degv(g(a,γ,v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' On the other hand, g(a0,γ0,v) = λ(a0,γ0)g∗(a0,γ0,v) = λ(a0,γ0) gcd(f1(a0,γ0,v),fZ(a0,γ0,Z,v)) and, since λ(a0,γ0) ≠ 0, this implies that g(a0,γ0,v) = gcd(f1(a0,γ0,v),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=',fr(a0,γ0,v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' □ Our next step is to analyze the squarefreeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let f ∈ F[v]∖F be squarefree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let R be the discriminant of f w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' v and let A be the leading coefficient of f w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We define the open subset Ωsqfree(f) ∶= Ωdef(f) ∩ ΩnonZ(R) ∩ ΩNonZ(A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let f ∈ F[v] ∖ F be squarefree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' If a0 ∈ Ωsqfree(f), then degv(f(a,γ,v)) = degv(f(a0,γ0,v)) and f(a0,γ0,v) is squareefree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Since a0 ∈ Ωdef(f), by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='5, f(a0,γ0,v) is well–defined and, since a ∈ ΩNonZ(A)), f(a0,γ0,v) ≠ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Furthermore, one has the equality of the degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Moreover, since a0 ∈ ΩnonZ(R), also by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='5, R(a0,γ0,v) is well-defined and non–zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Since A(a0,γ0,v) ≠ 0, by [39, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='1], the discrimininant of f(a0,γ0,v) is not zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Then, by [39, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='1], f(a0,γ0,v) is squarefree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Specialization of the curve defining polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' In this subsection, we deal with the specialization of defining polynomials of irreducible plane curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let G ∈ F[x,y] ∖ F be irreducible over F of total degree d and let Gh ∈ F[x,y,z] be its homogenization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' In the following, let G be written as (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='4) G = gd(x,y) + ⋯ + g0(x,y) where gi is either the zero polynomial or a form of degree i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We associate to G the open subset (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='4)) ΩG ∶= Ωdef(G) ∩ ΩnonZ(gd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let G be as above and let a0 ∈ ΩG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Then (1) G(a0,x,t) is well–defined and deg(G(a0,x,y)) = deg(G);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' (2) G(a0,x,y)h = Gh(a0,x,y,z);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' (3) the partial derivatives of Gh, of any order, specialize properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Since a0 ∈ Ωdef(G), by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='5, G(a0,x,y) is well–defined and, since a0 ∈ ΩnonZ(gd), the equality on the degree holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Since G(a0,x,y) is well–defined, the other statements directly follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' □ RATIONALITY AND PARAMETRIZATIONS OF ALGEBRAIC CURVES UNDER SPECIALIZATIONS 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Specialization of families of points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let us now deal with the specialization of conju- gate families of points associated to a curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' More precisely, let G and Gh be as in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We will study the specialization of families in the standard decomposition of the singular locus of C(Gh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' For this purpose, we observe that, for each a0 ∈ S such that G(a0,x,y) /∈ K, G(a0,x,y) defines an affine plane curve over K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let us denote by C(G) the first curve and by C(G,a0) the second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' The conjugate families of C(Gh) will be over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' When we specialize a we need to have a reference field where the conjugation of the points is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' This motivates the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' For a0 ∈ S, we define Ka0 as the smallest subfield of K containing the coefficients of G(a0,γ0,x,y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Moreover, if F = {(f1 ∶ f2 ∶ f3)}m(t) is an F–conjugate family, and a0 ∈ S is such that γ0,f1(a0,t),f2(a0,t),f3(a0,t),m(a0,t) are well–defined, we denote by F(a0) the specialization of F at a0 (and γ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let F(Gh) be an F–standard decomposition of the singular locus of C(Gh) obtained using the process described in Subsection A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let F(Gh) decompose as (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='5) F(Gh) = ⋃ m(t)∈Aa {(f1,m(t) ∶ f2,m(t) ∶ 1)}m(t) ∪ ⋃ m(t)∈A∞ {(L1,m(t) ∶ L2,m(t) ∶ 0)}m(t) where fi,m,m(t) ∈ F[t], Li,m ∈ K[t] (recall that the transformation L, in the standard decomposition process in Subsection A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='1, can be taken over K) with deg(Li,m) ≤ 1, gcd(L1,m,L2,m) = 1, and m(t) irreducible over F, and where Aa and A∞ are finite sets of irreducible polynomials in F[t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' By abuse of notation, we will write F ∈ F(Gh) for such a component F of F(Gh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let F ∶= {(f1,m(t) ∶ f2,m(t) ∶ 1)}m(t) ∈ F(Gh) be an irreducible F-conjugate family of affine singularities of C(Gh) (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='5)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let A be the product of the leading coefficient of m w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' t and the leading coefficients w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' t of f1,m,f2,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We associate to F the open set Ωdef(F) ∶= Ωdef(f1,m) ∩ Ωdef(f2,m) ∩ Ωsqfree(m) ∩ ΩNonZ(A)) ∩ ΩG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let F ∶= {(L1,m(t) ∶ L2,m(t) ∶ 0)}m(t) ∈ F(Gh) be an irreducible F-conjugate family of singularities of C(Gh) at infinity (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='5)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let A be the leading coefficient of m w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We associate to F the open set Ωdef(F) ∶= Ωsqfree(m) ∩ ΩNonZ(A)) ∩ ΩG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We start our analysis with a technical lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let H,m ∈ F[t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let H = R mod m, and let A be the leading coefficient of m w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' If H(a0,t),m(a0,t) are well–defined and A(a0) ≠ 0, then H(a0,t) = R(a0,t) mod m(a0,t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let Q be the quotient of H by m w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' So, H = Q ⋅ m + R with degt(R) < degt(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Since H(a0),m(a0,t) is well–defined, γ0 is well–defined or all polynomials are independent of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Since A(a0) ≠ 0, then Q(a0,t),R(a0,t) are well–defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Moreover, degt(m(a,t)) = degt(m(a0,t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Then H(a0,t) = Q(a0,t)m(a0,t) + R(a0,t) with degt(R(a0,t)) ≤ degt(R) < degt(m) = degt(m(a0,t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' □ 10 RATIONALITY AND PARAMETRIZATIONS OF ALGEBRAIC CURVES UNDER SPECIALIZATIONS Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let F ∈ F(Gh) be an irreducible F–family of C(Gh) (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='5)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' If a0 ∈ Ωdef(F), then F(a0) is a Ka0-conjugate family of points of C(Gh,a0) and #(F) = #(F(a0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let F = {(f1,m(t) ∶ f2,m(t) ∶ 1)}m(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let us first show that F(a0) is a Ka0-conjugate family of points of C(Gh,a0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Since a0 ∈ Ωdef(fi,m) and a0 ∈ Ωsqrfree(m) ⊂ Ωdef(m), we have that γ0, fi,m(a0,t) and m(a0,t) are well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Furthermore, since a0 ∈ ΩNonZ(A)), the degree of all non-constant polynomials fi,m and m is preserved under the specialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' In addition, since a0 ∈ Ωsqfree(m), it holds that m(a0,t) is squarefree (see Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='13) and condition (3) in Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='1 holds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' note that conditions (1) and (2) in Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='1 hold trivially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Furthermore, note that, after specialization, all polynomials are over Ka0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' So, F(a0) is a family over Ka0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' It remains to prove that the points in F(a0) are in the specialized curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Since a0 ∈ ΩG, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='15, it holds that G(a0,x,y)h = Gh(a0,x,y,z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let T(a,t) = Gh(a,f1,m,f2,m,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Since F is a family of points in C(Gh), it holds that T = 0 mod m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Since Gh(a0,x,y,z) and fi,m(a0,t) are well–defined, then T(a0,t) is well–defined too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We know that m(a0,t) is well–defined and that the leading coefficient of m in t does not vanish after specialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Therefore, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='18, Gh(a0,f1,m(a0,t),f2,m(a0,t),1) = 0 modulo m(a0,t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Hence, F(a0) is a Ka0-conjugate family of points of C(Gh,a0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' If F ∶= {(L1,m(t) ∶ L2,m(t) ∶ 0)}m(t) all the arguments above apply and conditions (1) and (2) in Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='1 also hold because Li,m do not depend on a or γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We already know that F(a0) is a Ka0-conjugate family of points of C(Gh,a0), and degt(m) = degt(m(a0,t)), where m is the defining polynomial of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' It remains to prove that #(F(a0)) = #(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let L be the K–linear change of coordinates transforming C(G) in regular position;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' see Step (1) in the standard decomposition process described in Subsection A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Then, #(F) = #(L−1(F)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Furthermore, L−1(F) is in the form appearing either in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='2) or in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Therefore, #(F) = degt(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Since L is over K, we may apply it to F(a0) and L−1(F(a0)) will be of the form either {(t ∶ B(a0,t) ∶ 1)}m(a0,t) or {(1 ∶ t ∶ 0)}m(a0,t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' In both cases, #(F(a0)) = #(L−1(F(a0))) = degt(m(a0,t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Now, the result follows using that degt(m) = degt(m(a0,t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Given an F–conjugate family F ∈ F(Gh) (see equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='5)), and a0 ∈ Ωdef(F) (see Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='17), we observe that, even though F is irreducible, F(a0) may be reducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We will be interested in working with irreducible specialized families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' So, factoring over Ka0 the defining polynomial of F(a0), the family will be decomposed as F(a0) = ⋃ i∈I Fi where Fi is an irreducible Ka0–family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We will refer to Fi as the irreducible subfamilies of F(a0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' In the sequel we analyze the multiplicity of families of singularities under specializations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let F ∈ F(Gh) (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='5)) be an irreducible F-conjugate family of r-fold points of C(Gh) with defining polynomial m(t), and let H∗ be one of the order r derivatives of Gh such that H∗(F) ≠ 0 modulo m(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let H(a,t) be the reduction of H∗(F) modulo m(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let R(a) ∶= rest(H(t),m(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We define the open subset Ωmult(F) ∶= Ωdef(F) ∩ ΩnonZ(H) ∩ ΩnonZ(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' RATIONALITY AND PARAMETRIZATIONS OF ALGEBRAIC CURVES UNDER SPECIALIZATIONS 11 Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We observe that in Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='21, m is irreducible (note that F belongs to a standard decomposition of the singular locus) over F, H ∈ F[t] ∖{0} and degt(H) < degt(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Therefore, gcd(m,H) = 1 and hence R ≠ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let F ∈ F(Gh) (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='5)) be an irreducible F-conjugate family of r-fold points of C(Gh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' If a0 ∈ Ωmult(F), then every irreducible subfamily of F(a0) (see Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='20) is a Ka0–conjugate family of r-fold points of C(Gh,a0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let F be expressed as F = {(f1 ∶ f2 ∶ λ)}m(t) where f1,f2,m ∈ F[t], λ ∈ {0,1}, m irreducible over F, and such that, for the case λ = 0, degt(fi) ≤ 1 and gcd(f1,f2) = 1 (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='5)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let H∗ and H be as in Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='21, and let Fi, with defining polynomial mi, be an irreducible subfamily of F(a0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Since a0 ∈ Ωdef(F), by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='19, F(a0) is a Ka0–conjugate family of points of C(Gh,a0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' in particular fi(a0,t) and m(a0,t) are well–defined and the leading coefficient of m w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' t does not vanish at a0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Moreover, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='15, since a0 ∈ ΩG, all partial derivatives specialize properly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' note that Ωdef(F) ⊂ ΩG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Therefore, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='18, the multiplicity of Fi is at least r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Moreover, since a0 ∈ ΩG, H∗(a0,f1(a0,t),f2(a0,t),λ) is well– defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' So, again by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='18, H∗(a0,f1(a0,t),f2(a0,t),λ) = H(a0,t) modulo m(a0,t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Furthermore, since a0 ∈ ΩnonZ(H), then H(a0,t) ≠ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Moreover, since the leading coefficient of m w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' t does not vanish at a0, we have that rest(H(a0,t),m(a0,t)) = µR(a0) for some non- zero constant µ (see Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='1 in [39]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' So, since a0 ∈ ΩnonZ(R), rest(H(a0,t),m(a0,t)) ≠ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Therefore, gcd(H(a0,t),m(a0,t)) = 1 and hence H(a0,t) ≠ 0 mod mi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Summarizing, the multiplicity of Fi is r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' □ In the last part of this subsection, we deal with the tangents to C(Gh) at an irreducible F-conjugate family;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' since the family F is assumed to be irreducible, one may think on the tangents at F to the curve C(Gh F) (see Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='2(3) in Subsection A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let F, with defining polynomial m(t), be an irreducible F-conjugate family of r-fold points of C(Gh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let Fm be the quotient field of F[t]/ < m(t) >.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' The defining tangent polynomial of C(Gh) at F is defined as the homogenous polynomial T ∈ Fm[x,y,z] of degree r which factors over the algebraic closure of Fm into the tangents, with the according multiplicities, of C(Gh) at F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Similarly, we introduce the defining tangent polynomial to an specialized curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' (1) Let us assume that F = {(f1 ∶ f2 ∶ 1)}m(t) ∈ F(Gh) (see equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='5)) is a family of r–fold points with irreducible m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' similarly if the family is at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Then T is the reduction of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='6) T ∗(a,t,x,y,z) = r ∑ i=0 (r i) ∂rG ∂ix∂r−iy(f1,f2)(x − f1z)i(y − f2z)r−i modulo m(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' (2) Let F be as in Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We observe that F is a family of ordinary r–fold points if and only if T is squarefree over Fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' In the sequel, we assume w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' that there is no tangent of C(Gh F) at F independent of x and for two different tangents T1(x,y,z),T2(x,y,z) it holds that T1(x,1,1) ≠ T2(x,1,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Note that, if this is not the case, one can apply a linear change over K (and thus invariant under specializations 12 RATIONALITY AND PARAMETRIZATIONS OF ALGEBRAIC CURVES UNDER SPECIALIZATIONS of the parameters a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Then, the ordinary character of the family is readable from the squarefreeness of T(a,t,x,1,1) over Fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let F, with defining polynomial m(t), be an irreducible F-conjugate family of ordinary r-fold points of C(Gh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let T be the defining tangent polynomial of F, where we assume w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' that the hypotheses in Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='25 (2) are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let D(a,t) be the reduction modulo m(t) of the discriminant w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' x of T(a,t,x,1,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let N(a) = rest(D,m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let A(a,t) be the leading coefficient of T(a,t,x,1,1) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' x and let R(a) = rest(A,m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let S(a,x) = rest(T(a,t,x,0,0),m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We define the set Ωord(F) = Ωmult(F) ∩ ΩnonZ(R) ∩ ΩnonZ(N) ∩ ΩnonZ(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' In relation to Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='26, we observe the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' (1) By construction, degt(A) < degt(m) and clearly A is not zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Since m is irreducible, gcd(A,m) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Hence, R ≠ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' (2) Since F is ordinary and the two hypotheses in Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='25 (2) are satisfied, D ≠ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Because degt(D) < degt(m) and m is irreducible, gcd(m,H) = 1 and hence, N ≠ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' (3) T(a,t,x,y,z) has a factor in Fm[y,z] if and only if T(a,t,x,0,0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' This fol- lows from the fact that the tangents are of degree one and thus, T(a,t,x,y,z) = ∏(Ai(a,t)x+Bi(a,t)y+Ci(a,t)z) for some Ai,Bi,Ci ∈ Fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Thus, under our assump- tion that T(a,t,x,y,z) does not have a factor independent of x, T(a,t,x,0,0) ≠ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Since m is irreducible, and degt(T(a,t,x,0,0)) < degt(m), it follows that S ≠ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let F ∈ F(Gh) (see equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='5)) be an irreducible F-conjugate family of ordinary r-fold points of C(Gh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' If a0 ∈ Ωord(F), then every irreducible subfamily of F(a0) (see Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='20) is a Ka0–conjugate family of ordinary r-fold points of C(Gh,a0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Since a0 ∈ Ωord(F) ⊂ Ωmult(F) ⊂ Ωdef(F), then degt(m(a,t)) = degt(m(a0,t)) and, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='23, every irreducible subfamily of F(a0) is a Ka0–conjugate family of r-fold points of C(Gh,a0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let us prove that all points in F(a0) are ordinary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let T ∗ be as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='6), or similarly if the family is at infinity, and let T be the reduction of T ∗ modulo m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Since a0 ∈ Ωord(F) ⊂ Ωmult(F) ⊂ Ωdef(F) ⊂ ΩG, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='15, T ∗ specializes properly at a0, and since degt(m(a,t)) = degt(m(a0,t)), by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='18, T also specializes properly at a0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Now, let P be a point in F(a0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Then, there exists a root t0 of m(a0,t) such that P is obtained by specializing F at a0 and t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Since P belongs to one of the irreducible subfamilies, P is an r–fold point of the curve C(Gh,a0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Because of the discussion above, E(x,y,z) ∶= T(a0,t0,x,y,z) is the defining tangent polynomial of C(Gh,a0) at P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' It remains to prove that E is squareefree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' First, let us see that there is no factor of E independent of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Assume that e(y,z) is a factor of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Then E(x,0,0) = 0, and (a0,t0,x,0,0) is a common zero of T(a,t,x,y,z) and m(a,t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Therefore, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='3 in [39], S(a0,x) = 0 which contradicts that a0 ∈ ΩnonZ(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Thus, it is sufficient to prove the squarefreeness of E(x,1,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let us assume that it is not squarefree, then its discriminant is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' That is, the discriminant of T(a0,t0,x,1,1) is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' On the other hand, a0 ∈ ΩnonZ(R), R(a0) ≠ 0 and thus, A(a0,t0) ≠ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' By [39, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='3] and the fact that m(a0,t0) = 0, it follows that D(a0,t0) = 0 and consequently, N(a0) = 0, in contradiction to a0 ∈ ΩnonZ(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' □ As a consequence of Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='19, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='23, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='28, and taking into account that Ωord(F) ⊂ Ωmult(F) ⊂ Ωdef(F), we get the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' RATIONALITY AND PARAMETRIZATIONS OF ALGEBRAIC CURVES UNDER SPECIALIZATIONS 13 Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let F ∈ F(Gh) (see equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='5)) be an irreducible F-conjugate family of ordinary r-fold points of C(Gh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' If a0 ∈ Ωord(F), then all points in F(a0) are ordinary r-fold points of C(Gh,a0) and #(F) = #(F(a0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Preservation of the Genus We consider a polynomial (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='1) F(a,γ,x,y) ∈ K[a,γ][x,y] ∖ K[a,γ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' F, as a non–constant polymomial in F[x,y], defines an affine plane curve over F that we assume irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' As introduced in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='3, for each a0 ∈ S such that F(a0,γ0,x,y) /∈ K, we denote by C(F,a0) the curve C(F(a0,γ0,x,y));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' similarly for C(F h,a0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Also, we denote by Ka0 the ground field of C(F,a0) (see Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Our goal is to analyze the relation between the genus of C(F) and the genus of C(F,a0) under the assumption that C(F,a0) is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Ordinary singular locus case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We start our analysis assuming that C(F h) has only ordinary singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let F(F h) be an F–standard decomposition of the singular locus of C(F h) obtained by using the process described in Subsection A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let F(F h) decompose as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='5) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='2) F(F h) = ⋃ m(t)∈Aa {(f1,m(t) ∶ f2,m(t) ∶ 1)}m(t) ∪ ⋃ m(t)∈A∞ {(L1,m(t) ∶ L2,m(t) ∶ 0)}m(t) where fi,m ∈ F[t], Li,m ∈ K[t] with gcd(L1,m,L2,m) = 1 and deg(Li,m) ≤ 1, and Aa, A∞ are finite sets of irreducible polynomials in F[t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We start with the following definition, where sing(C(F h)) denotes the singular locus of C(F h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We define the open subset (see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='2) and Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='14 and Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='26) ΩsingOrd(F h) ∶= ⎧⎪⎪⎪⎨⎪⎪⎪⎩ ⋂ F∈F(F h) Ωord(F) if sing(C(F h)) ≠ ∅ ΩF if sing(C(F h)) = ∅ Then, the following result holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let a0 ∈ ΩsingOrd(F h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' If C(F,a0) is irreducible, then genus(C(F)) ≥ genus(C(F,a0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let d be the degree of C(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' If sing(C(F h)) = ∅, then a0 ∈ ΩF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' So, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='15, deg(F(a0,γ0,x,y)) = d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Since C(F,a0) is irreducible, by (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='4), one has that genus(C(F)) = (d − 1)(d − 2) 2 ≥ genus(C(F,a0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' If sing(C(F h)) ≠ ∅, then a0 ∈ ΩsingOrd(F h) ⊂ ΩF and deg(F(a0,γ0,x,y)) = d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' By Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='29, all elements in sing(C(F h)) have the same multiplicity and character as their corre- sponding elements in sing(C(F h,a0)) after specialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' New singularities, however, may appear in sing(C(F h,a0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' So, reasoning as above with the genus formula in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='4), or (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='8), we get the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' □ The next result is a direct consequence of the previous theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' 14 RATIONALITY AND PARAMETRIZATIONS OF ALGEBRAIC CURVES UNDER SPECIALIZATIONS Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let C(F) be a rational curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let a0 ∈ ΩSingOrd(F h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' If C(F,a0) is irreducible, then C(F,a0) is rational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' The inequality in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='2 comes from the fact that, using ΩSingOrd(F h), we cannot ensure that sing(C(F h,a0)) does not include new singularities apart from those coming from the specialization of the singular locus of C(F h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' To control this phenomenon, we will ensure that certain Gr¨obner bases behave properly under specializations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' By, exercises 7, 8, pages 315–316 in [6], or by Proposition 1, page 308 in [6], we know that there exists an open Zariski set such the Gr¨obner basis specializes properly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' in fact, a description of this open subset is also available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' For a more general analysis of Gr¨obner bases with parametric coefficients we refer to [17] and [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' On the other hand, since we are working with bivariate polynomials in F[x,y], the open subset above can be determined by using resultants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' This motivates the next definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let I be an ideal in F[v], where v is tuple of variables, generated by G ⊂ F[v].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let G be a Gr¨obner basis of G w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' some order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We define ΩspGB(G) ⊂ S as a non-empty open subset such that for every a0 ∈ ΩspGB(G) it holds that {g(a0,γ0,v)∣g ∈ G} is a Gr¨obner basis, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' the same order, of the ideal generated by {g(a0,γ0,v)∣g ∈ G } in Ka0[v].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Now, we focus our attention on the standard decomposition of the singular locus of C(F h) described in Subsection A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' In the first step, if necessary, we apply a K linear change of coordinates to ensure that the curve is in regular position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Hence, this linear transformation it is not affected by the specializations of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Therefore, for our reasonings, we may assume w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' that F is already in regular position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Next, let G1 be a Gr¨obner basis of w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' the lexicographic order with x < y, and let G2 be a Gr¨obner basis of the same ideal w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' the lexicographic order with y < x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let {f(a,γ,x)} = G1 ∩ F[x], {g(a,γ,y)} = G2 ∩ F[y], ˜f = f/gcd(f, ∂f ∂x) and ˜g = g/gcd(g,gcd(g, ∂g ∂x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Finally, let G3 ∶= {A(a,x),y − B(a,x)}, with A square-free and deg(B) < deg(A), be the normed reduced Gr¨obner basis w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' the lexicographic order with x < y of .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Then, we introduce the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' With the notation introduced above, let (see also Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='3, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='6, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='12) Ω1 = ΩnonZ(U) ∩ Ωgcd(f, ∂f ∂x ) ∩ Ωsqfree( ˜f), Ω2 = ΩnonZ(V ) ∩ Ωgcd(g, ∂g ∂y ) ∩ Ωsqfree(˜g) where U and V are the leading coefficients of f and g w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' x and y, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We define the open subset Ωsinga(F) ∶= 3 ⋂ i=1 ΩspGB(Gi) ⋂ q∈G1∖{f} ΩnonZ(Wq,y) ⋂ q∈G2∖{g} ΩnonZ(Wq,x) 2 ⋂ i=1 Ωi ∩ Ωsqfree(A) where Wq,y denotes the leading coefficient of q w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' similarly with Wq,x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' In addition, let G(a,γ,y,z) ∶= F h(a,γ,1,y,z), let U(a,γ,t) = gcd(G(a,γ,t,0),Gy(a,γ,t,0),Gz(a,γ,t,0)) and ˜U(a,γ,t) ∶= U/gcd(U,U′), where U ′ is the derivative of U w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let Ω(0∶1∶0) ∶= S if (0 ∶ 1 ∶ 0) ∈ sing(C(F h)) and else Ω(0∶1∶0) ∶= ΩnonZ(J(a,γ,0,1,0)) where J is one the first derivatives of F h not vanishing at (0 ∶ 1 ∶ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We define the open subset (see Definitions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='9 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='12) Ωsing∞(F) ∶= Ωgcd(G(a,γ,t,0),Gy(a,γ,t,0),Gz(a,γ,t,0)) ∩ Ωgcd(U,U′) ∩ Ω(0∶1∶0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Then, we define (see Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='1) ΩgenusOrd(F h) ∶= ΩsingOrd(F h) ∩ Ωsinga(F) ∩ Ωsing∞(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' RATIONALITY AND PARAMETRIZATIONS OF ALGEBRAIC CURVES UNDER SPECIALIZATIONS 15 Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Note that G1 ∩ F[x] = {f}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' So all polynomials in G1 ∖ {f} do depend on y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' similarly for G2 ∖ {g}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' The idea of controlling the coefficients Wq,x and Wq,y in Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='5 is to ensure that the elimination ideal of the specialized Gr¨obner basis does not include additional generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' In the following lemma, we see that the cardinality of the singular locus, as a set, is preserved under specializations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let a0 ∈ Ωsinga(F) ∩ Ωsing∞(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Then #(sing(C(F h)) = #(sing(C(F h,a0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let F 0(x,y) ∶= F(a0,γ0,x,y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let G0 1 be a Gr¨obner basis of < F 0,F 0 x,F 0 y > w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' the lexicographic order x < y, and let G0 2 be a Gr¨obner basis of the same ideal w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' the lexicographic order y < x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Since a0 ∈ ΩspGB(G1) ∩ ⋂ q∈G1∖{f} ΩnonZ(Wq,y) ∩ Ω1, then {f(a0,γ0,x)} = G0 1 ∩ Ka0[x] and degx(f(a,γ,x)) = degx(f(a0,γ0,x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Similarly, {g(a0,γ0,x)} = G0 2 ∩Ka0[y] and degy(f(a,γ,y)) = degy(f(a0,γ0,y)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Since a0 ∈ Ω1, by Corol- lary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='8, gcd(f, ∂f ∂x)(a0,γ0,x) = gcd(f(a0,γ0,x), ∂f(a0,γ0,x) ∂x ) and degx(gcd(f, ∂f ∂x)(a0,γ0,x)) = degx(gcd(f, ∂f ∂x)(a,γ,x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Thus, ˜f(a0,γ0,x) = f(a0,γ0,x) gcd(f(a0,γ0,x), ∂f(a0,γ0,x) ∂x ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='13, it holds that degx( ˜f(a,γ,x)) = degx( ˜f(a0,γ0,x)) and ˜f(a0,γ0,x) is square- free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Similarly for g and ˜g since a0 ∈ Ω2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' In addition, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='15, F 0 x(x,y) = Fx(a0,γ0,x,y) and F 0 y (x,y) = Fy(a0,γ0,x,y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Thus, √ = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Since a0 ∈ ΩspGB(G3), {A(a0,γ0,x),y − B(a0,γ0,x)} is a Gr¨obner basis of √ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Since a0 ∈ Ωsqfree(A), by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='13, A(a0,γ0,x) is squarefree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Therefore, the number of affine singularities of C(F,a0) is degx(A(a0,γ0,x)) and degx(A(a,γ,x)) is the number of affine singularities of C(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='13, we get that degx(A(a0,γ0,x)) = degx(A(a,γ,x)) and, hence, C(F) and C(F,a0) have the same number of affine singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' It remains to prove that the number of singularities at infinity is also the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' First we observe that, if (0 ∶ 1 ∶ 0) ∈ sing(C(F h)), then (0 ∶ 1 ∶ 0) ∈ sing(C(F h,a0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Moreover, since a0 ∈ Ω(0∶1∶0), if (0 ∶ 1 ∶ 0) /∈ sing(C(F h)), then (0 ∶ 1 ∶ 0) /∈ sing(C(F h,a0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' For the remaining singularities at infinity, denote by Σ the set of the singularities of the form (1 ∶ µ ∶ 0) ∈ sing(C(F h));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' similarly let Σ0 be the set of singularities of this type in sing(C(F h,a0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let G0(y,z) ∶= G(a0,γ0,y,z) (see Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='5), U 0(t) ∶= gcd(G0(t,0),G0 y(t,0),G0 z(t,0)) and ˜U 0(t) ∶= U 0/gcd(U 0,(U 0)′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Then, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='3) #(Σ) = degt( ˜U) and #(Σ0) = degt( ˜U 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Since a0 ∈ Ωgcd(G(a,γ,t,0),Gy(a,γ,t,0),Gz(a,γ,t,0)), by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='11, it holds that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='4) U 0(t) = U(a0,γ0,t) and degt(U 0(t)) = degt(U(a,γ,t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' 16 RATIONALITY AND PARAMETRIZATIONS OF ALGEBRAIC CURVES UNDER SPECIALIZATIONS Let D(a,γ,t) = gcd(U,U′) and D0(t) = gcd(U 0,(U 0)′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Since a0 ∈ Ωgcd(U,U′), by Corol- lary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='8, one has that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='5) D0(t) = D(a0,γ0,t) and degt(D0(t)) = degt(D(a,γ,t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Now, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='3), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='4) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='5), we get that #(Σ) = #(Σ0) and this concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' □ Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let a0 ∈ ΩgenusOrd(F h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' If C(F,a0) is irreducible, then genus(C(F)) = genus(C(F,a0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Since a0 ∈ Ωsinga(F) ∩ Ωsing∞(F), by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='7, it holds that #(sing(C(F h)) = #(sing(C(F h,a0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' On the other hand, since a0 ∈ ΩSingOrd(F h), by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='29, we know that each r–fold in sing(C(F h)) generates an ordinary r-fold in sing(C(F h,a0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Therefore, applying (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='4), we conclude the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' General case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let F be as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' But now, differently to the case of Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='1, we do not introduce any assumption on the singular locus of the irreducible curve C(F h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' The key of our analysis is to reduce the general case to the case studied in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' For this purpose, we recall that any irreducible curve is birationally equivalent to a curve having only ordinary singularities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' [37, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='] or [30, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='] for a more computational description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' This transformation, say ϕ, can be seen as a finite sequence of blowups of the irreducible families of non-ordinary singularities and, hence, as a finite sequence of compositions of quadratic Cremona transformations and linear transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Now, our goal is to find an open subset Ωblowup of S such that, when a is specialized in Ωblowup, the birationality of ϕ is preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' For this purpose, let F0(x,y) = F(x,y) and let F(F h 0 ) be a F-conjugate irreducible family of non-ordinary singularities of C(F h 0 ) with defining polynomial m1(t1) ∈ F[t1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let Fm1 be the quotient field of F[t1]/, that is, Fm1 = F(t1) with m1(t1) as minimal polynomial of t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Then we apply a linear transformation L1, given by a matrix M1 ∈ M3×3(Fm1), and the Cremona transformation Q1 = (yz ∶ xz ∶ xy) as described in the blow up basic step in Subsection A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='1 of the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Denote by ∆1 the determinant det(M1) and let C(F h 1 ) be the curve over Fm1 obtained after the quadratic transformation Q1 ○ L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Note that F h 1 , the quadratic transformation of F h 0 , is the cofactor of F h 0 (Q1(L1)) not being divisible by neither x, y nor z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We repeat the above process for F h 1 (t1,x,y,z), F h 2 (t1,t2,x,y,z),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=',F h r (t1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=',tr,x,y,z) until all singularities of C(F h r ) are ordinary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Then ϕ(a,γ,t1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=',tr,x,y,z) = (Qr ○ Lr) ○ ⋯ ○ (Q1 ○ L1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Note that F h r is defined over F(t1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=',tr) and C(F h r ) over the algebraic closure of F(t1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=',tr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' In addition, F(t1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=',tr) = K(a,γ,t1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=',tr) = L(γ,t1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=',tr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' So, we consider a primitive element of the extension over L, say γ∗, and we work over L(γ∗) = L(γ,t1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=',tr);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' note that results in Section 3 apply to this new frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' In this situation, let us denote by ∆ = ∆1⋯∆r ∈ L(γ∗) the product of the determinants of the linear transformations L1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=',Lm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' In addition, let M ∶= {all entries of Mi ∈ M3×3(L(γ∗))}i∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=',r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' and let B ∶= {F h i (Qi+1(Li+1))(a,γ∗,0,y,z),F h i (Qi+1(Li+1))(a,γ∗,x,0,z), F h i (Qi+1(Li+1))(a,γ∗,x,y,0)}i∈{0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=',r−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' RATIONALITY AND PARAMETRIZATIONS OF ALGEBRAIC CURVES UNDER SPECIALIZATIONS 17 Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' With the notation introduced above, we define the set Ωblowup(F) ∶= ΩF ∩ ⋂ h∈M Ωdef(h) ∩ ΩnonZ(∆) ∩ ⋂ h∈B ΩnonZ(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' The previous observations lead to the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let a0 ∈ Ωblowup(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Then ϕ(a0,(γ∗)0,x,y,z) is birational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Since a0 ∈ Ωblowup(F) ⊂ ⋂h∈M Ωdef(h) ∩ ΩnonZ(∆), all Qi ○ Li are well–defined, and birational, when a is specialized as a0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' So ϕ(a0,(γ∗)0,x,y,z) is birational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let a0 ∈ Ωblowup(F) and let ϕ0 ∶= ϕ(a0,(γ∗)0,x,y,z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' If F0(a0,γ0,x,y,z) is irreducible, then Fr(a0,(γ∗)0,x,y,z) is the quadratic transformation of F0(a0,γ0,x,y,z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' First we observe that because of (the proof of) Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='10 each ϕi ∶= Qi ○ Li is well defined at a0 and it is birational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let us prove the result by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' By hy- pothesis F h 0 (a0,γ0,x,y,z) is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We have the equality F h 0 (ϕi) = xn1yn2zn3F h 1 for some ni ∈ N and that neither x, y nor z divides F h 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' So, F0(ϕi)(a0,(γ∗)0,x,y,z) = xn1yn2zn3F h 1 (a0,(γ∗)0,x,y,z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Moreover, since a0 ∈ ⋂h∈B ΩnonZ(h), we know that neither x, y nor z divides F1(a0,(γ∗)0,x,y,z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Furthermore, since ϕi is birational when specialized at a0 and F h 0 (a0,γ0,x,y,z) is irreducible, we have that F h 1 (a0,(γ∗)0,x,y,z) is also irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Thus, F h 1 (a0,(γ∗)0,x,y,z) is the quadratic transformation of F h 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Now, the i-induction step is reasoned analogously using that, by induction, F h i (a0,(γ∗)0,x,y,z) is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' □ With this, we can now give an open set where the genus is preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let F ∈ F[x,y] be as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='1), and let G ∈ F(γ∗)[x,y] be the polynomial obtained after the blowup process of F h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We define the set Ωgenus(F) ∶= Ωblowup(F) ∩ ΩgenusOrd(Gh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let F ∈ F[x,y] be as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='1), and let a0 ∈ Ωgenus(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' If C(F,a0) is irre- ducible, then genus(C(F)) = genus(C(F,a0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Since Gh is the quadratic transformation of F h, we have that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='6) genus(C(F h(a,x,y,z))) = genus(C(Gh(a,γ∗,x,y,z))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let ϕ0 denote the map ϕ(a0,(γ∗)0,x,y,z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Since a0 ∈ Ωblowup(F), by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='10, ϕ0 is birational and, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='11, Gh(a0,(γ∗)0,x,y,z) is the quadratic transformation of F h(a0,γ0,x,y,z) via ϕ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Therefore, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='7) genus(C(F h(a0,γ0,x,y,z))) = genus(C(Gh(a0,(γ∗)0,x,y,z))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Moreover, since a0 ∈ ΩgenusOrd(Gh), by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='8, it holds that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='8) genus(C(Gh(a,γ∗x,y,z))) = genus(C(Gh(a0,(γ∗)0,x,y,z))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Now, the proof follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='6), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='7) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' □ 18 RATIONALITY AND PARAMETRIZATIONS OF ALGEBRAIC CURVES UNDER SPECIALIZATIONS 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Birational Parametrization of Parametric Rational Curves In Section 4, and more precisely in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='8 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='13, we have described open subsets of S where the genus of the curve is preserved under specializations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' even in Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='3 the particular case of genus zero was treated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Nevertheless, in all these results the additional condition that the specialized polynomial is irreducible over K was required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Avoiding the irreducibility is in general a difficult problem related to the Hilbert irreducibility problem (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' [31]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' More precisely, there is no algorithm known that finds for given irreducible F ∈ F[x,y] the specializations a0 ∈ S such that F(a0,x,y) is reducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Nevertheless, in this section, we see how in the case of genus zero the problem can be solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Furthermore, we described an open subset where the specialized parametrization parametrizes the specialized curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' For this purpose, throughout this section, let assume that F ∈ F[x,y] is as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='1) and additionally assume that C(F) is rational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Moreover, let us assume that P is a proper (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' birational) parametrization of C(F) which can be computed, for instance, by the algorithm described in Subsection A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='2 in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Note that, in general, one may need to extend F with an algebraic element δ of degree two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' If #(a) = 1 and deg(γ) = 1, or degx,y(F) is odd, then no extension of F is required (see Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='5 and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' So, we may consider a primitive element of L(γ,δ), say γ∗, and express our parametrization in L(γ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Throughout this section, by abuse of notation, let F denote the field L(γ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let us write the proper parametrization P of C(F) as (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='1) P(a,γ,t) = (p1 q1 , p2 q2 ) ∈ F(t)2 ∖ F2 where we assume that P is in reduced form, that is gcd(p1,q1) = gcd(p2,q2) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let us start with the simple case of degree one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let F = A2(a,γ)x+A1(a,γ)y+A0(a,γ) ∈ F[x,y]∖F, and let P be expressed as P(a,γ,t) = (λ1t + λ0,µ1t + µ0) ∈ F(t)2 ∖ F2 be a proper polynomial parametrization of C(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We define the set (see Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='3 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='14) Ωproper(P) ∶= ΩF ∩ Ωdef(λ1t+λ0) ∩ Ωdef(µ1t+µ0) ∩ (ΩnonZ(λ1) ∪ ΩnonZ(µ1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let F and P be as in Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Then, for every a0 ∈ Ωproper(P), it holds that P(a0,t) is a proper polynomial parametrization of C(F,a0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Since a0 ∈ ΩF , by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='15, F(a0,γ0,x,y) is well defined and C(F,a0) is an affine line, obviously irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Since a0 ∈ Ωdef(λ1t+λ0) ∩ Ωdef(µ1t+µ0) then P(a0,γ0,t) is well–defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Moreover, since a0 ∈ (ΩnonZ(λ1) ∪ ΩnonZ(µ1)), P(a0,γ0,t) is a polynomial parametrization, clearly proper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Furthermore, F(a0,γ0,P(a0,γ0,t)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Thus, since F(a0,γ0,x,y) is irreducible, we conclude that P(a0,γ0,t) is a proper parametrization of C(F,a0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' □ Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let us use the notation in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' If a0 ∈ S ∖ Ωproper(P) then: (1) If a0 /∈ ΩF ∖ Ωdef(F), it holds that F(a0,γ0,x,y) = A0(a0,γ0) ∈ K, and hence C(F,a0) does not define an affine curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' (2) If a0 /∈ Ωdef(λ1t+λ0) but a0 ∈ ΩF (similarly if a0 /∈ Ωdef(µ1t+µ0)), the specialization P(a0,γ0,t) is not well–defined even though C(F,a0) is a line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Clearly, in this case, one has that C(F,a0) is rational and a proper parametrization of the specialized line can be provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' RATIONALITY AND PARAMETRIZATIONS OF ALGEBRAIC CURVES UNDER SPECIALIZATIONS 19 (3) If a0 /∈ (ΩnonZ(λ1) ∪ ΩnonZ(µ1)) but a0 ∈ ΩF ∩ Ωdef(λ1t+λ0) ∩ Ωdef(µ1t+µ0), then P(a0,γ0,t) ∈ K 2 and hence it is not a parametrization although C(F,a0) is a line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Again, as in the previous case, one can easily provide a polynomial parametrization of the specialized line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' In the sequel, we assume that C(F) is not a line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Then, we generalize the open subset in Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='1 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' (1) Let Ω1 ∶= ΩF (see Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' (2) Ω2 ∶= Ωdef(p1) ∩ Ωdef(p2) ∩ ΩnonZ(q1) ∩ ΩnonZ(q2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' (3) We consider the polynomials Gi = pi(h)qi(t) − pi(t)qi(h) ∈ F[h][t] ∖ {0} for i ∈ {1,2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let Ω3 ∶= Ωgcd(G1,G2) (see Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' (4) Let Ω4 ∶= Ωgcd(p1,q1) ∩Ωgcd(p2,q2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' note that pi,qj ∈ F[t] ⊂ F[h,t] and, since C(F) is not a line, the pi and gi are are non–zero (see Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We define Ωproper(P) as Ωproper(P) = 4 ⋂ i=1 Ωi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' The following theorem generalizes Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let a0 ∈ Ωproper(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Then C(F,a0) is a rational affine curve in K 2 properly parametrized by P(a0,δ0,t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' If deg(F) = 1, the result follows from Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let deg(F) > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Since a0 ∈ Ω1, then F(a0,γ0,x,y) is well–defined and deg(C(F)) = deg(C(F,a0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' In particular C(F,a0) is an affine curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' On the other hand, since a0 ∈ Ω2, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='5 (1), we have that P(a0,γ0,t) is well–defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' In addition, since a0 ∈ Ω4, the leading coefficients of p1,p2,q1,q2 do not vanish at a0 (see Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Consequently the degree of all numerators and denominators of P after specialization are preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Furthermore, by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='8, and using that pi/qi are in reduced form, we get that pi(a0,γ0,t)/qi(a0,γ0,t) are also in reduced form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Therefore, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='2) degt (pi(a0,γ0,t) qi(a0,γ0,t)) = degt (pi(a,γ,t) qi(a,γ,t)) for i ∈ {1,2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' In particular, P(a0,γ0,t) /∈ K 2, and hence P(a0,γ0,t) is a parametrization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Moreover, F(a0,γ0,P(a0,δ0,t)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Thus, P(a0,γ0,t) parametrizes the curve defined by one of the factors, say H(x,y), of F(a0,γ0,x,y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let us see that indeed C(F,a0) = C(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let Gi(a,γ,h,t) as in Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='4 (3), and let G ∶= gcd(G1(a,γ,h,t),G2(a,γ,h,t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let ˜Gi(h,t) be the corresponding polynomials associated, as in Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='4 (3), to P(a0,γ0,t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let ˜G ∶= gcd( ˜G1, ˜G2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Since pi(a0,γ0,t)/qi(a0,γ0,t) are in reduced form, no simplification of the rational functions have been required, and therefore ˜Gi(h,t) = Gi(a0,γ0,h,t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Moreover, since a0 ∈ Ω3, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='7, it holds that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='3) degt(G(a0,γ0,h,t)) = degt( ˜G(h,t)), and degt(G(a0,γ0,h,t)) = degt(G(a,γ,h,t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' By [29, Theorem 3], since P(a,γ,t) is proper, we have that degt(G(a,γ,h,t)) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Therefore, it holds that degt( ˜G(h,t)) = degt(G(a0,γ0,h,t)) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Again by [29, Theorem 3], P(a0,γ0,t) is proper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' On the other 20 RATIONALITY AND PARAMETRIZATIONS OF ALGEBRAIC CURVES UNDER SPECIALIZATIONS hand, by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='2), degt(P(a,γ,t)) = degt(P(a0,γ0,t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Therefore, by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='21 in [30], we have that max{degx(F(a,γ,x,y)),degy(F(a,γ,x,y)} = degt(P(a,γ,t)) = degt(P(a0,γ0,t)) = max{degx(H),degy(H)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Moreover, since F is not linear, by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='2), no component of P(a0,γ0,t) is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Applying again [30, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' ], we have that degx(H) = degt (p2(a0,γ0,t) q2(a0,γ0,t)) = degt (p2(a,γ,t) q2(a,γ,t)) = degx(F) degy(H) = degt (p1(a0,γ0,t) q1(a0,γ0,t)) = degt (p1(a,γ,t) q1(a,γ,t)) = degy(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Finally, since H(x,y) divides F(a0,γ0,x,y), one has that C(F,a0) = C(H), which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' □ Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let us analyze the behavior of F and/or P when specializing in S ∖ Ωproper(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' (1) If a0 ∈ S ∖ Ω1, since F ∈ K[a][x,y] (see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='1)), then F(a0,x,y) is always well–defined, and hence, deg{x,y}(F(a0,x,y)) < deg{x,y}(F(a,x,y)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' So, it can happen that either 0 < deg{x,y}(F(a0,x,y)) < deg{x,y}(F(a,x,y)), in which case C(F,a0) is an affine curve;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' or 0 = deg{x,y}(F(a0,x,y)), which implies that C(F,a0) is the empty set or K 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' (2) If a0 ∈ S ∖ Ω2, then P(a0,δ0,t) is not defined, and hence the specialization fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' (3) If a0 ∈ (S ∖ (Ω3 ∩ Ω4)) ∩ Ω1 ∩ Ω2, at least one of the following assertions hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' (a) P(a0,t) is not proper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' (b) P(a0,t) ∈ K 2 and hence, P(a0,t) is not a parametrization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' (c) P(a0,t) parametrizes a proper factor of F(a0,x,y), that is, C(F,a0) decomposes and one of its components is rational and parametrized by P(a0,t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' The next result follows from Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='5 and emphasizes the polynomiality of the parametrizaion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' If P is proper and polynomial and a0 ∈ Ωproper(P), then P(a0,γ0,t) parametrizes properly and polynomially C(F,a0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We now analyze the normality (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' the surjectivity, see [30]) of the parametrization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We recall that any parametrization can be reparametrized surjectively (see [30, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='26]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' This reparametrization requires, in our case, a new algebraic extension of F via a new algebraic element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Alternatively, one may reparametrize normally the specialized parametrizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' In the following we deal with the case where P is already normal and we want to preserve this property through the specializations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' For this purpose, we first introduce a new definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let P be as in 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' If P is normal, we define the set Ωnormal(P) ∶= { S if degt(p1) > degt(q1) or degt(p2) > degt(q2), Ωgcd(N1,N2) if degt(p1) ≤ degt(q1) and degt(p2) ≤ degt(q2) where (α1/β1,α2/β2) ∈ F2 is the critical point of P (see [30, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='24]) and, for i ∈ {1,2}, Ni = αiqi − βipi, RATIONALITY AND PARAMETRIZATIONS OF ALGEBRAIC CURVES UNDER SPECIALIZATIONS 21 Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let P be proper and normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' For a0 ∈ Ωproper(P) ∩ Ωnormal(P), P(a0,γ0,t) parametrizes properly and normally C(F,a0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' In the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='5 we have seen that, for a0 ∈ Ωproper(P), degt(pi(a,γ,t)) = degt(pi(a0,γ0,t)), similarly for qi, and that the rational functions in P(a0,γ0,t) are in re- duced form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Now, if degt(p1) > degt(q1) or degt(p2) > degt(q2), the result follows from Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='5 and [30, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' If degt(p1) ≤ degt(q1) and degt(p2) ≤ degt(q2), since a0 ∈ Ω2 in Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='4, we get that C ∶= (α1(a0,γ0) β1(a0,γ0), α2(a0,γ0) β2(a0,γ0)) is well defined and, by the above remark on the degrees, C is the critical point of P(a0,γ0,t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let ˜Ni ∶= αi(a0,γ0)qi(a0,γ0,t) − βi(a0,γ0)pi(a0,γ0) = Ni(a0,γ0,t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Now, since a0 ∈ Ωgcd(N1,N2), by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='8, one has that degt(gcd( ˜N1, ˜N2)) = degt(gcd(N1,N2)) > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' recall that P is normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Now, the result follows from Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='5 and [30, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' □ Let us illustrate these ideas in an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let us consider K = Q and F = L ∶= Q(a1,a2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let F(a, x, y) =((a5 1 + a5 2 + 3a2 1a2 2 − a1a2)y2 + (2a3 1a2 + (−9a2 − 1)a2 1 + 3a1 − 6a4 2 + a3 2)y + a2 2(a1 + 9a2 − 3))x3 + ((−3a3 1a2 2 − 6a4 1 + 3a2 1a2 − 6a1a2 2)y2 + 9((a2 + 2/9)a2 1 + (−(8a2)/9 − 1)a1 − a3 2/9 + 2a2 + 2/9)a1y + 3(a1 − 2/3)a2 2)x2 − 3(((a2 − 4)a1 − 2a2 2)a1y + a3 1/3 − 3a2 1 + (6a2 + 4/3)a1 − (8a2)/3)a1xy + ((a2 1a2 − 8)y − 3a2 1 + 2a1)a2 1y C(F) is a rational quintic that can be properly parametrized as (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='4) P(a,t) = ( ta1 + 2 t2a2 + t + a1 , t + 3 t3a1 + a2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' The field of parametrization is L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We determine the open subset Ωproper(P) (see Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='4)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let us deal with Ω1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Clearly Ωdef(F) = C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' The homogeneous component of F of maximum degree is (a5 1 + a5 2 + 3a2 1a2 2 − a1a2)x3y2 So, Ω1 ∶= C2 ∖ V(a5 1 + a5 2 + 3a2 1a2 2 − a1a2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' One has that Ω2 ∶= C2 ∖ {(0,0)} 22 RATIONALITY AND PARAMETRIZATIONS OF ALGEBRAIC CURVES UNDER SPECIALIZATIONS Note that, if a0 /∈ Ω2, the second component of P is not well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let us deal with Ω3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' The polynomials Gi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' G∗ i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' R are G1 = h2ta1a2 − ht2a1a2 + 2h2a2 − ha2 1 − 2t2a2 + ta2 1 + 2h − 2t G2 = h3ta1 − ht3a1 + 3h3a1 − 3t3a1 − ha2 + ta2 G = h − t G∗ 1 = (hta1 + 2h + 2t)a2 − a2 1 + 2 G∗ 2 = ((t + 3)h2 + (t2 + 3t)h + 3t2)a1 − a2 R = 3h4a3 1a2 2 − 2h4a2 1a2 2 + h3a4 1a2 + 6h3a2 1a2 2 + 3h2a4 1a2 − h2a2 1a3 2 − 2h3a2 1a2 − 2h2a3 1a2 +ha5 1 − 6h2a2 1a2 + 12h2a1a2 2 − 6ha3 1a2 − 4ha1a3 2 + 3a5 1 + 4h2a1a2 − 4ha3 1 + 12ha1a2 −12a3 1 − 4a3 2 + 4ha1 + 12a1 Moreover,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' A1 = −ha1a2 − 2a2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='A2 = −ha1 − 3a1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='B = −1 (see Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='6)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Hence, Ω1 = C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Moreover, ΩnonZ(A1) = C2 ∖ V(a2), ΩnonZ(A2) = C2 ∖ V(a1) and ΩnonZ(B) = C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' On the other hand, ΩnonZ(R) can be expressed as C2 ∖ {(0,0),(± √ 2,0)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Therefore, Ω3 = Ωgcd(G1,G2) = C2∩(C2 ∖ V(a1))∩(C2 ∖ V(a2))∩(C2 ∖ {(0,0),(± √ 2,0)}) = C2∖V(a1a2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Finally, we deal with Ω4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We have p1 = ta1 + 2 p2 = t + 3 q1 = t2a2 + t + a1 q2 = t3a1 + a2 gcd(p1,q1) = 1 gcd(p2,q2) = 1 rest(p1,q1) = a3 1 − 2a1 + 4a2 rest(p2,q2) = a2 − 27a1 Therefore, Ω4 = C2 ∖ V(a1a2(a3 1 − 2a1 + 4a2)(a2 − 27a1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Summarizing (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' 1, left) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='5) Ωproper(P) = C2 ∖ V(a1a2(a5 1 + a5 2 + 3a2 1a2 2 − a1a2)(a3 1 − 2a1 + 4a2)(a2 − 27a1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Decomposition of S The goal in this section is to provide an algorithm decomposing the space S so that in each subset of the decomposition we can give information on the genus of the corresponding specialized curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let F be as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='1), irreducible over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We first compute the genus of C(F h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let g ∶= genus(C(F h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Furthermore, if g = 0, let P(a,γ,t) be, as in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='1), a proper parametrization of C(F h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We consider the open subset (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='1) Σ ∶= { Ωgenus(F) if g > 0 (see Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='12)) Ωproper(P) if g = 0 (see Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='4)) At this level of the process we know that (see Theorems (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='13) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='5)) (1) If g > 0, then for a0 ∈ Σ it holds that C(F,a0) is either reducible or its genus is g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' (2) If g = 0, then for a0 ∈ Σ it holds that C(F,a0) is rational and P(a0,γ0,t) parametrizes properly C(F,a0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' RATIONALITY AND PARAMETRIZATIONS OF ALGEBRAIC CURVES UNDER SPECIALIZATIONS 23 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Left: Plot of the real part of the closed set defining Ωproper(P) in Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Right: Plot of the real part of the closed set defining Ωgenus(F) in Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='3 In the following, we analyze the specializations when working in the closed set (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='2) Z ∶= S ∖ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' First, let us discuss the computational issues that may appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let A ⊂ K[a] be a set of generators of Z, and let I be the ideal generated by A in K[a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We consider the prime decomposition of I I = ℓ ⋃ j=1 Ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Now, for each prime ideal J ∈ {I1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=',Iℓ} we consider the quotient field of K[a]/J;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' we denote it by LJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Elements in LJ are quotients of equivalence classes of K[a]/J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We will assume that elements in K[a]/J are always expressed by means of a canonical representative of the class in the following sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We fix a Gr¨obner basis G of J w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' some fixed order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Then, the elements in K[a]/J are uniquely represented by their normal form w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' G (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' 1 and Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' 13, Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' 2, Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' in [6]) and, hence, elements in LJ are represented as the quotient of the canonical representatives of their numerators and denominators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' So, by abuse of notation, we will identify, via the canonical representation, the elements in LJ with elements in K(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' In addition, we consider an algebraic element γJ over LJ and we denote by FJ the field FJ ∶= LJ(γJ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We observe that FJ is a computable field with a polynomial factorization algorithm avail- able;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' zero test and basic arithmetic (addition, multiplication and inverse computation) can be carried out e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' by taking the normal forms w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' a Gr¨obner basis of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' For the polyno- mial factorization we refer to (see Section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='2 and Appendix B in [36], see also [35]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' As a particular case, as in Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='10 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='2, if V(J) is a rational variety, one may work over K(Q(λ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=',λm)) instead of FJ, where Q(λ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=',λm) is a parametrization of V(J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Concerning specializations, instead of working in S (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='1)), we take the parameter values in the irreducible variety V(J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Then, for a0 ∈ V(J) ⊂ S, and f ∈ K[a]/J, we denote by f(a0) the specialization at a0 of the equivalence class of f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' note that since a0 ∈ V(J) the 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' 2 3 224 RATIONALITY AND PARAMETRIZATIONS OF ALGEBRAIC CURVES UNDER SPECIALIZATIONS specialization does not depend on the representative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Similarly, if f ∶= p/q ∈ FJ and q(a0) ≠ 0, then f(a0) ∶= p(a0)/q(a0) ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' In this situation, for each prime ideal J ∈ {I1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=',Iℓ} we consider the polynomial F in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='1) as a polynomial in FJ[x,y].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' To emphasize this fact, we write FJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' First we check the irreducibility of FJ over the algebraic closure of FJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' If FJ is reducible we can either stop the decomposition over this closed subset, and claim that the specialization over V(J) is reducible, or continue the process with each irreducible factor of FJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' For irreducible FJ, the process continues, as in the initial step, by computing the genus of C(FJ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Since in each iteration of the process the dimension of the variety V(J) decreases, we, at the end, reach the zero–dimensional case, and the decomposition ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let us say that a specialization degenerates if either F(a0,γ0,x,y) is not well–defined or F(a0,γ0,x,y) ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' As a result of the process described above, we find a disjoint decomposition (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='3) S = ˙⋃i∈ISi such that, for every specialization a0 ∈ Si, one of the following holds (1) the specialization degenerates;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' (2) the genus is positive and preserved, or the specialized curve is reducible;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' (3) the genus is zero and a proper parametrization of C(F,a0) is provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let us remark that in the decomposition (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='3) we can take the union of those Si corresponding to each of the three items above;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' say S1,S2,S3 representing the corresponding item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' In this way, we can achieve a unique decomposition of the parameter space S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' The Si obtained in this way are constructible sets of S, and S2,S3 are a finite union of subsets Σ as in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='1) and S1 is a closed subset directly defined from the implicit equation F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Moreover, S3 can further be decomposed into a finite union of subsets S3,j such that for every j, there is a proper parametrization Pj which is well–defined for every a0 ∈ S3,j and specialized properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Finally, since Σ of F as in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='1) is open on non-empty, depending on the genus of F, either S2 or S3 is a dense subset of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We illustrate the previous ideas by continuing the analysis of Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' (Continuation of Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='10) Taking into account (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='5), the closed set Z (see (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='2)) decomposes as Z = V(a1) ∪ V(a2) ∪ V(a2 − 27a1) ∪ V(a3 1 − 2a1 + 4a2) ∪ V(a5 1 + a5 2 + 3a2 1a2 2 − a1a2) ⊂ Q 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We start with J1 ∶= and V1 ∶= V(J1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Since V1 is rational, surjectively parametrized by Q1 ∶= (0,λ), we work over the field Q(λ)[x,y].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We have that FJ1 ∶= λ2x2 (λ3xy2 − 6λ2xy + λxy + 9λx − 3x − 2) and therefore all specializations in V(a1) lead to a reducible curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Additionally, one may distinguish the cases λ = 0, that corresponds to the point (0,0), where the specialization degenerates, and λ ≠ 0 where C(F,a0) decomposes to the union of a double line and a rational cubic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' The analysis for J2 ∶=, and V2 ∶= V(J2) looks similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Since V2 is rational, parametrized by Q2 ∶= (λ,0), we work over the field Q(λ)[x,y].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We have that FJ2 ∶=λy(λ4x3y − 6λ3x2y − λ3x + 2λ2x2 + 12λ2xy − λx3 − 3λ3 + 9λ2x − 9λx2 + 3x3 + 2λ2 − 4λx − 8λy + 2x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' RATIONALITY AND PARAMETRIZATIONS OF ALGEBRAIC CURVES UNDER SPECIALIZATIONS 25 Thus, all specializations in V2 lead to a reducible curve;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' note that Q2 is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' The case λ = 0 is covered above, and for λ ≠ 0, the specialization C(F,a0) decomposes to the union of a line and a rational quartic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let us study J3 ∶= and V3 ∶= V(J3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Again, V3 is rational, parametrized by Q3 ∶= (λ,27λ), and we work over the field Q(λ)[x,y].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We have that FJ3 ∶=λ(244λx + 3λ − 3x − 2)(58807λ3x2y2 − 732λ3xy2 + 732λ2x2y2 + 9λ3y2 − 13068λ2x2y + 464λ2xy2 + 9λx2y2 + 81λ2xy + 6λ2y2 − 81λx2y − 6λxy2 − λ2y + 729λx2 − 106λxy + 4λy2 − x2y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' The analysis of V3 is identical to V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let us study J4 ∶= and V4 ∶= V(J4) that is again rational and it is properly and surjectively parametrized by Q4 ∶= (λ,−1 4λ3 + 1 2λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We work over the field Q(λ)[x,y].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We have that FJ4 ∶=λ(λ5y − 2λ3y + 12λ2 − 8λ + 32y)(λ9x3y − 8λ7x3y + 12λ6x3 + 24λ5x3y + 8λ5x3 − 32λ4x3y − 72λ4x3 − 32λ3x3y − 16λ4x2 − 32λ3x3 + 192λ3x2y + 144λ2x3 + 16λx3y + 64λ2x2 − 384λ2xy + 32λx3 − 96x3 + 256λy − 64x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Again V4 behaves as V2 and V3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Finally, let us analyze J5 ∶= and V5 ∶= V(J5) which is a rational quintic and properly and surjectively parametrized as Q5(λ) ∶= ( (5λ − 1)(5λ − 2)4 3125λ4 − 3750λ3 + 1750λ2 − 375λ + 31,− (5λ − 2)(5λ − 1)4 3125λ4 − 3750λ3 + 1750λ2 − 375λ + 31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Those values for which the parametrization is not defined, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' the poles, play no role in this analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' The polynomial FJ5 is FJ5(λ,x,y) = F(Q5(λ),x,y) ∈ Q(λ)[x,y].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' It holds that deg(C(FJ5)) = 4, genus(C(FJ5)) = 0 and a proper surjective parametrization is PJ5(λ,t) ∶= P(Q(λ),t) (see (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='4)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' So, we get (see Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='4) Ωproper(PJ5) ∶= V5 ∖ {PJ5(λ0)∣f(λ0) = 0} where f ∶= p1 p2 p3 p4 and p1 ∶= 5λ − 1, p2 ∶= 5λ − 2, p3 ∶= 25λ2 − 15λ + 3, p4 ∶= 390625λ8 − 937500λ7 + 1343750λ6 − 1237500λ5 + 711875λ4 − 253500λ3 +54475λ2 − 6495λ + 331.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' By Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='5, for every a0 ∈ Ωproper(PJ5) it holds that C(F,a0) is rationally parametrized by P(a0,t) (see (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='4)) or, equivalently, by PJ5(Q−1(a0),t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Now, let us analyze the curve in Z5 ∶= V5 ∖ Ωproper(PJ5) = {PJ5(λ0)∣f(λ0) = 0} (see (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We define a0 1 ∶= (0,0) = PJ5(λ0), where λ0 is a root of p1 p2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' a0 2 ∶= (−1,−1) = PJ5(λ0), where λ0 is a root of p3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' and observe that p4 generates 8 points on the curve that we denote by a0 i ,i ∈ {3,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=',10} and which correspond to PJ5(λ0) where λ0 is one of the roots of p4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Thus, Z = {a0 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=',a0 10}, C(F,a0 1) = C2, and, for i ∈ {2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=',10}, C(F,a0 i ) are rational cubics parametrized by PJ5(a0 i ,t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Summarizing, S decomposes as (see (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='3) and Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='1) S =(S1 ∶= {(0,0)}) ∪ (S2 ∶= ∪4 i=1Vi ∖ {(0,0)}) 26 RATIONALITY AND PARAMETRIZATIONS OF ALGEBRAIC CURVES UNDER SPECIALIZATIONS ∪ (S3,1 ∶= Ωproper(P)) ∪ (S3,2 ∶= Ωproper(PJ5)) ∪ (S3,3 ∶= {a0 2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=',a0 10}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' For a0 ∈ S1, C(F,a0) degenerates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' For a0 ∈ S2, C(F,a0) is reducible (note that a0 1 ∈ S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' For a0 ∈ S3,1, the specialized curve C(F,a0) is a quintic parametrized by P(a0,t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' for a0 ∈ S3,2, C(F,a0) is a quartic parametrized by PJ5(a0,t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' and for a0 ∈ S3,3, C(F,a0) is a cubic parametrized by PJ5(a0,t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let K = Q and F = L = Q(a1,a2,a3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We consider F = x3 + x2a1 + y3 + a2a3 ∈ F[x,y].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' One has that genus(C(F)) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Using the ideas in Section 4, we compute Ωgenus(F) (see Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We observe that since C(F) is an elliptic cubic, no blowup is required and, hence, Ωgenus(F) = ΩgenusOrd(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Indeed, one gets that (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' 1, right) Ωgenus(F) ∶= C3 ∖ V(−a2a3 (−4a13 − 27a2a3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Thus, by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='8, for every a0 ∈ Ωgenus(F), C(F,a0) is either reducible or it is a genus 1 cubic curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let us analyze the specializations in Z ∶= C3 ∖ Ωgenus(F) = V(−a2a3 (−4a13 − 27a2a3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let J1 ∶= and V1 ∶= V(J1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Then, FJ1 ∶= x3+a1x2+y3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We observe that genus(C(FJ1)) = 0 and PJ1 ∶= (− t3a1 t3 + 1,− t2a1 t3 + 1) is a proper parametrization of C(FJ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Applying Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='4 to FJ1 and PJ1 we get that Ωproper(PJ1) ∶= V1 ∖ V(a1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Therefore, by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='5, for all a0 ∈ Ωproper(PJ1) it holds that C(F,a0) is a rational curve parametrized by PJ1(a0,t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' However, for the remaining case, namely the points (0,0,µ) for µ ∈ C, C(F,(0,0,µ)) decomposes as the product of three lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let J2 ∶= and V2 ∶= V(J2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Now, the situation is identical to the previous case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let J3 ∶=<−4a13 − 27a2a3> and V3 ∶= V(J3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' The surface V3 can be properly parametrized as Q(λ1,λ2) = (λ1,λ2,−4λ13 27λ2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We observe that Q is not surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Indeed, Q(C2) = V3 ∖{(0,0,µ)∣µ ∈ C} (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Remark 3 in [26]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' However, the specializations in {(0,0,µ)∣µ ∈ C} have already been analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' So, we treat the case Q(C2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Then, FJ3 can be taken as FJ3 ∶= F(Q(λ1,λ2),t) = x3 + x2λ1 + y3 − 4λ13 27 , where (λ1,λ2) ∈ C2 ∖ {(0,0)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' It holds that genus(C(FJ3)) = 0 and PJ3 ∶= (t3λ1 − 2λ1 3t3 + 3 , t2λ1 t3 + 1) is a proper parametrization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Applying Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='4 to FJ3 and PJ3, we get that Ωproper(PJ3) = V3 ∖ {(0,0,µ)∣µ ∈ C}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Therefore, by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='5, for all a0 ∈ Ωproper(PJ3) it holds that C(F,a0) is a rational curve parametrized by PJ3(a0,t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Summarizing, S decomposes as (see (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='3)) S = (S2 ∶= Ωgenus(F) ∪ {(0,0,µ)∣µ ∈ C}) ∪ (S3 ∶= Ωproper(PJ1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' RATIONALITY AND PARAMETRIZATIONS OF ALGEBRAIC CURVES UNDER SPECIALIZATIONS 27 For a0 ∈ S2,C(F,a0) is either reducible or an elliptic curve;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' and for a0 ∈ S3,C(F,a0) is a rational cubic parametrized by PJ3(a0,t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Some Illustrating Applications In this section, we illustrate by examples some possible applications of the theory developed in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' In the first example, given a surface, we consider the problem of determining its rational level curves, if any.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Level curves of a surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let S be the surface defined over C by the polynomial F = x6 − 5x4y + 3x4z − y5 + 2y4z − y3z2 − x3z + 5x2y2 − 7x2yz + 3x2z2 + y2z − 2yz2 + z3 − x2 where F ∈ Q(z)[x,y].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' So, with the terminology of the paper, a = z, K = Q and L = F = Q(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' With this interpretation, the idea is to analyze the genus of C(F) ⊂ Q(z) 2 under specializations in S ∶= C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' For this purpose, we first compute a standard decomposition of the singular locus as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Following the steps described in Subsection A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='1, we get F(F h) = {(0 ∶ z ∶ 1)}m1=t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Moreover, one can easily check that the family consists in one ordinary double point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' There- fore, one gets that genus(C(F)) = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Since the singularities are all ordinary, we get that (see Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='12) Ωgenus(F) = ΩgenusOrd(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Moreover (see Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='5), ΩgenusOrd(F) = C ∖ {−1,0,1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Using Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='13, for z0 ∈ ΩgenusOrd(F), C(F,(x,y,z0)) is either reducible or its genus is 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' In any case, no rational level curve appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' For the elements in Z ∶= C ∖ ΩgenusOrd(F), we get that C(F,(x,y,±1)) are irreducible of genus 7 and C(F,(x,y,0)) is irreducible of genus 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Indeed, C(F,(x,y,0)) can be parametrized by (t5,t6 − t2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' In the second example, we consider the linear homotopy deformation of two curves and we analyze the genus of each instance curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Linear homotopy deformation of curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let us consider the linear homotopy between the Fermat cubic curve and the unit circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' That is we consider the polynomial F = (1 − λ)(x3 + y3 − 1) + λ(x2 + y2 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We consider F as a polynomial in Q(λ)[x,y] and we analyze the genus behavior through the deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' So, with the terminology of the paper, a = λ, K = Q and L = F = Q(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Now, the idea is to study the genus of C(F) under specializations in S ∶= C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' or, in particular, in the real interval [0,1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' For this purpose, we first observe that genus(C(F)) = 1 and hence (see Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='12), Ωgenus(F) = ΩgenusOrd(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We get that ΩgenusOrd(F) = C ∖ V(g) where g(λ) = − (2λ3 − 5λ2 + 7λ − 3)(λ6 − 4λ5 + 15λ4 − 29λ3 + 43λ2 − 33λ + 9)(4λ − 3)(−1 + λ)(λ − 3) (2λ9 + 27λ8 + 5049λ7 − 40068λ6 + 148716λ5 − 315657λ4 + 398763λ3 − 295245λ2 + 118098λ − 19683) (8λ3 − 27λ2 + 54λ − 27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Using Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='13, for λ0 ∈ ΩgenusOrd(F), C(F,(x,y,λ0)) is either reducible or its genus is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' In any case, no rational deformation instance appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' For the elements in Z = C ∖ 28 RATIONALITY AND PARAMETRIZATIONS OF ALGEBRAIC CURVES UNDER SPECIALIZATIONS ΩgenusOrd(F), we get that C(F,(x,y,1)) is rational, the specialized cubics C(F,(x,y,3/4)) and C(F,(x,y,3)) factor as a union of a line and a conic, and for all the other cases the genus remains one (see Fig 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Left: Plot of the real part of different instances of the deforma- tion in Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Right: Plot of the real part of C(F,(x,y,3/4)) and C(F,(x,y,3)) in Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let O be a connected open subset of C and let Mer(O) be the field of meromorphic functions in O (see [14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We consider a polynomial equation of the form (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='1) ∑ i,j∈I fi,j(t)xiyj = 0 where I is a finite subset of N2, and where fi,j ∈ Mer(O).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let f be the tuple with all the functions fi,j appearing in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' The question now is to decide, and indeed compute, whether there exists rational solutions of the equation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' that is p,q ∈ C(f) such that ∑i,j∈I fi,j(t)piqj = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We may proceed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We consider the polynomial F(a,x,y) resulting from the formal replacement in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='1) of each function fi,j by a parameter ak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Now, with the terminology of the paper, we take K = C(f) and L = F = K(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Then, we decompose S (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='1)) as described in Section 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' note that the computations can be carried out over C(a) instead of over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Then, if any subset in the decomposition has genus zero, and the functions f belong to it, we obtain a (family of) rational solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let us see a particular example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Rational solution of functional algebraic equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We consider the functional algebraic equation (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='2) x2y3f4 1 − x2y2f4 2 f3 − 2xy3f2 1 f2 + 2xyf1f2 2 f2 3 + y3f2 2 − f2 1 f3 3 = 0, where f = (f1,f2,f3) ∶= (sin(t),cos(t),et).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We associate to (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='2) the curve C(F) where F = x2y3a4 1 − x2y2a4 2a3 − 2xy3a2 1a2 + 2xya1a2 2a2 3 + y3a2 2 − a2 1a3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' It holds that genus(C(F)) = 0 and that a proper parametrization is (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='3) P(a,W) = (W 3a1 + a2 Wa2 2 + a2 1 , a3 W 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' 3 N 3 2 3 33 2 3 0 2 1 3RATIONALITY AND PARAMETRIZATIONS OF ALGEBRAIC CURVES UNDER SPECIALIZATIONS 29 The open subset in Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='4 turns to be Ωproper(P) = C(f) 3 ∖ VC(f) (a1a2 (a1 − a2)(a6 1 + a5 1a2 + a4 1a2 2 + a3 1a3 2 + a2 1a4 2 + a1a5 2 + a6 2)a3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Since f ∈ Ωproper(P), by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='5, we have that (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='4) {x = W 3 sin(t) + cos(t) W (cos2 (t)) + sin2 (t),y = et W 2 }, for every W ∈ C(f) such that (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='4) is well–defined, is a rational solution of (7.' 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polynomials and applications to the Hough transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Journal of Algebra, 486:328–359, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' [35] Wang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Elimination Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Texts and Monographs in Symbolic Computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Springer-Verlag, Vi- enna, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' [36] Wang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Elimination Practice: Software Tools and Applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Imperial College Press, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' [37] Walker, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Algebraic Curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Princeton Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Press (1950).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' [38] Weispfenning, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Comprehensive Gr¨obner Bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Symbolic Computation, 14:1–29, 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' [39] Winkler, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Polynomial Algorithms in Computer Algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Springer–Verlag, Wien New York, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Rational Curves In this appendix we recall the main steps to compute the genus of an irreducible plane curve and, in the affirmative case, how to parametrize a proper rational curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' There exist different methods to that goal: the adjoint curve based method (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' [37] and [30]), the method based on the anticanonical divisor (see [13]) or the method based on Puiseux expansions (see [20]), among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' In this paper we will follow the adjoint curve based method (see [37]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' more precisely, we will follow the symbolic approaches in Chapters 3,4,5 in [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' A similar treatment of the problem can be performed using the other algorithmic approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Throughout this appendix, let G ∈ K[x,y] ∖ K be irreducible over K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Genus computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We denote the genus as either genus(C(G)) or genus(C(Gh)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' The key tool for our symbolic computation of the genus is the notion of conjugate family of points (see Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='15 in [30]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We adapt the definition to our purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' The set F = {(f1(α) ∶ f2(α) ∶ f3(α))∣m(α) = 0} ⊂ P2(K) is called a K–conjugate family of points if (1) at least one of the polynomials fi is not zero, (2) f1,f2,f3,m ∈ K[t] and gcd(f1,f2,f3,m) = 1, RATIONALITY AND PARAMETRIZATIONS OF ALGEBRAIC CURVES UNDER SPECIALIZATIONS 31 (3) degt(m) > 0 and m is squarefree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' The family is represented as F ∶= {(f1(t) ∶ f2(t) ∶ f3(t))}m(t) and m(t) is called the defining polynomial of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We say that F is a family of points of C(Gh) if Gh(f1,f2,f3) = 0 modulo m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' If m(t) is irreducible over K, we say that F is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' (1) We will assume that families are always represented in canonical form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' That is, if F ∶= {(f1(t) ∶ f2(t) ∶ f3(t))}m(t) then f1,f2,f3 are reduced modulo m(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' (2) Note that a single point (a ∶ b ∶ c) ∈ P2(K) can be seen as a K-conjugate family, for instance as {(a ∶ b ∶ c)}t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' (3) Note that, factoring the defining polynomial m(t) over K, every family decomposes as a finite union of irreducible families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' So, in general, we may assume that families are irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Observe that, for H ∈ K[x,y,z] and F ∶= {(f1(t) ∶ f2(t) ∶ f3(t))}m(t) irreducible, H(f1,f2,f3) ≠ 0 mod m(t) iff H does not vanish at any point in the family F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Moreover, if F is irreducible, then Km ∶= K[t]/ is an integral domain, and we can see (f1(t) ∶ f2(t) ∶ f3(t)) as a single point in the curve defined by Gh over Km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let us denote this curve by C(Gh F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' In that situation, by abuse of notation, we will say that F is a point of C(Gh F) meaning that (f1(t) ∶ f2(t) ∶ f3(t)) ∈ C(Gh F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' In [30] it is proved that the singular locus of C(Gh) can be decomposed as a finite union of irreducible families of K-conjugate points such that in each family the multiplicity and the singularity character (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' whether the singularity is ordinary or non–ordinary) is preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' The set of all conjugate families appearing in the union above is called a K–standard decom- position of the singular locus of C(Gh), or of C(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' In the following, we describe a process to determine a K–standard decomposition of the singular locus;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' see also [30] page 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' First, we introduce the notion of regular position: Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We say that the affine plane curve C(G) is in regular position w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' x if (1) the coefficient of ydeg(G) in G is a non-zero constant;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' and (2) G(a,bi) = Gx(a,bi) = 0 with i ∈ {1,2} implies that b1 = b2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' If C(G) is not in regular position, we may apply a linear change of coordinates over K such that G is transformed into regular position (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Lemma 2 in [10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' More precisely, for this purpose, one may perform a change of coordinates of the form {x = ¯x + q¯y,y = ¯y} where q is taken in an open subset defined by means of subresultants and by the homogeneous component of maximum degree in G (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Remark 3 in [10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Condition (1) in Definition (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='3) is easy to check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let us now deal with the second condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let us consider the ideal J generated by {G,Gx} in K[x,y].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Taking into account that G is irreducible over K, one has that J is zero–dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Therefore, using the Shape lemma (see [39] page 194), the normed reduced Gr¨obner basis of √ J, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' the lexicographic order x < y, is of the form {u(x),y − v(x)}, with u square-free and deg(v) < deg(u), if and only if √ J, or equivalently J, satisfies condition (2) in Definition (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' It remains to have a computational approach to determine √ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' This 32 RATIONALITY AND PARAMETRIZATIONS OF ALGEBRAIC CURVES UNDER SPECIALIZATIONS can be achieved, for instance, using Seidenberg lemma (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' [22] or [15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' More precisely, if J ∩ K[x] = and J ∩ K[y] =, then (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='1) √ J =, where ˜f = f/gcd(f,f′) and ˜g = g/gcd(g,g′) and where f′ and g′ denotes the derivatives of f and g w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' x and y, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Then, the process for computing a standard decomposition of the singular locus of C(Gh) is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Standard decomposition of the singular locus Step 1: Regular position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' If C(G) is not in regular position, apply an affine linear change of coordinates over K, say L, such that C(G) is transformed into regular position (see comments above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Step 2: Families of singularities at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Factor Gh(x,y,0) over K, Gh(x,y,0) = ∏ i∈I gi(x,y)ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Note that none of the polynomials gi is x because C(G) is in regular position w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Then, the set of points at infinity decomposes in irreducible K–conjugate families as ⋃ i∈I {(1 ∶ t ∶ 0)}gi(1,t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Now, a family {(1 ∶ t ∶ 0)}gi(1,t) is singular if and only if Gh x(1,t,0) = Gh y(1,t,0) = Gh z(1,t,0) = 0 modulo gi(1,t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let I be the set containing all gi(1,t) defining singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Then, the irreducible families of K–conjugate singularities at infinity are (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='2) ⋃ m∈I {(1 ∶ t ∶ 0)}m(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Step 3: Families of affine singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let I be the ideal generated by {G,Gx,Gy}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Since √ I is regular w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' x because of condition (2) in Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='3, and zero–dimensional, by the discussion above on the Shape lemma, the normed reduced Gr¨obner basis w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' the lexicographic order with x < y of √ I is of the form {A(x),y − B(x)} with A square-free and deg(B) < deg(A);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' recall that if I ∩ K[x] = and I ∩ K[y] =, then √ I =, where ˜f = f/gcd(f,f′), ˜g = g/gcd(g,g′), and f′ and g′ denote the derivatives of f and g w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' x and y, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' So, each irreducible factor m(x) of A(x) over K generates the irreducible family {(t ∶ B(t) ∶ 1)}m(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Therefore, if A denotes the set of all irreducible factors of A, the affine singularities decompose in irreducible K–families as (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='3) ⋃ a∈A {(t ∶ B(t) ∶ 1)}m(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Step 4: Standard decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Applying L−1 to (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='2) and to (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='3) we get an K–standard decomposition of the singular locus of C(Gh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We observe that the irreducible families of affine singularities will be of the form {(f1(t) ∶ f2(t) ∶ 1)}m(t) and the singularities at infinity of the form {(L1(t) ∶ L2(t) ∶ 0)}m(t) with degt(Li) ≤ 1 and gcd(L1,L2) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let F be a K–standard decomposition of the singular locus of C(Gh) and let F ∈ F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' note that by construction, F is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Then, we denote by mult(F) the multiplicity of the RATIONALITY AND PARAMETRIZATIONS OF ALGEBRAIC CURVES UNDER SPECIALIZATIONS 33 points in F, as points of C(Gh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' In addition, since the character of the singularity is invariant within the family we will speak about ordinary and non-ordinary families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' The multiplicity of an irreducible family F = {(f1 ∶ f2 ∶ f3)}m(t) can be computed by determining the greatest non-negative integer r such that all partial derivatives of Gh of order less than r vanish at (f1 ∶ f2 ∶ f3) modulo m(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' The character of F can be decided by analyzing the squarefreeness, modulo m(t), of the polynomial defining the tangents to C(Gh) at (f1 ∶ f2 ∶ f3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Alternatively, one can work with the curve C(Gh F) (see Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='2 (3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Now, the family F in C(Gh) turns to be one point in C(Gh F), namely (f1 ∶ f2 ∶ f3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Then, the character and multiplicity of F is the character and multiplicity of (f1 ∶ f2 ∶ f3) as a point in C(Gh F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Once the singularities have been detected, one proceeds to recursively, and separately, blow up each non–ordinary singularity via the neighboring singularities (see [37] or [30] for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let us briefly recall how the first iteration step works, first for a single point, and afterwards for an irreducible family of conjugate points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' In the following, let p ∈ C(Gh) be a non–ordinary singular point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Blow up basic step Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Apply a linear change of coordinates L such that: p = L(0 ∶ 0 ∶ 1), none of the tangent to C(Gh(L)) at (0 ∶ 0 ∶ 1) is one of the lines x = 0, and y = 0, and for v ∈ {x,y,z} no point in (C(Gh(L)) ∖ C(v)) ∖ {(0 ∶ 0 ∶ 1)} is a singularity of C(Gh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Apply the Cremona transformation Q ∶= (yz ∶ xz ∶ xy) to C(Gh(L)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Then, Gh(L(Q)) factors as Gh(Q(L)) = xn1yn2zn3G∗ for some natural numbers n1,n2,n3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We call G∗ the quadratic transform of Gh w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' The first neighboring of p is defined as (C(G∗) ∩ C(z)) ∖ {(1 ∶ 0 ∶ 0),(0 ∶ 1 ∶ 0)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' The points, and their multiplicities, in the neighborhood, are in one to one correspondence with the tangents, and their multiplicities, to C(Gh) at p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' The (first) neighboring singularities of p are the neighboring points being singularities of C(G∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' The process continues till no non-ordinary neighboring point appears in the neighborhoods and until all non-ordinary singularities of C(Gh) have been blowed up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We will refer to the set of all singularities and neighboring singularities as the neighboring graph of C(Gh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' In this situation, the genus can be computed as (recall that d = deg(C(G))) (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='4) genus(C(G)) = (d − 1)(d − 2) 2 − 1 2 ∑mult(p)(mult(p) − 1) where the sum is taking over all the points in the neighboring graph of C(Gh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' If we work with families, say F = {(f1 ∶ f2 ∶ f3)}m0(t0) is a non-ordinary irreducible family of singularities, we may introduce the curve C(Gh F) (see above) and repeat the process with the single point pF ∶= (f1 ∶ f2 ∶ f3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' In this way the first neighboring of pF is decomposed into irreducible Km0–conjugate families;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' recall that Km0 is the quotient field of K[t0]/).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' These irreducible Km0–conjugate families are of the form (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='5) {(t1 ∶ 1 ∶ 0)}m1(t0,t1) where t1 is a new variable and m1(t0,t1) ∈ Km0[t1] ∖ {t1} is irreducible over Km0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' In this situation, the method continues by applying the same process to the irreducible families of non-ordinary singularities in the first neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' The irreducible families in the second 34 RATIONALITY AND PARAMETRIZATIONS OF ALGEBRAIC CURVES UNDER SPECIALIZATIONS neighborhood will be of the form (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='6) {(t2 ∶ 1 ∶ 0)}m2(t0,t1,t2) where t2 is a new variable and m2 ∈ Km0,m1[t2]∖{t2} is irreducible over Km0,m1, where Km0,m1 denotes the quotient field of Km0[t1]/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' In general, the irreducible families in the i-th neighborhood will be of the form (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='7) {(ti ∶ 1 ∶ 0)}mi(t0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=',ti).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' By abuse of notation, we will say that the families of the form in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='5), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='6), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='7) are K– conjugate families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Applying this process till no non-ordinary neighboring conjugate family appears in the neighborhoods and until all non-ordinary families of C(Gh) have been blown up, we get a decomposition that we call a K–standard decomposition of the neighboring graph of singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' In this situation, the genus can be computed as (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='8) genus(C(G)) = (d − 1)(d − 2) 2 − 1 2 ∑ F∈N #(F)mult(F)(mult(F) − 1) where N is a standard decomposition of the neighboring graph of C(Gh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Parametrization algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Once the genus of C(Gh) has been computed, if it is zero, we may derive a rational parametrization of the curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' There are different approaches to achieve a rational parametrization, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' [13], [27], [28], [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Here we will use a simplified version of the algebraically optimal algorithm in [28] where Hilbert–Hurwitz theorem (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' in [30]) is applied directly and recursively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let N be a K–standard decomposition of the neighboring graph of singularities of C(Gh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We consider the linear system Ad−2(C(Gh)) of adjoint curves to C(Gh) of degree d −2 (recall that d = deg(C(G)));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' that is, the linear system of all (d −2)–degree curves having each r-fold point of C(Gh), including the neighboring ones, as a point of multiplicity at least r − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' In other words, Ad−2(C(Gh)) is the linear system of curves of degree d−2 defined by the effective divisor ∑ F∈N ∑ p∈F (mult(p) − 1)p, where the multiplicity is w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' the curve where F belongs to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' In practice, the linear con- ditions to construct the linear system can be derived by working modulo the corresponding defining polynomials of the families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' In the following, we outline the process for computing Ad−2(C(Gh)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Adjoints computation Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We identify the set of all projective curves, including multiple component curves, of fixed degree d − 2, with the projective space Vd−2 ∶= P (d−2)(d+1) 2 (K) via their coefficients, after fixing an order of the monomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' By abuse of notation, we will refer to the elements in Vd−2 by either their tuple of coefficients, or by the associated {x,y,z}–form, or by the corresponding curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let H(Λ,x,y,z) denote the {x,y,z}–homogeneous polyno- mial of degree d − 2 defining a generic element in Vd−2, where Λ is a tuple of undetermined coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' For each irreducible K–family F ∶= {(f1 ∶ f2 ∶ f3)}m0(t0), with r ∶= mult(F), in the standard decomposition of the singular locus of C(Gh), and for each partial derivative M RATIONALITY AND PARAMETRIZATIONS OF ALGEBRAIC CURVES UNDER SPECIALIZATIONS 35 of H of order r − 2 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' the variables {x,y,z}, compute M(Λ,f1,f2,f3) mod m0(t0) and collect in a set S of its non–zero coefficients w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' For each neighborhood and for each irreducible K–family of neighboring singularities F ∶= {(ti ∶ 1 ∶ 0)}mi(t0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=',ti), with r ∶= mult(F), compute (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='9) (⋯(N(Λ,t1,1,0) mod mi)⋯) mod m1 where N is every partial derivative of order r−2 of N∗, being N∗ the form obtained applying to H the same blow up process (linear changes of coordinates, Cremona transformation and quadratic transformation, see Subsection A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='1) as the one to reach the curve where the neigh- boring family F belongs to;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' in [27, Theorem 6] appears an alternative method to derive (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Include in S all non–zero coefficients w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' t of the polynomials in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Solve the homogeneous linear system of equations {L(Λ) = 0}L∈S and substitute the result in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' The form resulting from this substitution defines Ad−2(C(Gh)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' We observe that the defining polynomial of Ad−2(C(Gh)) has coefficients in K(Λ) and hence the ground field is not extended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Hilbert-Hurwitz theorem (see [9] or [30, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='8]) ensures that for almost all (φ1,φ2,φ3) ∈ Ad−2(C(Gh))3 the mapping (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='10) H ≡ (y1 ∶ y2 ∶ y3) = (φ1(x,y,z),φ2(x,y,z),φ3(x,y,z)) transforms birationally C(Gh) to an irreducible curve of degree d − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Note that, because of Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='4, the map in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='10) is defined over K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Furthermore, since the map is birational, the genus is preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Thus, applying successively Hilbert–Hurwitz theorem one gets either a conic (if d is even) or a line (if d is odd) K–birationally equivalent to C(Gh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' After these considerations, we can outline the parametrization algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Parametrization computation Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let G be as above so that the genus of C(Gh) is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Set M ∶= G, ρ ∶= deg(M), and T as the identity map in P2(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' While ρ > 2 do (1) Compute Aρ−2(C(M)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' (2) Take (φ1,φ2,φ3) ∈ Ad−2(C(Gh))3, so that the mapping H in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='10) is birational over C(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' (3) Replace M by M(H −1), ρ by deg(M), and T by H ○ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Parametrize birationally C(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Let Q(t) be the output parametrization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Step 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Return T −1(Q(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' (1) If d is odd one may stop the loop in Step 3 when ρ = 3 since cubics can be easily parametrize over the ground field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' For parametrizing a conic or a cubic see [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' (2) The direct classical parametrization algorithm, see [37], [30], for C(Gh), in addition to the computation of the adjoints, needs d−2 simple points of C(Gh) (indeed, a more sophisticated method in [27] shows that only one simple point is enough).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' Therefore, and alternative to Step 5 is: compute d − 2 simple points either on the conic, or on the line, for instance, using Q(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' apply T to these simple points;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} +page_content=' and now use the direct parametrization algorithm for C(Gh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE4T4oBgHgl3EQfKwzR/content/2301.04933v1.pdf'} diff --git a/SNE0T4oBgHgl3EQfUgB0/vector_store/index.pkl b/SNE0T4oBgHgl3EQfUgB0/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..2ef305d67aa3af0474f1e2a6a4afaf09f6121151 --- /dev/null +++ b/SNE0T4oBgHgl3EQfUgB0/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:ce7beedceaaeafbc554080a84608975a3ad4b3055888e523e8f79c368562ff34 +size 103677 diff --git a/UdAzT4oBgHgl3EQfX_ww/content/tmp_files/2301.01326v1.pdf.txt b/UdAzT4oBgHgl3EQfX_ww/content/tmp_files/2301.01326v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a2ac93bfb9a15bc70840ae647b71da7c9dad2f66 --- /dev/null +++ b/UdAzT4oBgHgl3EQfX_ww/content/tmp_files/2301.01326v1.pdf.txt @@ -0,0 +1,1581 @@ +Astronomy & Astrophysics manuscript no. art72_final +©ESO 2023 +January 5, 2023 +The height of convective plumes in the red supergiant µ Cep⋆ +A. López Ariste1, M. Wavasseur1, Ph. Mathias1, A. Lèbre2, B. Tessore3, S. Georgiev2, 4 +1 IRAP, Université de Toulouse, CNRS, CNES, UPS. 14, Av. E. Belin. 31400 Toulouse, France +2 LUPM, Université de Montpellier, CNRS, Place Eugène Bataillon, 34095 Montpellier, France +3 Université Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France +4 Institute of Astronomy and NAO, Bulgarian Academy of Science, 1784 Sofia, Bulgaria +Received ...; accepted ... +ABSTRACT +Aims. We seek to understand convection in red supergiants and the mechanisms that trigger the mass loss from cool evolved stars. +Methods. Linear spectropolarimetry of the atomic lines of the spectrum of µ Cep reveals information well outside the wavelength +range expected from previous models. This is interpreted as structures in expansion that are visible in the front hemisphere and +sometimes also in the back hemisphere. We model the plasma distribution together with its associated velocities through an inversion +algorithm to fit the observed linear polarization. +Results. We find that supposing the existence of plasma beyond the limb rising high enough to be visible above it can explain the +observed linear polarization signatures as well as their evolution in time. From this we are able to infer the geometric heights of the +convective plumes and establish that this hot plasma rises to at least 1.1 R∗. +Conclusions. µ Cep appears to be in an active phase in which plasma rises often above 1.1 R∗ . We generalize this result to all red +supergiants in a similarly evolved stage, which at certain epochs may easily send plasma to greater heights, as µ Cep appears to be +doing at present. Plasma rising to such heights can easily escape the stellar gravity. +1. Introduction +Despite their large impact on stellar and galactic evolution, the +properties of outflows from red supergiants (RSGs) are not well +characterized. In particular, the role of convection is still poorly +understood, partly because their structures are difficult to ob- +serve directly through interferometry. In this work, we propose a +view of the convection structure of the RSG µ Cep through im- +ages reconstructed from spectropolarimetric data. Convection is +also of interest to those studying mass loss from these evolved +stars because these convective processes are thought to con- +tribute. +These have been the motivations for several campaigns of +observation of the red supergiant µ Cep at the Telescope Bernard +Lyot (TBL) with both spectropolarimeters Narval and Neo- +Narval. These observations were made in parallel to those of +Betelgeuse presented by Aurière et al. (2016), Mathias et al. +(2018), and López Ariste et al. (2018), though less frequently. +Nevertheless, as for Betelgeuse, the observations of µ Cep were +predominantly aimed at measuring the linear polarization in the +atomic lines of its spectrum. +An observed net linear polarization was interpreted as depo- +larization —during line formation— of the spectrum continuum, +which is itself polarized by Rayleigh scattering (Aurière et al. +2016). Using this hypothesis of the physical origin of the ob- +served linear polarization, two-dimensional images of the pho- +tosphere of Betelgeuse were inferred, images that could be fa- +vorably compared to co-temporal images of this star made us- +ing interferometric techniques (López Ariste et al. 2018). These +successful comparisons suggest that the many approximations +involved in this new imaging technique, which uses spectro- +polarimetry, are appropriate and have spurred a further step in +⋆ Based on observations obtained at the Télescope Bernard Lyot +(TBL) at Observatoire du Pic du Midi, CNRS/INSU and Université de +Toulouse, France. +the technique: recently López Ariste et al. (2022) published the +first three-dimensional images of Betelgeuse. In addition to the +satisfying images, this technique provided some interesting data: +the spatial and temporal scales of the convective patterns were +measured, and the characteristic velocities of the raising plasma +were determined. These velocities are much higher than the adia- +batic estimates, easily reaching values of 40km s−1(López Ariste +et al. 2018; Stothers 2010). More interestingly, López Ariste +et al. (2022) observed that in several observed cases of rising +hot plasma, the velocity was constant with height, suggesting +the presence of a force counter-acting gravity in the photospheric +layers. These large velocities, sometimes reaching 60km s−1, are +comparable to the escape velocity at tantalizingly low heights +(1.5R∗). If at any time the hot plasma reaches the escape veloc- +ity, it will escape the gravity of the star and, cooling down, may +be the origin of the clumpy dust clouds seen around Betelgeuse +(Montargès et al. 2021). This fast, rising plasma may also be the +source of mass loss in these stars. +However, in order to reach this interesting result, a critical +piece of information is missing. The technique presented and +used by López Ariste et al. (2022) to build three-dimensional +images of the photosphere of Betelgeuse is unable to determine +the geometric height of the successive layers imaged. The tech- +nique only provides the ordering of the layers, from the deep +atmosphere up to higher layers, but not the geometric distance +between them. Here, we present a means to measure or at least +estimate this geometric height. +In Section 2 we present the set of observations of µ Cep +collected with Narval and Neo-Narval at the TBL from 2015 +through 2022. In Section 3, we describe the spectral features in +the linear polarization of µ Cep that cannot be explained with +the model used to image Betelgeuse, and propose a modifica- +tion of the model. We propose that, from time to time, convec- +tive plumes are powerful enough to rise to sufficient heights that +they can be seen beyond the geometric horizon of the star. This +Article number, page 1 of 11 +arXiv:2301.01326v1 [astro-ph.SR] 3 Jan 2023 + +A&A proofs: manuscript no. art72_final +cannot be a permanent feature, but it may happen from time to +time, an aspect that is critical to the plausibility of this propo- +sition. We also discuss how this modification affects previous +results for Betelgeuse, if we assume that a unique model serves +all RSGs. In Section 4 we build an inversion code based on this +modified model, where the usual description of the brightness +variation across the disk is supplemented with the presence of up +to five clouds of plasma visible beyond the horizon of the star. +The measured linear polarization degrees and angles allow us to +determine how far beyond the horizon this plasma is and there- +fore how high it must be to be visible from Earth. It is in this way +that we can determine a minimum height for these structures. In +Section 5, we build a time evolution of one of those plumes that +we were lucky to follow in 2021 from rise to fall. In Section 6, +we put these measurements in context, in particular with respect +to the measured plasma velocities. We confirm that it is highly +possible that the most powerful of these convective plumes are +high enough to escape the gravity of the star at the observed ve- +locities. +2. Spectropolarimetric data from Narval and +Neo-Narval +µ Cep is an M2-type RSG with stellar parameters (Tef f = 3750K +and log g = −0.36 ) very similar to those of Betelgeuse, while +its mass (25M⊙) and radius (1420 R⊙) on the other hand may +be larger (Levesque et al. 2005). Tessore et al. (2017) first de- +tected strong linear polarization features (both in Stokes Q and +U) associated to atomic lines. +We began observing µ Cep in linear polarimetry in July 2017 +with the Telescope Bernard Lyot at Pic du Midi (France,TBL). +Until August 2019, the Narval spectropolarimeter was used. +After an upgrade, starting in September 2019, Neo-Narval re- +observed µ Cep in May 2020 and regular observations have been +conducted ever since. This long series allowed us to follow the +entire lifetime of one of these convective plumes (see Sect. 5). +Narval and Neo-Narval have been described before in the +literature. These descriptions are extensive for the case of Narval +(Donati et al. 2006), with a succinct description of the changes of +Neo-Narval provided by López Ariste et al. (2022). As stressed +in this latter publication, we note the continuity of data quality +from the instrument through its upgrade, and handle Narval and +Neo-Narval data as a unique dataset with no further reference to +the instrument used. +We performed relatively short exposures (about 3 minutes +per polarimetric sequence) in order to ensure a peak signal-to- +noise ratio (S/N) of about 2000 in Stokes I per velocity bin. A +list of the observations of µ Cep is presented in Table A, corre- +sponding to all those studied in this work. A least-squares de- +convolution (LSD) procedure (Donati et al. 2006) is applied to +the reduced spectra. Atomic lines from an appropriate list (Au- +rière et al. 2016) are summed after rescaling of the wavelength +binning. The result is a single spectral profile for both Stokes I +and the observed Stokes parameter. The whole set of Stokes Q, +U, and I profiles thus obtained is shown in Fig.1 in the form of +an image with time on the vertical axis. +3. Redshifted linear polarization features. +The model used to interpret the linear polarization observed in +the atomic lines of Betelgeuse and µ Cep and, in general, of +all red supergiants assumes a nonrotating convective star. This +model and the implicit approximations involved were described +in detail by López Ariste et al. (2018) and López Ariste et al. +(2022). From the point of view of the physical origin of the lin- +ear polarization, our model assumes that what we observe is the +depolarization of the continuum by atomic lines due to Rayleigh +scattering. A key diagnostic of the trustworthiness of this inter- +pretation of the polarization is that all lines must show similar +polarization, independent of their quantum structure. In partic- +ular, the Na D1 and D2 lines must show similar signals to one +another. This was seen to be the case for Betelgeuse (Aurière +et al. 2016), CE Tau, and now for µ Cep, the target of the present +study. Once the physical origin of this polarization is confirmed, +the model focuses on the distinct spatial origin of the spectral +features seen in the linear polarization profiles. +The observed linear polarization profiles characteristically +show several distinctive lobes inside every atomic line. In the ab- +sence of rotation, the wavelength position of each of those lobes +is assumed to be due to convection. The brightest plasma is as- +sumed to be rising, and the cooler, darker plasma sinks. At first +approximation, most light comes from the brighter regions and is +therefore Doppler shifted by the projection onto the line of sight +of the convective velocity at which the plasma rises. Thus, bright +hot plasma at disk center will emit light in the blue wing, while +bright hot plasma at the limb will emit light in the red wing, at +a wavelength which will coincide with the velocity of the center +of mass of the star with respect to the Sun. Dark, sinking plasma +would be redshifted with respect to this red wing, but its low in- +tensity translates into a tiny signal to be added in the further red +wing of the observed profile. In this model, the spectral profile +of an atomic line is framed by two velocities. One of these ve- +locities is the heliocentric velocity V∗, which limits the red wing +of the polarization profile. Plasma at the stellar limb will emit +light at or near to this red wavelength. The second of these ve- +locities is the maximum velocity of the plasma in the convective +flow, Vp, which limits the blue wing of the profile. Plasma ris- +ing at this maximum velocity at disk center will emit light at the +bluest wavelengths. In the absence of rotation, all other velocity +fields, such as micro- and macroturbulent velocities or thermal +broadening, are assumed to be isotropic and would just broaden +the signals. Such broadening, added to instrumental effects, is +seen as a minimum width for all observed polarization signals, a +width that is much smaller than the span of velocities attributed +to convection. +The two velocities, V∗ and Vp, that limit the observed pro- +files are parameters of the model and should be determined a +priori. As discussed by López Ariste et al. (2022), this a priori +determination is done by inspection of the whole set of avail- +able observations. In Fig. 1, the two velocities are represented as +vertical lines on top of the pile up of the observed profiles. Our +choice for these two velocities, namely V∗ = +35 km s−1for the +velocity of the center of mass of the star and Vp = −70 km s−1for +the maximum velocity of the convecting plasma, can be judged +with respect to the wavelength span of the polarization signals. +We note that while V∗ is measured in the heliocentric reference +system, we are giving the value of Vp in the star’s own reference +system. In the heliocentric reference system used in Fig. 1, we +find Vp at +35 − 70 = −35km s−1. As Vp has a meaning in terms +of the physics of convection of the star, it is useful to keep its +value in the reference system of the star, even at the risk of some +confusion when looking at Fig.1. +The choice of these values is not free from criticism. Judging +from Fig.1 alone, it appears as if the red limit V∗ has been placed +in the middle of the polarization signal rather than at its red edge. +As these two velocity limits cannot be directly measured, we +can only advance the arguments that justify our choice for these +Article number, page 2 of 11 + +A. López Ariste et al.: Raising plumes in µ Cep +two parameters. These arguments are qualitatively similar to the +ones used by López Ariste et al. (2018) and that López Ariste +et al. (2022) justified to be acceptable within 10 km s−1. Part of +this justification lies in the fact that accepting the model and the +values of these velocities results in images of Betelgeuse or CE +Tau, another observed RSG, that are comparable with contempo- +raneous images inferred by interferometers (López Ariste et al. +2018). However, in the present case, we lack any such interfero- +metric images for µ Cep, and contrary to the previously studied +Betelgeuse and CE Tau, there is a considerable amount of sig- +nal redward from V∗. It is worth examining the arguments that +justify this choice. +It is obvious that Vp, the maximum velocity of the plasma +represented by the blue line in Fig.1 (which, we reiterate, is +in the heliocentric reference system, while the value of Vp is +given in the star’s reference system), must encompass the most +blueshifted signals observed over the years. Accordingly, added +to V∗, this velocity must be somewhere beyond -20km s−1in +Fig.1. We have chosen -35km s−1to include the extended wings +of the signals observed. In our model, we have no explanation +whatsoever for any signal blueward from Vp. We therefore have +to make sure that there is no signal beyond this limit, and this +fixes minimum values of Vp. +In our model, Vp is interpreted as the velocity of the rising +plasma during convection in the reference frame of the star. This +interpretation sets further constraints on its maximum value. Our +choice has been, for µ Cep, to set this maximum velocity at Vp = +−70 km s−1. This is already a large velocity for rising plasma and +is roughly seven times the speed of sound in the atmosphere of µ +Cep. Numerical simulations (Freytag et al. 2002; Chiavassa et al. +2011) produce supersonic flows for convective patterns, which +were confirmed by López Ariste et al. (2018). But a value of +Vp = −70 km s−1is 50% to 100% larger than any published figure +so far, based on either observations or numerical simulations. +However, as Vp is fixed on its blue side by the extent of the +polarization signal, accepting these large plasma velocities is the +only way to place V∗ as far to the red as possible. As V∗+Vp is set +at -35 km s−1, the velocity of the center of mass must therefore +be V∗ = +35km s−1. In spite of the large value of V∗, this choice +still leaves lots of redshifted signal beyond the limit of the model. +Once again, including those signals in our convection model by +shifting V∗ to higher values would imply accepting convection +velocities larger than 70km s−1and this seems unphysical. Also, +once more, diminishing the maximum velocity Vp also seems +unphysical, as blueshifted signatures would be left unexplained +beyond the maximum velocity of our model. This sequence of +arguments justifies our choices up to 10km s−1, and leaves large +amounts of signal beyond the red limit of V∗. +Another argument justifying the choice of V∗ = 35km s−1is +apparent from the Stokes I variation. From the LSD profile, one +can compute a mean heliocentric radial velocity from the profile +Gaussian fit. This mean velocity is found to be < v >= 22km s−1 +in the heliocentric reference frame, and from profile to profile +varies with an amplitude of about 4 km s−1. For Betelgeuse, the +corresponding quantities are respectively < v >= 21km s−1and +4km s−1. The radial velocity of Betelgeuse was estimated to be +about V∗ = 40km s−1. Supposing the same parameters for both +these RSGs, that is, a number of granules number of similar +order and the same temperature contrast, then the V∗− < v > +values should also be similar, meaning a value of the order +of 40km s−1, or, within the 10km s−1uncertainty, the adopted +V∗ = 35km s−1value. To conclude this discussion of the choice +of the value of these two velocities, we must add that several +Fig. 1. Pile-up of the Stokes Q (left), U (center) and I (right) profiles +over the whole time series. For illustrative purposes, every observation +has been made to span 15 days on the vertical direction. The blue and +red vertical lines mark the maximum plasma velocity Vp and the radial +velocity of the center of mass of the star V∗ respectively (see main text +for definitions). Velocities are measured in the heliocentric reference +system. +velocities have been tested inside the range allowed by those 10 +km s−1, without significant differences in the results. +A strict interpretation of these velocity limits implies that no +polarization signal can be seen in our modeled profiles at wave- +lengths redder than V∗, the limit given by the velocity of the +center-of-mass of the star for each atomic spectral line. Polariza- +tion signals beyond this limit would come, in this model, from +plasma moving away from the observer and towards the center +of the star. Plasma sinking towards the core of the star is as- +sumed to be cool, dark plasma. This rigorous interpretation must +be softened somehow, as cold, dark plasma still emits some light +and plasma may start sinking while still being bright enough to +contribute to the net spectral line profile; but these are always +small contributions, and we are not expecting any large signals +on the red side of the velocity limit. After inspection of the pro- +files of Betelgeuse published by Aurière et al. (2016), Mathias +et al. (2018), López Ariste et al. (2018) and López Ariste et al. +(2022) we can confirm that this is the case, and that the hypoth- +esized model can confidently describe all the available observa- +tions. However, this is not the case for µ Cep. +In Fig. 2, we plot the observed spectra of µ Cep collected +since September 6, 2015. The observations are plotted as dotted +lines. We return to the continuous lines in different colors further +below. At this point, we focus our attention on the strong Q sig- +nal peaking at about +50 km s−1. This is actually the strongest +polarization of the observed profiles on that date and it is found +to the red of the limiting velocity V∗ = +35 km s−1, and is indi- +cated by the vertical dashed line. +Such strong signal in the red wing of the profiles of atomic +lines is unlike anything observed to this day in Betelgeuse. The +many more observations of linear polarization in the spectra of +Betelgeuse are better illustrated by the profile of Stokes U shown +in the right plot of Fig.2. A strong peak is seen in the blue wing +and is attributed to bright plasma near the center of the stellar +disk; another (negative) peak is seen near V∗, and attributed to +bright rising plasma coming from regions near the stellar limb; +and a small (positive) signal is seen beyond V∗ corresponding, +Article number, page 3 of 11 + +Stokes Q +Stokes U +Stokes I +01/2022 +09/2020 +06/2019 +02/2018 +10/2016 +07/2015 +-50 +50 +100 +50 +50 +100 +-50 +50 +100 +Velocity (km/s) +Velocity (km/s) +Velocity (km/s)A&A proofs: manuscript no. art72_final +Fig. 2. Observed linear polarization of µ Cep on September 6, 2015. The observed Stokes Q is plotted on the left, and Stokes U on the right, as +dots in both cases. Continuous lines represent the best fit from the assumed model (green line) with separated contributions from the front disk +brightness distribution (red) and the two plumes beyond the limb (black), visible at those wavelengths when they do not constitute the whole +contribution to the final fit in green. The upper (orange) profile shows the normalized intensity profile. The vertical dashed lines give the two +limiting velocities, Vp and V∗. +hypothetically, to sinking dark plasma. The observed Stokes U +profile is, in this manner, qualitatively explained and, after in- +version, the inferred image confirms this basic description of the +visible structures. Such a model would also explain the small +lobe seen in the blue wing of the Stokes Q profile and the larger +negative peak near the red boundary (red line of Fig. 2). The +respective amplitudes and signs of these peaks in Q and U will +constrain the position and brightness of the different bright struc- +tures over the disk. However, this model has no explanation +whatsoever for the strong signal on the red side of the red bound- +ary of the Q profile. Such strong signal cannot be attributed to +dark sinking plasma, for there would be no explanation for its +large amplitude. The amplitude could either be due to the amount +of photons or the polarization degree of those photons. To inter- +pret such a large amplitude would require either a brighter re- +gion or a more polarized region. It appears contradictory to say +that the sinking plasma is brighter than the rising plasma, and so +we are only left with the possibility that this is sinking plasma +with an enormous polarization degree. Implicit in our model is +that polarization degree is directly related to the height of the +plasma. One possible explanation that our model would have +for this strong polarization peak would therefore be that this is +a huge cloud of cold plasma sinking from large heights that is +much larger than any other structure in the atmosphere, because +height must compensate for the loss of signal due to the lower +emissivity of this cool dark plasma. +The presence of that unexpected strong peak is forcing the +model towards extreme scenarios. +Another possibility is that our determination of V∗, the veloc- +ity of the center of mass of the star, is wrong. It suffices to shift +this limit a further 35 km s−1to the red, up to V∗ = +70km s−1, +for the entire peak to fit inside the limits of the model. However, +this scenario entails unpleasant conclusions as well. Shifting this +limit to the red without touching the blue limit would mean that +the maximum velocity of the convective flows in µ Cep would +increase to a staggering 100 km s−1. This is an uncomfortably +large number for the convective flows, about ten times the speed +of sound. The problems with a modified red boundary do not end +here. We expect there to always be some signal coming from near +the limb, because statistically there is a large probability of find- +ing a bright structure somewhere along the long circumference. +This is the case with the actual red boundary limit plotted in Fig. +2: both Stokes Q and U profiles show signal near the limb. How- +ever, if we were to accept this boundary shift, Stokes Q would +still have the strong peak that would be attributed to a near-limb +structure, but no comparable signal is visible anywhere near that +limit for Stokes U. In order to produce such signal imbalance +between Q and U, one would need to imagine a stellar disk with +a continuous dark band along and inside the limb except at one +position where a bright structure would give the observed Stokes +Q signal. While not impossible, this appears as a strange disposi- +tion of structures on µ Cep, something never seen on Betelgeuse. +In addition, this 70 km s−1value is clearly outside the I profile, +meaning that this latter would have no link with the heliocentric +star velocity, which also seems difficult to accept. +One year later, in October 2016, the observations shown in +Fig.3 have drastically changed. Between the velocity boundaries, +the polarization signals keep providing an image of changing +bright, convective structures, but there is always signal at the +qualitatively expected places, even if that signal has changed +in amplitude, ratio, and position. This is interpreted as bright +structures that have moved over the disk; some have appeared +anew, others disappeared, but there is always signal coming from +around disk center and visible around the blue wing, and signal +coming from the limbs and visible around the red wing, but on +the correct side of the boundary V∗. The big change is that at +this date, and contrary to the observations in 2015, there are no +conspicuous signals on the wrong, red side of V∗. There are al- +ways small amplitude signals, both in Q and U. Because of their +small amplitude, they can be comfortably assigned to dark sink- +ing plasma, or perhaps a small error in the determination of V∗, +an error of at most 10 km s−1, which is consistent with the rough +arguments used in its determination. However, there is no large +peak visible, which casts doubt on the validity of the model. +These two observations of µ Cep show an expected signal +between the velocity boundaries that, while changing, is always +Article number, page 4 of 11 + +2015/09/06 +Stokes Q +StokesU +off limb sources +1.0 +front hemisphere +0.0015 +0.8 +0.0010 +Polarization +Intensity +0.6 +0.0005 +0.4 +0.0000 +0.2 +-0.0005 +- +1 +-0.0010 +0.0 +-60 +-20 +0 +20 +40 +60 +80 +100-60 +40 +-20 +0 +20 +40 +60 +80 +100 +Wavelength(km/s) +Wavelength (km/s)A. López Ariste et al.: Raising plumes in µ Cep +Fig. 3. Observed linear polarization of µ Cep on October 21, 2016. The same color codes are used for Stokes Q and U as in Fig. 2. The velocity +boundaries given by the dashed lines are common to all the observations of µ Cep. +there. It can be explained as it was for Betelgeuse: by a spatially +inhomogeneous distribution of bright patterns that have been in- +terpreted as convection. However, these observations also show +a new signal that appears and disappears in time, and that, if +we accept the velocity boundaries, corresponds to bright plasma +moving away from the observer. +The conclusions drawn from these qualitative arguments +are definitively confirmed by the inversion codes developed by +López Ariste et al. (2018) and López Ariste et al. (2022): the +model used to fit the observed polarized spectra of Betelgeuse +and to infer the published images is unable to produce a solu- +tion for the spectrum of µ Cep on September 2015, though it +provides a solution for the observations of October 2016. Un- +willing to discard a model that has been successful with Betel- +geuse, we propose an addition to this model that can explain +the intermittent appearance of strong signals on the red side of +the red velocity boundary, which are illustrated in Fig. 2. We +propose that the bright convective structures inferred for Betel- +geuse and µ Cep and present over the whole star —and also in +the back hemisphere—, may raise plasma high enough for it to +become visible above the stellar limb. We refer to this high ris- +ing hot plasma as plumes. When these plumes are on the front +hemisphere, they produce the signals between the two velocity +boundaries and the basic model is able to explain them. Similar +convective bright structures must also occur on the back hemi- +sphere, but they are usually hidden by the stellar limb. From +time to time, one of these bright structures in the back hemi- +sphere may push plasma high enough for it to become visible to +us above the limb. This plasma rises in a radial direction, but as +it is in the back hemisphere of the star, we see it redshifted, mov- +ing away from us beyond the red velocity boundary; it is bright +plasma nevertheless, and so we expect it to have similar polariza- +tion amplitudes to plasma in the front hemisphere in symmetric +geometries. Plasma is not usually expected to rise high enough, +and so we often expect to see nothing beyond V∗. This has been +the case for all available observations of Betelgeuse and also for +µ Cep in October 2016. However, from time to time this may +happen, producing the signal illustrated in Fig. 2. When this is +observed, it cannot happen all over the stellar limb, but only at +particular polar angles, thus explaining the single peak that is +visible only in Stokes Q. +Becoming visible over the limb depends on the distance to +that limb of the bright structure. The further a structure is from +the limb, the higher it has to rise to become visible. This suggests +that we can determine the height of one of those structures as +the minimum height at which, by geometry, they become visible +above the limb. This measurement of a minimum height for the +rising plasma to become visible is going to be our main result. +4. Inversions with a modified model +In accordance with the suggested modification of the model pro- +posed in the previous section upon inspection of those polariza- +tion signals beyond the red velocity boundary, we built an inver- +sion code to fit the observed spectra of µ Cep. The core of this +inversion code is identical to the one described by López Ariste +et al. (2018). Mathematically, it is a Marquardt-Levemberg al- +gorithm that fits the observed Stokes Q and U profiles with syn- +thetic profiles computed from a distribution of brightness over +the surface of the star. On the front hemisphere, this distribu- +tion of brightness is described by a linear combination of spher- +ical harmonics up to sixth order. The blue velocity boundary +is the maximum velocity of the rising plasma. The brightest +point over the disk at any particular realization of the model is +supposed to move radially at that maximum velocity. All other +points over the disk have a brightness described by the spher- +ical harmonics, and a velocity which is mathematically related +to its brightness, meaning that the resulting brightness contrast +and velocities roughly match the solar case (see the Appendix +in López Ariste et al. 2022). The polarization emitted by a point +over the hemisphere is proportional to that brightness, but also to +the squared sine of the scattering angle, as expected for Rayleigh +scattering. The ratio of polarizations between Q and U is given +by the tangent of half the polar angle position of the point. Its +wavelength is determined by its velocity projected onto the line +of sight and therefore depends on the distance to the center of the +disk. Mathematically, the model uses, as parameters, the coeffi- +cients of a brightness distribution written in terms of spherical +Article number, page 5 of 11 + +2016/10/21 +Stokes Q +Stokes U +off limb sources +1.0 +front hemisphere +0.0010 +Ft +0.8 +0.0005 +Intensity +Polarization +0.6 +0.0000 +0.4 +0.0005 +0.2 +-0.0010 +-- +- +- +- +0.0 +-0.0015 +-60 +-40 +-20 +20 +40 +60 +80 +100-60 +40 +-20 +20 +40 +60 +80 +100 +Wavelength(km/s) +Wavelength (km/s)A&A proofs: manuscript no. art72_final +harmonics as +B(µ, χ) = +���������� +� +ℓ=0,ℓmax +m=−ℓ,+ℓ +am +ℓ ym +ℓ (µ, χ) +���������� +, +(1) +with ℓmax = 6, and µ and χ being the angle to disk center and +the polar angle with respect to celestial north, respectively. This +brightness distribution results in the emission of net polarization +described in terms of Stokes parameters as +Qdisk(v) = +� +µ,χ,vz +B(µ, χ) sin2 µ cos 2χe−(v−vz)2/σ2, +(2) +Udisk(v) = +� +µ,χ,vz +B(µ, χ) sin2 µ sin 2χe−(v−vz)2/σ2, +(3) +where vz = V(µ, χ) cos µ, with V(µ, χ) being the plasma veloc- +ity at that point, and proportional to the brightness and lim- +ited by the maximum speed of the plasma Vp. Each emission +is broadened with a Gaussian profile of fixed width σ = 6 +km s−1(i.e., 10 km s−1FWHM), which represents both instrumen- +tal and thermal broadenings. This is a rough description of the +basic model, for which many more details are given and scru- +tinized by López Ariste et al. (2018) and López Ariste et al. +(2022). In addition to this basic model, we assume the pres- +ence of one or several sources of polarization beyond the limb. +When adding new parameters to the model to describe those new +sources of polarization, one should be careful not to overload the +inversion algorithm with more new unknowns than available new +information. Therefore, it would be unreasonable to try to pro- +vide a description of the continuous distribution of brightness +in the back hemisphere, because only a very limited amount of +that plasma will be contributing to the observed spectra. It is +tempting to try to propose a description of the brightness in a +ring above the limb. Unfortunately, we have not found a proper +mathematical description for such a ring. One of the difficulties +is that, as we assume that it is uncommon for plumes to rise high +enough to become visible above the limb, we are expecting con- +tributions from at most a small range of polar angles, the rest +contributing zero. Any orthogonal family of functions trying to +describe this paucity of sources requires an excessive number +of parameters. We finally opted for a simplistic description in +terms of a small number of discrete sources. Each one of these +discrete sources over the limb is described by its polar angle χ, +its angular distance to the limb θ, and a brightness value Z (see +cartoon in Fig.4). Its polarization is given, as in the case of any +other emitting point in the front hemisphere, by the scaled prod- +uct of its brightness and the squared sine of the scattering angle, +this scattering angle being geometrically related to the distance +to the limb. +Qoff (v) = +� +i=0,N +Zi sin2 θi cos 2χie−(v−vz)2/σ2, +(4) +Uoff (v) = +� +i=0,N +Zi sin2 θi sin 2χie−(v−vz)2/σ2. +(5) +The radial velocity of the rising plasma is identically given +as a function of brightness. Its redshifted wavelength is analo- +gously given by the projection of this velocity onto the line of +sight, a projection which is again geometrically dependent on +the distance to the limb. Each one of these discrete over-the-limb +sources produces a polarization peak in Stokes Q and U that is +broadened by a Gaussian profile with a full width at half maxi- +mum (FWHM) of 10 km s−1. This FWHM is supposed to encom- +pass both the instrumental resolution and various stellar broad- +ening mechanisms, that is, thermal, microturbulent, and so on. +Fig. 4. Cartoon defining the parameters of a discrete source (yellow +sphere) in the back hemisphere (in gray) beyond the plane of the sky +(bluish plane). Celestial north is up, in the plane of the sky. The image of +the front hemisphere corresponds to the inferred brightness distribution +of µ Cep in September, 2015. +Finally, both sources of polarization, Qdisk, Udisk, and Qoff , Uoff +are added. +The last parameter to be determined is the number N of dis- +crete sources to be allowed. We find it unpractical to leave this +number unbound, and prefer to fix it. We attempted from N = 0 +up to N = 5 discrete sources. As expected, having zero sources +allows us to recover the basic model, unable to reproduce the +anomalous signals on the red wing. On the other end, we find +that beyond four sources we are not learning anything new from +the inversion results, and the algorithm becomes unstable, and +presents convergence issues. This can be safely understood as +an excessive number of new parameters given the available in- +formation. Between one and four sources is therefore the right +number of sources that we can safely infer. Interestingly, we also +find that for any individual observation of our long dataset, the +inferred value of the polar angle of all the sources is similar, +even if the intensity and height of each one of them is different. +This means that the solution found by the inversion algorithm +proposes that, at a given polar angle, there are several sources +of polarization at different heights and with different intensities. +That is, the inferred sources clump together on the same region +above the limb. This can be interpreted as one single but ex- +tended source over the limb of µ Cep at the time of the observa- +tion. This result appears to justify our intuition that such events +of high-rising plasma are not common. From this conclusion, +one may expect one single source in our model to be sufficient +to describe the observed polarization profiles in the red wing, +but we find that this is not the case and that we need a minimum +of two sources to reproduce the basic spectral features observed. +This may be an indication that even if there is a unique object +beyond the limb, it has sufficient structure that our description in +terms of a Gaussian profile per source is inadequate. Using two +or more sources becomes a simple manner of better describing +Article number, page 6 of 11 + +Xi +theta +SunA. López Ariste et al.: Raising plumes in µ Cep +the extent and structure of the emitting region. Because of this +result, we present inversions in this work with just two sources. +This has the advantage of capturing the important physical pa- +rameter for our work, the main distance of the bright structure +to the limb, while easing the constraints on the inversion algo- +rithm. The observed structure is often spectrally broader than +twice the FWHM of 10 km s−1of every discrete source. The fit is +therefore approximative. By increasing the number of sources, +we improve this fit, but do not bring any further information. +The above developments are illustrated in Figs. 2 and 3. Both +figures show on top of the observed profiles the solution found +by the inversion code as a green continuous line. This solution is +made of three different contributions. The basic model describ- +ing the front hemisphere as a linear combination of spherical +harmonics is plotted in red, and is fully coincident with, and hid- +den behind, the full solution between the two dashed lines that +limit the contribution of the front hemisphere. This basic model +can only be seen as a tail of small signal on the red side of the red +velocity boundary. This small signal, as mentioned above, is the +contribution from the dark sinking plasma, and is insufficient to +explain the observed polarization peak in September 2015. How- +ever, it is almost sufficient to explain the entirety of the redshifted +signal in October 2016. The two other contributions combined +are shown as a black continuous curve, and correspond to two +discrete sources above the limb. Again, this black line is only +visible when it does not fully coincide with the final solution +plotted in green. As explained, limiting the number of sources to +just two results in an approximative fit of the redshifted signal. +The full solution profile clearly shows two peaks on the red wing, +coinciding with the maxima of the two sources, a feature ab- +sent in the observations. There is also a clear tail further towards +the red in the observations that cannot be captured with just two +sources. Adding more sources would correct these missed fits, +but the parameters of the added sources will not change signifi- +cantly. In September 2015, the two sources over the limb bring +signal comparable to anything else over the front disk. In Octo- +ber 2016, the two sources appear as small contributions that may +drop to zero if just the red velocity boundary is shifted a few +km s−1 towards the red. The modified model is therefore able to +capture both those cases with important sources over the limb as +well as those cases with negligible contributions. +We used this model, in conjunction with two sources above +the limb, to invert the whole available dataset of linearly polar- +ized spectra of µ Cep presented in Sect. 2. Imaging from lin- +ear spectropolarimetry is subject to a certain number of ambi- +guities: Several images, with different distributions of brightness +are compatible with the same observed polarized spectra, that is, +they are possible alternative solutions of the inversion problem. +These latter images are not completely unrelated. The most com- +mon ambiguity appears between two images that are identical +but rotated 180 degrees with respect to one another. A compar- +ison with images of Betelgeuse made with interferometric tech- +niques allows us to determine which of these two rotated images +is the one that better corresponds to reality. However, we do not +have interferometric images for all dates, and none for µ Cep. +Because of this, in the case of Betelgeuse, the best solution for +a date with available interferometric images is propagated as the +initial solution to the next date, which encourages the inversion +code to stay in the group of solutions sharing choices among the +possible ambiguities that better compared to interferometric im- +ages at one particular date. Similarly, for µ Cep, we inverted the +first available date without constraints. But for next dates, the so- +lution of the previous available date was used as initial condition. +This ensures a certain time coherence in the series of images. +The inversion code provides values for the polar angle and +distance to the limb for the two sources over time. The distance +to the limb θ is directly converted into the minimum height h +above the stellar surface for this source in the back hemisphere +to be visible above the limb: +h = 1 − cos θ +cos θ +R∗. +(6) +Presented in this manner, the results of our inversions are shown +in Fig. 5 +Over the last six years, µ Cep appears to have produced three +events in the back hemisphere with plasma being lifted to con- +siderable heights. The first of these events was ongoing when +our observations started in September 2015. It had completely +disappeared when the star was re-observed in late spring 2016. +The next event started one year later, between January and April +2017, and by January 2018 plasma had reached heights of at least +1.1R∗ and perhaps higher. This plasma appeared at polar angles +of 100 degrees and after the winter blind window, the plasma was +still at the same position and at even greater heights of 1.15R∗. +Over the spring of 2019, the emitting plasma was seen to be de- +scending in height, until it disappeared by the summer of that +year. The beginning of the observations with Neo-Narval at the +beginning of 2020 showed µCep to be still quiet, with no partic- +ular signals on the red wing. But this situation changed by the +end of the year, with the rapid rise of a new clump of plasma at +a polar angle of 0 degrees —and therefore unrelated to the pre- +vious one—, which in less than one month reached heights of +at least 1.175R∗. The maximum height reached by this event ap- +pears to be quite ephemeral. As fast as it rises, it disappears. But +as it disappears, we are left with a low-lying clump that persists +throughout 2022. Optimistically, we may interpret this as a large +event of rising plasma inside of which there is a small clump at +high speed reaching even higher heights in a short time before +disappearing, perhaps due to a quick cooling, while the rest of +the rising plasma is still visible. In all these events, the value of +the polar angle of the two sources is quite similar, as can be seen +in the right plot of Fig. 5. As said above, we interpret this result +as proof that there is a unique source above the limb but more +extended and complex than what our model with two Gaussians +can reproduce. +Our 8 years of observations of linear polarization of Betel- +geuse have not produced any single event sufficiently large to +require a modification of the inversion model. In 5 years of ob- +servations, µ Cep has produced three such events. It is possi- +ble that this is due to the slightly different stellar parameters of +these two stars. The fundamental parameters of µ Cep recently +determined by Montargès et al. (2019) show a star similar to +Betelgeuse within error bars. Rather than invoking fundamental +differences between the two stars, we speculate that µ Cep may +be at present in a Decin stage (Decin et al. 2006), as suggested +by Montargès et al. (2019), with common episodes of mass loss, +while Betelgeuse may rather be in a quiet stage with rare and +separated events of this kind. This is simply speculation. At this +point, we lack any clear scenario explaining why and when a red +supergiant will enter into a Decin stage, if such episodes exist at +all. Further observations in time will be needed to see whether +or not µ Cep stops producing these events1. +In the fall of 2019, Betelgeuse suffered a large dimming that +has been attributed to the formation of a dense dust cloud al- +most directly along our line of sight (Montargès et al. 2021). +1 It must be said that Decin et al. (2006) estimate the duration of such +episodes in the tens of years. +Article number, page 7 of 11 + +A&A proofs: manuscript no. art72_final +Fig. 5. Plots of the height of the two sources visible above the limb and of their polar angle for the observations of µ Cep over the last six years. +For heights below 0.05, the source is considered to be absent and the corresponding value of the polar angle is made transparent. +López Ariste et al. (2022) suggested that these mass-loss events +are triggered from fast-rising plasma in the photosphere reach- +ing the escape velocity at a certain height. The suggestion made +by these latter authors stems from the measurement of plasma +velocities that are constant with height and sufficiently large to +be comparable to escape velocities at the estimated heights of +these structures. It is tempting to see this event in Betelgeuse as +one example of the more common events in µ Cep of plasma +rising sufficiently high to be visible above the limb. But in the +case of Betelgeuse, this event happened in the front hemisphere, +rather than in the back hemisphere as in µ Cep. If we accept +that Betelgeuse is at present in a quiet stage of mass loss, un- +like µ Cep, events where plasma is ejected from the star appear +to still happen. Just by chance, in Betelgeuse, lately, they have +not been happening in the regions near the limb, but rather in +the front disk, the ultimate example being the one that produced +the large dust cloud involved in the great dimming of 2019. In +µ Cep on the other hand, three such events have taken place in +regions around the limb, making it visible to our spectropolari- +metric measurements. +5. Follow-up of a convective plume above the limb +Figure 6 shows a time series of spectropolarimetric observations +of µ Cep starting on September 15, 2020, and ending on May +1, 2021. The first five observations during the fall and winter of +2020 show the rapid rise of a convective plume above the celes- +tial north limb of the star. This plume can easily be identified in +the inferred heights shown on Fig. 5. Such behavior can also be +seen directly in the profiles as a red peak with negative ampli- +tude in Stokes Q that, day after day, shifts to redder and redder +wavelengths, meaning that its projection over the line of sight is +greater and greater. Our interpretation of this is that plasma be- +yond the limb is rising. First, the parts nearer to the limb become +visible above the limb and, as time goes on, plasma farther and +farther from the limb becomes visible as it reaches the height at +which this is geometrically possible. The plume, which is cen- +tered well beyond the limb, is rising over a period of 3 months. +We lose track of the star from January through April 2021, and in +the first observation in May the structure has almost completely +disappeared: the polarization beyond the red velocity boundary +is small and centered very near the limit, as if only the regions +closer to the limb were still emitting light. The plume has disap- +peared. We have chosen this event to illustrate how the rise of the +plume can be estimated from direct visual inspection of the pro- +files, before the inversion code confirms the interpretation. The +rise of the plume is quite fast, and similarly fast is its disappear- +ance, as there is barely any signal of its presence at the opening +of the observing window in the following spring. +It may be tempting to say that the plume fell back into the +star, but we have no signature of this. We must recall that, any +plasma falling back into the back hemisphere would produce +a blueshifted signal that would melt into the signals of rising +plasma from the front hemisphere. We have no manner to disen- +tangle to two origins of polarization. +The event of 2018 is better followed during its disappear- +ance. What we observe is that the signal is still highly visible in +the red wing, meaning that the emitted plasma is still rising in the +back hemisphere, but its height is lower and lower. We interpret +this as follows: the upper parts of the plume of plasma, while still +rising, stop emitting light in the atomic lines measured here. This +may be because as it cools down, its brightness diminishes, or +because atomic lines are no longer excited. Translating our tech- +nique to molecular lines, if feasible, would shed light on this. In +both cases, the top of the plume cools down first and stops emit- +ting light. We only measure light from the lower parts, which +are still hot enough and still raising. This process continues until +only the lower parts of the plume are emitting measurable sig- +nals. Therefore, at the end of these episodes, we do not see the +plume falling, but just disappearing from our sensing window of +atomic lines in what we interpret as a cooling down phase that +starts from the top of the plume. +6. Discussion on the height of the observed +structures +Looking back at Fig. 5 we see that the structure followed in Fig. +6 reached a minimum height of 1.175R∗ during those 3 months. +This could be higher, because, using geometry, we can only give +the lower bound of this height. Taking 1000 − 1200R⊙ as the +radius of the star, this rise requires an average velocity of 15 +Article number, page 8 of 11 + +0.175 +Source 1 +Polar angle from celestial North (°) +0.150 +Source 2 +250 +Height above limb (R+) +0.125 +200 +0.100 +150 +0.075 +100 +0.050 +50 +0.025 +0 +0.000 +2016/01/01 +2017/01/01 +2018/01/01 +2019/01/01 +2020/01/01 +2021/01/01 +2022/01/01 +2016/01/01 +2017/01/01 +2018/01/01 +2019/01/01 +2020/01/01 +2021/01/01 +2022/01/01A. López Ariste et al.: Raising plumes in µ Cep +Fig. 6. Time series of spectropolarimetric observations of µ Cep corresponding to all dates from September 15, 2020, through May 1, 2021, +showing the rise and fall of a convective plume. The meaning of curve colors and styles is the same as in Fig.2. +to 20 km s−1, maintained constant for 3 months. This velocity +fits comfortably with the velocity limit Vp = 70km s−1in µ Cep +determined for the basic model of the front hemisphere. These +are minimum velocities, as we can only determine minimum +heights. +The possibility of detecting this kind of plume over the limb, +as offered by µ Cep, is exceptional. The observation of three such +events in over five years may suggest that there is some abnor- +mal convective activity in this star, at least when compared with +Betelgeuse. Nevertheless, we stick to the assumption that both +stars represent different cases of the same physics and that it is +just the relatively short span of the observations available that +explains the observed differences, and not any fundamental dif- +ference between the behaviors of these two RSGs. Building on +this assumption, the measurement of a geometric height made +on these structures is deemed typical of convective plasma fea- +tures in RSGs, and we generalise it to all other structures im- +aged on such objects with spectropolarimetry. We consider the +value of 1.1R∗ as the typical height of the plasma in the atmo- +spheres of RSGs hot enough to emit atomic spectral lines. Mak- +ing the link with López Ariste et al. (2022), we consider that this +measured geometric height, recovered from spectropolarimetry +of the deepest atomic lines in the spectrum, must correspond typ- +ically to the height of their uppermost layer. Figure 9 of this lat- +ter publication must extend therefore up to 1.1R∗. It is only by +considering that this upper layer is visible for several months at +that height that in this latter work it is assumed that the observed +structures may well have reached 1.3R∗. +7. Conclusion +Spectropolarimetric observations of the RSG µ Cep show spec- +tral features in linear polarization that were not observed in the +better studied Betelgeuse. Such spectral features are much more +redshifted than any other signal and are not permanent features: +on certain dates, the observed spectra are qualitatively identical +to those of Betelgeuse. We argue that the origin of those unex- +pected spectral features is convective plumes in the back hemi- +sphere of the star that rise high enough to be visible above the +stellar limb. +This hypothesis allows us to conserve the inversion algo- +rithms and model that have successfully been applied to Betel- +geuse and that have produced images comparable to those from +interferometry. However, such a basic model must be extended +to allow for temporary sources of polarized light above the stel- +lar limb. We produced an inversion algorithm using this extended +model and successfully fitted the observed profiles, including the +unexpected new features. This model assumes the presence of a +small number of discrete sources over the limb. Although about +four such sources are required to correctly fit the profiles, we re- +alize that, at any given date, all those sources appear to be com- +bined to describe a unique but extended source on the star. This +observation allowed us to reduce the number of sources over the +limb to just two. While the fit with just two sources is not as +good as it would with four sources, we still capture the main +parameters of the sources and stabilize the convergence of the +code, which can be automatically launched to handle the whole +dataset available. +The inversion results produce the polar angle position and +the height of those sources. This gives us access, for the first +Article number, page 9 of 11 + +2021/01/13 +Stokes Q +Stokes U +off limb sources +1.0 +front hemisphere +0.0015 +Fit +0.8 +0.0010 +Polarization +Intensit +0.6 +0.0005 +0.4 +0.0000 +0.2 +- +- +- +-0.0005 +0.0 +-60 +-40 +-20 +20 +40 +60 +80 +100-60 +40 +-20 +20 +40 +60 +80 +100 +Wavelength(km/s) +Wavelength (km/s)2021/05/01 +Stokes Q +StokesU +off limb sources +1.0 +fronthemisphere +0.0010 +Ft +0.8 +0.0005 +Polarization +Intensity +0.6 +0.0000 +0.4 +-0.0005 +0.2 +- +0.0010 +0.0 +-60 +-40 +-20 +0 +20 +40 +60 +80 +100-60 +40 +-20 +0 +20 +40 +60 +80 +100 +Wavelength(km/s) +Wavelength (km/s)2020/09/15 +Stokes Q +StokesU +off limb sources +1.0 +front hemisphere +0.0015 +Ft +0.8 +0.0010 +Polarization +Intensit +0.6 +0.0005 +0.4 +0.0000 +0.2 +-- +- +0.0005 +0.0 +-60-40 +-20 +0 +20 +40 +60 +80 +100-60 +40 +-20 +0 +20 +40 +60 +80 +100 +Wavelength(km/s) +Wavelength (km/s)2020/10/16 +Stokes Q +StokesU +off limb sources +1.0 +front hemisphere +Ft +0.0015 +0.8 +0.0010 +Polarization +0.0005 +0.4 +0.0000 +0.2 +0.0005 +- +0.0 +-60-40 +-20 +0 +20 +40 +60 +80 +100-60 +40 +-20 +0 +20 +40 +60 +80 +100 +Wavelength(km/s) +Wavelength (km/s)2020/11/21 +Stokes Q +Stokes U +off limb sources +1.0 +front hemisphere +Fit +0.0015 +0.8 +0.0010 +Polarization +Intensit +0.6 +0.0005 +0.4 +0.0000 +0.2 +0.0005 +- +- +- +0.0 +-60-40 +-20 +0 +20 +40 +60 +80 +100-60 +40 +-20 +20 +40 +60 +80 +100 +Wavelength(km/s) +Wavelength(km/s)2020/12/18 +Stokes Q +Stokes U +off limb sources +1.0 +front hemisphere +Fit +0.0015 +0.8 +0.0010 +Polarization +Intensit +0.6 +0.0005 +0.4 +0.0000 +0.2 +-- +- +-0.0005 +0.0 +-60 +-20 +0 +20 +40 +60 +80 +100-60 +40 +-20 +0 +20 +40 +60 +80 +100 +Wavelength(km/s) +Wavelength (km/s)A&A proofs: manuscript no. art72_final +time, to a geometric height for the convective structures detected +through spectropolarimetry. +Three events of plasma rising over the limb have been ob- +served during the six years of observation of µ Cep with Nar- +val and Neo-Narval at the TBL at Pic du Midi. Two of those +events were tracked during their rapid rising phase and into their +disappearance. The characteristic heights reach 1.1R∗ and even +1.175R∗ in the last observed event. We consider this a typical +value of the heights of the convective structures observed in the +photosphere of RSGs, and López Ariste et al. (2022) use this +value to calculate a geometric height from the three-dimensional +images of Betelgeuse. Thanks to this measurement, these authors +demonstrated that the measured velocities in the plasma are very +near the escape velocity of Betelgeuse and that this rising plasma +is likely a contributor to the mass loss of these stars. +Acknowledgements. This work was supported by the "Programme National de +Physique Stellaire" (PNPS) of CNRS/INSU co-funded by CEA and CNES. S.G. +acknowledges support under the Erasmus+ EU program for doctoral mobility. +S.G. acknowledges partial support by the Bulgarian NSF project DN 18/2. +References +Aurière, M., López Ariste, A., Mathias, P., et al. 2016, Astronomy and Astro- +physics, 591, A119 +Chiavassa, A., Freytag, B., Masseron, T., & Plez, B. 2011, Astronomy and As- +trophysics, 535, A22 +Decin, L., Hony, S., de Koter, A., et al. 2006, Astronomy & Astrophysics, 456, +549 +Donati, J. F., Catala, C., Landstreet, J. D., & Petit, P. 2006, 358, 362, conference +Name: Solar Polarization 4 +Freytag, B., Steffen, M., & Dorch, B. 2002, Astronomische Nachrichten, 323, +213 +Levesque, E. M., Massey, P., Olsen, K. A. G., et al. 2005, 207, 182.13, conference +Name: American Astronomical Society Meeting Abstracts +López Ariste, A., Georgiev, S., Mathias, P., et al. 2022, Astronomy and Astro- +physics, 661, A91 +López Ariste, A., Mathias, P., Tessore, B., et al. 2018, Astronomy and Astro- +physics, 620, A199 +Mathias, P., Aurière, M., López Ariste, A., et al. 2018, Astronomy and Astro- +physics, 615, A116 +Montargès, M., Cannon, E., Lagadec, E., et al. 2021, Nature, 594, 365 +Montargès, M., Homan, W., Keller, D., et al. 2019, Monthly Notices of the Royal +Astronomical Society, 485, 2417 +Stothers, R. B. 2010, The Astrophysical Journal, 725, 1170 +Tessore, B., Lèbre, A., Morin, J., et al. 2017, Astronomy and Astrophysics, 603, +A129 +Article number, page 10 of 11 + +A. López Ariste et al.: Raising plumes in µ Cep +Appendix A: Log of Observations +Table A.1. Log of Narval and Neo-Narval observations of µ Cep and +polarimetric measurements since July 2015. +Date +Julian date +Stokes +Sequence +July 10, 2015 +7214.578 +8U+8Q +September 05, 2015 +7271.504 +2U+2Q +November 10, 2015 +7337.462 +2Q+2U +May 16, 2016 +7525.609 +2Q+2U +June 08, 2016 +7548.628 +2U +June 16, 2016 +7556.604 +2Q +June 21, 2016 +7561.586 +2U +June 27, 2016 +7567.569 +2Q +July 05, 2016 +7575.566 +2Q+2U +July 15, 2016 +7585.517 +2Q+2U +August 02, 2016 +7603.448 +2U+2Q +September 01, 2016 +7633.418 +2Q+2U +September 28, 2016 +7660.436 +2Q+2U +October 07, 2016 +7669.366 +4Q+2U +November 27, 2016 +7720.273 +2Q+2U +December 18, 2016 +7741.265 +4Q+4U +January 07, 2017 +7761.276 +2Q+2U +April 08, 2017 +7852.658 +2Q+2U +April 15, 2017 +7859.664 +2Q+2U +April 22, 2017 +7866.669 +2Q+2Q +May 07, 2017 +7881.614 +2Q+2U +May 20, 2017 +7894.562 +2Q+2U +June 01, 2017 +7906.57 +2Q+2U +June 11, 2017 +7916.545 +2Q+2U +June 16, 2017 +7921.578 +2Q+2U +July 02, 2017 +7937.627 +2Q+2U +July 11, 2017 +7946.616 +2Q+2U +July 31, 2017 +7966.561 +2Q+2U +August 08, 2017 +7974.536 +2Q+2U +August 13, 2017 +7979.45 +2Q+2U +August 21, 2017 +7987.524 +2Q+2U +September 02, 2017 +7999.445 +2Q+2U +September 05, 2017 +8002.412 +2Q+2U +September 13, 2017 +8010.449 +2Q+2U +September 20, 2017 +8017.504 +2Q+2U +September 27, 2017 +8024.37 +2Q+2U +October 02, 2017 +8029.469 +2Q+2U +October 07, 2017 +8034.342 +2Q+2U +October 12, 2017 +8039.359 +2Q+2U +October 30, 2017 +8057.341 +2Q+2U +November 07, 2017 +8065.285 +2Q+2U +November 14, 2017 +8072.337 +2Q+2U +November 19, 2017 +8077.255 +2Q+2U +November 26, 2017 +8084.269 +2Q+2U +December 04, 2017 +8092.275 +2Q+2U +May 17, 2018 +8256.624 +2Q+2U +June 14, 2018 +8284.607 +2Q+2U +June 30, 2018 +8300.503 +2Q+2U +July 22, 2018 +8322.619 +2Q+2U +Date +Julian date +Stokes +Sequence +August 13, 2018 +8344.677 +2Q+2U +September 26, 2018 +8388.478 +2Q+2U +October 25, 2018 +8417.442 +2Q+2U +November 14, 2018 +8437.322 +2Q+2U +December 10, 2018 +8463.313 +2Q+2U +January 04, 2019 +8488.239 +2Q+2U +January 16, 2019 +8500.281 +2Q+2U +March 21, 2019 +8564.685 +2Q+2U +May 05, 2019 +8609.594 +2Q+2U +June 01, 2019 +8636.633 +2Q+2U +June 18, 2019 +8653.622 +2Q+2U +July 18, 2019 +8683.555 +2Q+2U +August 02, 2019 +8698.589 +2Q+2U +August 15, 2019 +8711.498 +2Q+2U +August 30, 2019 +8726.552 +2Q+2U +January 06, 2020 +8855.288 +2Q+2U +May 17, 2020 +8987.572 +1Q+2U +June 22, 2020 +9023.621 +2Q+2U +July 05, 2020 +9036.583 +2Q+2U +July 24, 2020 +9055.576 +2Q+2U +August 22, 2020 +9084.518 +2Q+2U +September 15, 2020 +9108.483 +2Q+2U +October 16, 2020 +9139.313 +2U+2Q +November 21, 2020 +9175.329 +2Q+2U +December 18, 2020 +9202.256 +2Q+2U +January 13, 2021 +9228.271 +2Q+2U +May 01, 2021 +9337.647 +2Q+2U +May 26, 2021 +9361.603 +2Q+2U +June 14, 2021 +9380.536 +2Q+2U +July 10, 2021 +9406.619 +2Q+2U +August 07, 2021 +9434.455 +2Q+2U +August 19, 2021 +9446.59 +2Q+2U +September 04, 2021 +9462.43 +2Q+2U +October 06, 2021 +9494.441 +2Q+2U +October 13, 2021 +9501.405 +2Q+2U +November 09, 2021 +9528.363 +2Q+2U +December 13, 2021 +9562.279 +2Q+2U +December 22, 2021 +9571.252 +2Q+2U +January 11, 2022 +9591.264 +2Q+2U +Notes: Columns give the date, the heliocentric Julian date +(+2 450 000), and the observed Stokes sequence, that is, how +many observations of which Stokes parameter were made at that +date. An observation consists of four exposures with changing +polarimetric modulation that, after reduction, produce polariza- +tion spectra of either Stokes Q or U. Beyond a 2 year pro- +prietary embargo, all data are publicly available at PolarBase +(http://polarbase.irap.omp.eu/). +Article number, page 11 of 11 + diff --git a/UdAzT4oBgHgl3EQfX_ww/content/tmp_files/load_file.txt b/UdAzT4oBgHgl3EQfX_ww/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..bdaa4caebde531e09417f124657e07b9686818af --- /dev/null +++ b/UdAzT4oBgHgl3EQfX_ww/content/tmp_files/load_file.txt @@ -0,0 +1,766 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf,len=765 +page_content='Astronomy & Astrophysics manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' art72_final ©ESO 2023 January 5, 2023 The height of convective plumes in the red supergiant µ Cep⋆ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' López Ariste1, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Wavasseur1, Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Mathias1, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Lèbre2, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Tessore3, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Georgiev2, 4 1 IRAP, Université de Toulouse, CNRS, CNES, UPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' 14, Av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Belin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' 31400 Toulouse, France 2 LUPM, Université de Montpellier, CNRS, Place Eugène Bataillon, 34095 Montpellier, France 3 Université Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France 4 Institute of Astronomy and NAO, Bulgarian Academy of Science, 1784 Sofia, Bulgaria Received .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' accepted .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' ABSTRACT Aims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' We seek to understand convection in red supergiants and the mechanisms that trigger the mass loss from cool evolved stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Linear spectropolarimetry of the atomic lines of the spectrum of µ Cep reveals information well outside the wavelength range expected from previous models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' This is interpreted as structures in expansion that are visible in the front hemisphere and sometimes also in the back hemisphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' We model the plasma distribution together with its associated velocities through an inversion algorithm to fit the observed linear polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' We find that supposing the existence of plasma beyond the limb rising high enough to be visible above it can explain the observed linear polarization signatures as well as their evolution in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' From this we are able to infer the geometric heights of the convective plumes and establish that this hot plasma rises to at least 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='1 R∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' µ Cep appears to be in an active phase in which plasma rises often above 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='1 R∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' We generalize this result to all red supergiants in a similarly evolved stage, which at certain epochs may easily send plasma to greater heights, as µ Cep appears to be doing at present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Plasma rising to such heights can easily escape the stellar gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Introduction Despite their large impact on stellar and galactic evolution, the properties of outflows from red supergiants (RSGs) are not well characterized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' In particular, the role of convection is still poorly understood, partly because their structures are difficult to ob- serve directly through interferometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' In this work, we propose a view of the convection structure of the RSG µ Cep through im- ages reconstructed from spectropolarimetric data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Convection is also of interest to those studying mass loss from these evolved stars because these convective processes are thought to con- tribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' These have been the motivations for several campaigns of observation of the red supergiant µ Cep at the Telescope Bernard Lyot (TBL) with both spectropolarimeters Narval and Neo- Narval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' These observations were made in parallel to those of Betelgeuse presented by Aurière et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' (2016), Mathias et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' (2018), and López Ariste et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' (2018), though less frequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Nevertheless, as for Betelgeuse, the observations of µ Cep were predominantly aimed at measuring the linear polarization in the atomic lines of its spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' An observed net linear polarization was interpreted as depo- larization —during line formation— of the spectrum continuum, which is itself polarized by Rayleigh scattering (Aurière et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Using this hypothesis of the physical origin of the ob- served linear polarization, two-dimensional images of the pho- tosphere of Betelgeuse were inferred, images that could be fa- vorably compared to co-temporal images of this star made us- ing interferometric techniques (López Ariste et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' These successful comparisons suggest that the many approximations involved in this new imaging technique, which uses spectro- polarimetry, are appropriate and have spurred a further step in ⋆ Based on observations obtained at the Télescope Bernard Lyot (TBL) at Observatoire du Pic du Midi, CNRS/INSU and Université de Toulouse, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' the technique: recently López Ariste et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' (2022) published the first three-dimensional images of Betelgeuse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' In addition to the satisfying images, this technique provided some interesting data: the spatial and temporal scales of the convective patterns were measured, and the characteristic velocities of the raising plasma were determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' These velocities are much higher than the adia- batic estimates, easily reaching values of 40km s−1(López Ariste et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Stothers 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' More interestingly, López Ariste et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' (2022) observed that in several observed cases of rising hot plasma, the velocity was constant with height, suggesting the presence of a force counter-acting gravity in the photospheric layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' These large velocities, sometimes reaching 60km s−1, are comparable to the escape velocity at tantalizingly low heights (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='5R∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' If at any time the hot plasma reaches the escape veloc- ity, it will escape the gravity of the star and, cooling down, may be the origin of the clumpy dust clouds seen around Betelgeuse (Montargès et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' This fast, rising plasma may also be the source of mass loss in these stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' However, in order to reach this interesting result, a critical piece of information is missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' The technique presented and used by López Ariste et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' (2022) to build three-dimensional images of the photosphere of Betelgeuse is unable to determine the geometric height of the successive layers imaged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' The tech- nique only provides the ordering of the layers, from the deep atmosphere up to higher layers, but not the geometric distance between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Here, we present a means to measure or at least estimate this geometric height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' In Section 2 we present the set of observations of µ Cep collected with Narval and Neo-Narval at the TBL from 2015 through 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' In Section 3, we describe the spectral features in the linear polarization of µ Cep that cannot be explained with the model used to image Betelgeuse, and propose a modifica- tion of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' We propose that, from time to time, convec- tive plumes are powerful enough to rise to sufficient heights that they can be seen beyond the geometric horizon of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' This Article number, page 1 of 11 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='01326v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='SR] 3 Jan 2023 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' art72_final cannot be a permanent feature, but it may happen from time to time, an aspect that is critical to the plausibility of this propo- sition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' We also discuss how this modification affects previous results for Betelgeuse, if we assume that a unique model serves all RSGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' In Section 4 we build an inversion code based on this modified model, where the usual description of the brightness variation across the disk is supplemented with the presence of up to five clouds of plasma visible beyond the horizon of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' The measured linear polarization degrees and angles allow us to determine how far beyond the horizon this plasma is and there- fore how high it must be to be visible from Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' It is in this way that we can determine a minimum height for these structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' In Section 5, we build a time evolution of one of those plumes that we were lucky to follow in 2021 from rise to fall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' In Section 6, we put these measurements in context, in particular with respect to the measured plasma velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' We confirm that it is highly possible that the most powerful of these convective plumes are high enough to escape the gravity of the star at the observed ve- locities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Spectropolarimetric data from Narval and Neo-Narval µ Cep is an M2-type RSG with stellar parameters (Tef f = 3750K and log g = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='36 ) very similar to those of Betelgeuse, while its mass (25M⊙) and radius (1420 R⊙) on the other hand may be larger (Levesque et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Tessore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' (2017) first de- tected strong linear polarization features (both in Stokes Q and U) associated to atomic lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' We began observing µ Cep in linear polarimetry in July 2017 with the Telescope Bernard Lyot at Pic du Midi (France,TBL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Until August 2019, the Narval spectropolarimeter was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' After an upgrade, starting in September 2019, Neo-Narval re- observed µ Cep in May 2020 and regular observations have been conducted ever since.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' This long series allowed us to follow the entire lifetime of one of these convective plumes (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Narval and Neo-Narval have been described before in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' These descriptions are extensive for the case of Narval (Donati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' 2006), with a succinct description of the changes of Neo-Narval provided by López Ariste et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' As stressed in this latter publication, we note the continuity of data quality from the instrument through its upgrade, and handle Narval and Neo-Narval data as a unique dataset with no further reference to the instrument used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' We performed relatively short exposures (about 3 minutes per polarimetric sequence) in order to ensure a peak signal-to- noise ratio (S/N) of about 2000 in Stokes I per velocity bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' A list of the observations of µ Cep is presented in Table A, corre- sponding to all those studied in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' A least-squares de- convolution (LSD) procedure (Donati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' 2006) is applied to the reduced spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Atomic lines from an appropriate list (Au- rière et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' 2016) are summed after rescaling of the wavelength binning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' The result is a single spectral profile for both Stokes I and the observed Stokes parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' The whole set of Stokes Q, U, and I profiles thus obtained is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='1 in the form of an image with time on the vertical axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Redshifted linear polarization features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' The model used to interpret the linear polarization observed in the atomic lines of Betelgeuse and µ Cep and, in general, of all red supergiants assumes a nonrotating convective star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' This model and the implicit approximations involved were described in detail by López Ariste et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' (2018) and López Ariste et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' From the point of view of the physical origin of the lin- ear polarization, our model assumes that what we observe is the depolarization of the continuum by atomic lines due to Rayleigh scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' A key diagnostic of the trustworthiness of this inter- pretation of the polarization is that all lines must show similar polarization, independent of their quantum structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' In partic- ular, the Na D1 and D2 lines must show similar signals to one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' This was seen to be the case for Betelgeuse (Aurière et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' 2016), CE Tau, and now for µ Cep, the target of the present study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Once the physical origin of this polarization is confirmed, the model focuses on the distinct spatial origin of the spectral features seen in the linear polarization profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' The observed linear polarization profiles characteristically show several distinctive lobes inside every atomic line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' In the ab- sence of rotation, the wavelength position of each of those lobes is assumed to be due to convection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' The brightest plasma is as- sumed to be rising, and the cooler, darker plasma sinks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' At first approximation, most light comes from the brighter regions and is therefore Doppler shifted by the projection onto the line of sight of the convective velocity at which the plasma rises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Thus, bright hot plasma at disk center will emit light in the blue wing, while bright hot plasma at the limb will emit light in the red wing, at a wavelength which will coincide with the velocity of the center of mass of the star with respect to the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Dark, sinking plasma would be redshifted with respect to this red wing, but its low in- tensity translates into a tiny signal to be added in the further red wing of the observed profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' In this model, the spectral profile of an atomic line is framed by two velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' One of these ve- locities is the heliocentric velocity V∗, which limits the red wing of the polarization profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Plasma at the stellar limb will emit light at or near to this red wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' The second of these ve- locities is the maximum velocity of the plasma in the convective flow, Vp, which limits the blue wing of the profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Plasma ris- ing at this maximum velocity at disk center will emit light at the bluest wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' In the absence of rotation, all other velocity fields, such as micro- and macroturbulent velocities or thermal broadening, are assumed to be isotropic and would just broaden the signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Such broadening, added to instrumental effects, is seen as a minimum width for all observed polarization signals, a width that is much smaller than the span of velocities attributed to convection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' The two velocities, V∗ and Vp, that limit the observed pro- files are parameters of the model and should be determined a priori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' As discussed by López Ariste et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' (2022), this a priori determination is done by inspection of the whole set of avail- able observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' 1, the two velocities are represented as vertical lines on top of the pile up of the observed profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Our choice for these two velocities, namely V∗ = +35 km s−1for the velocity of the center of mass of the star and Vp = −70 km s−1for the maximum velocity of the convecting plasma, can be judged with respect to the wavelength span of the polarization signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' We note that while V∗ is measured in the heliocentric reference system, we are giving the value of Vp in the star’s own reference system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' In the heliocentric reference system used in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' 1, we find Vp at +35 − 70 = −35km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' As Vp has a meaning in terms of the physics of convection of the star, it is useful to keep its value in the reference system of the star, even at the risk of some confusion when looking at Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' The choice of these values is not free from criticism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Judging from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='1 alone, it appears as if the red limit V∗ has been placed in the middle of the polarization signal rather than at its red edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' As these two velocity limits cannot be directly measured, we can only advance the arguments that justify our choice for these Article number, page 2 of 11 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' López Ariste et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' : Raising plumes in µ Cep two parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' These arguments are qualitatively similar to the ones used by López Ariste et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' (2018) and that López Ariste et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' (2022) justified to be acceptable within 10 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Part of this justification lies in the fact that accepting the model and the values of these velocities results in images of Betelgeuse or CE Tau, another observed RSG, that are comparable with contempo- raneous images inferred by interferometers (López Ariste et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' However, in the present case, we lack any such interfero- metric images for µ Cep, and contrary to the previously studied Betelgeuse and CE Tau, there is a considerable amount of sig- nal redward from V∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' It is worth examining the arguments that justify this choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' It is obvious that Vp, the maximum velocity of the plasma represented by the blue line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='1 (which, we reiterate, is in the heliocentric reference system, while the value of Vp is given in the star’s reference system), must encompass the most blueshifted signals observed over the years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Accordingly, added to V∗, this velocity must be somewhere beyond -20km s−1in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' We have chosen -35km s−1to include the extended wings of the signals observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' In our model, we have no explanation whatsoever for any signal blueward from Vp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' We therefore have to make sure that there is no signal beyond this limit, and this fixes minimum values of Vp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' In our model, Vp is interpreted as the velocity of the rising plasma during convection in the reference frame of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' This interpretation sets further constraints on its maximum value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Our choice has been, for µ Cep, to set this maximum velocity at Vp = −70 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' This is already a large velocity for rising plasma and is roughly seven times the speed of sound in the atmosphere of µ Cep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Numerical simulations (Freytag et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Chiavassa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' 2011) produce supersonic flows for convective patterns, which were confirmed by López Ariste et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' But a value of Vp = −70 km s−1is 50% to 100% larger than any published figure so far, based on either observations or numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' However, as Vp is fixed on its blue side by the extent of the polarization signal, accepting these large plasma velocities is the only way to place V∗ as far to the red as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' As V∗+Vp is set at -35 km s−1, the velocity of the center of mass must therefore be V∗ = +35km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' In spite of the large value of V∗, this choice still leaves lots of redshifted signal beyond the limit of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Once again, including those signals in our convection model by shifting V∗ to higher values would imply accepting convection velocities larger than 70km s−1and this seems unphysical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Also, once more, diminishing the maximum velocity Vp also seems unphysical, as blueshifted signatures would be left unexplained beyond the maximum velocity of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' This sequence of arguments justifies our choices up to 10km s−1, and leaves large amounts of signal beyond the red limit of V∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Another argument justifying the choice of V∗ = 35km s−1is apparent from the Stokes I variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' From the LSD profile, one can compute a mean heliocentric radial velocity from the profile Gaussian fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' This mean velocity is found to be < v >= 22km s−1 in the heliocentric reference frame, and from profile to profile varies with an amplitude of about 4 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' For Betelgeuse, the corresponding quantities are respectively < v >= 21km s−1and 4km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' The radial velocity of Betelgeuse was estimated to be about V∗ = 40km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Supposing the same parameters for both these RSGs, that is, a number of granules number of similar order and the same temperature contrast, then the V∗− < v > values should also be similar, meaning a value of the order of 40km s−1, or, within the 10km s−1uncertainty, the adopted V∗ = 35km s−1value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' To conclude this discussion of the choice of the value of these two velocities, we must add that several Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Pile-up of the Stokes Q (left), U (center) and I (right) profiles over the whole time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' For illustrative purposes, every observation has been made to span 15 days on the vertical direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' The blue and red vertical lines mark the maximum plasma velocity Vp and the radial velocity of the center of mass of the star V∗ respectively (see main text for definitions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Velocities are measured in the heliocentric reference system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' velocities have been tested inside the range allowed by those 10 km s−1, without significant differences in the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' A strict interpretation of these velocity limits implies that no polarization signal can be seen in our modeled profiles at wave- lengths redder than V∗, the limit given by the velocity of the center-of-mass of the star for each atomic spectral line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Polariza- tion signals beyond this limit would come, in this model, from plasma moving away from the observer and towards the center of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Plasma sinking towards the core of the star is as- sumed to be cool, dark plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' This rigorous interpretation must be softened somehow, as cold, dark plasma still emits some light and plasma may start sinking while still being bright enough to contribute to the net spectral line profile;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' but these are always small contributions, and we are not expecting any large signals on the red side of the velocity limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' After inspection of the pro- files of Betelgeuse published by Aurière et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' (2016), Mathias et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' (2018), López Ariste et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' (2018) and López Ariste et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' (2022) we can confirm that this is the case, and that the hypoth- esized model can confidently describe all the available observa- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' However, this is not the case for µ Cep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' 2, we plot the observed spectra of µ Cep collected since September 6, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' The observations are plotted as dotted lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' We return to the continuous lines in different colors further below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' At this point, we focus our attention on the strong Q sig- nal peaking at about +50 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' This is actually the strongest polarization of the observed profiles on that date and it is found to the red of the limiting velocity V∗ = +35 km s−1, and is indi- cated by the vertical dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Such strong signal in the red wing of the profiles of atomic lines is unlike anything observed to this day in Betelgeuse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' The many more observations of linear polarization in the spectra of Betelgeuse are better illustrated by the profile of Stokes U shown in the right plot of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' A strong peak is seen in the blue wing and is attributed to bright plasma near the center of the stellar disk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' another (negative) peak is seen near V∗, and attributed to bright rising plasma coming from regions near the stellar limb;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' and a small (positive) signal is seen beyond V∗ corresponding, Article number, page 3 of 11 Stokes Q Stokes U Stokes I 01/2022 09/2020 06/2019 02/2018 10/2016 07/2015 50 50 100 50 50 100 50 50 100 Velocity (km/s) Velocity (km/s) Velocity (km/s)A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' art72_final Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Observed linear polarization of µ Cep on September 6, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' The observed Stokes Q is plotted on the left, and Stokes U on the right, as dots in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Continuous lines represent the best fit from the assumed model (green line) with separated contributions from the front disk brightness distribution (red) and the two plumes beyond the limb (black), visible at those wavelengths when they do not constitute the whole contribution to the final fit in green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' The upper (orange) profile shows the normalized intensity profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' The vertical dashed lines give the two limiting velocities, Vp and V∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' hypothetically, to sinking dark plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' The observed Stokes U profile is, in this manner, qualitatively explained and, after in- version, the inferred image confirms this basic description of the visible structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Such a model would also explain the small lobe seen in the blue wing of the Stokes Q profile and the larger negative peak near the red boundary (red line of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' The respective amplitudes and signs of these peaks in Q and U will constrain the position and brightness of the different bright struc- tures over the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' However, this model has no explanation whatsoever for the strong signal on the red side of the red bound- ary of the Q profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Such strong signal cannot be attributed to dark sinking plasma, for there would be no explanation for its large amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' The amplitude could either be due to the amount of photons or the polarization degree of those photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' To inter- pret such a large amplitude would require either a brighter re- gion or a more polarized region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' It appears contradictory to say that the sinking plasma is brighter than the rising plasma, and so we are only left with the possibility that this is sinking plasma with an enormous polarization degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Implicit in our model is that polarization degree is directly related to the height of the plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' One possible explanation that our model would have for this strong polarization peak would therefore be that this is a huge cloud of cold plasma sinking from large heights that is much larger than any other structure in the atmosphere, because height must compensate for the loss of signal due to the lower emissivity of this cool dark plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' The presence of that unexpected strong peak is forcing the model towards extreme scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Another possibility is that our determination of V∗, the veloc- ity of the center of mass of the star, is wrong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' It suffices to shift this limit a further 35 km s−1to the red, up to V∗ = +70km s−1, for the entire peak to fit inside the limits of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' However, this scenario entails unpleasant conclusions as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Shifting this limit to the red without touching the blue limit would mean that the maximum velocity of the convective flows in µ Cep would increase to a staggering 100 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' This is an uncomfortably large number for the convective flows, about ten times the speed of sound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' The problems with a modified red boundary do not end here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' We expect there to always be some signal coming from near the limb, because statistically there is a large probability of find- ing a bright structure somewhere along the long circumference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' This is the case with the actual red boundary limit plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' 2: both Stokes Q and U profiles show signal near the limb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' How- ever, if we were to accept this boundary shift, Stokes Q would still have the strong peak that would be attributed to a near-limb structure, but no comparable signal is visible anywhere near that limit for Stokes U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' In order to produce such signal imbalance between Q and U, one would need to imagine a stellar disk with a continuous dark band along and inside the limb except at one position where a bright structure would give the observed Stokes Q signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' While not impossible, this appears as a strange disposi- tion of structures on µ Cep, something never seen on Betelgeuse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' In addition, this 70 km s−1value is clearly outside the I profile, meaning that this latter would have no link with the heliocentric star velocity, which also seems difficult to accept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' One year later, in October 2016, the observations shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='3 have drastically changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Between the velocity boundaries, the polarization signals keep providing an image of changing bright, convective structures, but there is always signal at the qualitatively expected places, even if that signal has changed in amplitude, ratio, and position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' This is interpreted as bright structures that have moved over the disk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' some have appeared anew, others disappeared, but there is always signal coming from around disk center and visible around the blue wing, and signal coming from the limbs and visible around the red wing, but on the correct side of the boundary V∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' The big change is that at this date, and contrary to the observations in 2015, there are no conspicuous signals on the wrong, red side of V∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' There are al- ways small amplitude signals, both in Q and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Because of their small amplitude, they can be comfortably assigned to dark sink- ing plasma, or perhaps a small error in the determination of V∗, an error of at most 10 km s−1, which is consistent with the rough arguments used in its determination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' However, there is no large peak visible, which casts doubt on the validity of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' These two observations of µ Cep show an expected signal between the velocity boundaries that, while changing, is always Article number, page 4 of 11 2015/09/06 Stokes Q StokesU off limb sources 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='0 front hemisphere 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='0015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='0010 Polarization Intensity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='0005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='0005 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='0010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='0 60 20 0 20 40 60 80 100-60 40 20 0 20 40 60 80 100 Wavelength(km/s) Wavelength (km/s)A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' López Ariste et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' : Raising plumes in µ Cep Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Observed linear polarization of µ Cep on October 21, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' The same color codes are used for Stokes Q and U as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' The velocity boundaries given by the dashed lines are common to all the observations of µ Cep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' It can be explained as it was for Betelgeuse: by a spatially inhomogeneous distribution of bright patterns that have been in- terpreted as convection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' However, these observations also show a new signal that appears and disappears in time, and that, if we accept the velocity boundaries, corresponds to bright plasma moving away from the observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' The conclusions drawn from these qualitative arguments are definitively confirmed by the inversion codes developed by López Ariste et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' (2018) and López Ariste et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' (2022): the model used to fit the observed polarized spectra of Betelgeuse and to infer the published images is unable to produce a solu- tion for the spectrum of µ Cep on September 2015, though it provides a solution for the observations of October 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Un- willing to discard a model that has been successful with Betel- geuse, we propose an addition to this model that can explain the intermittent appearance of strong signals on the red side of the red velocity boundary, which are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' We propose that the bright convective structures inferred for Betel- geuse and µ Cep and present over the whole star —and also in the back hemisphere—, may raise plasma high enough for it to become visible above the stellar limb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' We refer to this high ris- ing hot plasma as plumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' When these plumes are on the front hemisphere, they produce the signals between the two velocity boundaries and the basic model is able to explain them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Similar convective bright structures must also occur on the back hemi- sphere, but they are usually hidden by the stellar limb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' From time to time, one of these bright structures in the back hemi- sphere may push plasma high enough for it to become visible to us above the limb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' This plasma rises in a radial direction, but as it is in the back hemisphere of the star, we see it redshifted, mov- ing away from us beyond the red velocity boundary;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' it is bright plasma nevertheless, and so we expect it to have similar polariza- tion amplitudes to plasma in the front hemisphere in symmetric geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Plasma is not usually expected to rise high enough, and so we often expect to see nothing beyond V∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' This has been the case for all available observations of Betelgeuse and also for µ Cep in October 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' However, from time to time this may happen, producing the signal illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' When this is observed, it cannot happen all over the stellar limb, but only at particular polar angles, thus explaining the single peak that is visible only in Stokes Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Becoming visible over the limb depends on the distance to that limb of the bright structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' The further a structure is from the limb, the higher it has to rise to become visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' This suggests that we can determine the height of one of those structures as the minimum height at which, by geometry, they become visible above the limb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' This measurement of a minimum height for the rising plasma to become visible is going to be our main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Inversions with a modified model In accordance with the suggested modification of the model pro- posed in the previous section upon inspection of those polariza- tion signals beyond the red velocity boundary, we built an inver- sion code to fit the observed spectra of µ Cep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' The core of this inversion code is identical to the one described by López Ariste et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Mathematically, it is a Marquardt-Levemberg al- gorithm that fits the observed Stokes Q and U profiles with syn- thetic profiles computed from a distribution of brightness over the surface of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' On the front hemisphere, this distribu- tion of brightness is described by a linear combination of spher- ical harmonics up to sixth order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' The blue velocity boundary is the maximum velocity of the rising plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' The brightest point over the disk at any particular realization of the model is supposed to move radially at that maximum velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' All other points over the disk have a brightness described by the spher- ical harmonics, and a velocity which is mathematically related to its brightness, meaning that the resulting brightness contrast and velocities roughly match the solar case (see the Appendix in López Ariste et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' The polarization emitted by a point over the hemisphere is proportional to that brightness, but also to the squared sine of the scattering angle, as expected for Rayleigh scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' The ratio of polarizations between Q and U is given by the tangent of half the polar angle position of the point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Its wavelength is determined by its velocity projected onto the line of sight and therefore depends on the distance to the center of the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Mathematically, the model uses, as parameters, the coeffi- cients of a brightness distribution written in terms of spherical Article number, page 5 of 11 2016/10/21 Stokes Q Stokes U off limb sources 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='0 front hemisphere 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='0010 Ft 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='0005 Intensity Polarization 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='0005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='0010 -- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='0015 60 40 20 20 40 60 80 100-60 40 20 20 40 60 80 100 Wavelength(km/s) Wavelength (km/s)A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' art72_final harmonics as B(µ, χ) = ���������� � ℓ=0,ℓmax m=−ℓ,+ℓ am ℓ ym ℓ (µ, χ) ���������� , (1) with ℓmax = 6, and µ and χ being the angle to disk center and the polar angle with respect to celestial north, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' This brightness distribution results in the emission of net polarization described in terms of Stokes parameters as Qdisk(v) = � µ,χ,vz B(µ, χ) sin2 µ cos 2χe−(v−vz)2/σ2, (2) Udisk(v) = � µ,χ,vz B(µ, χ) sin2 µ sin 2χe−(v−vz)2/σ2, (3) where vz = V(µ, χ) cos µ, with V(µ, χ) being the plasma veloc- ity at that point, and proportional to the brightness and lim- ited by the maximum speed of the plasma Vp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Each emission is broadened with a Gaussian profile of fixed width σ = 6 km s−1(i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=', 10 km s−1FWHM), which represents both instrumen- tal and thermal broadenings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' This is a rough description of the basic model, for which many more details are given and scru- tinized by López Ariste et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' (2018) and López Ariste et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' In addition to this basic model, we assume the pres- ence of one or several sources of polarization beyond the limb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' When adding new parameters to the model to describe those new sources of polarization, one should be careful not to overload the inversion algorithm with more new unknowns than available new information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Therefore, it would be unreasonable to try to pro- vide a description of the continuous distribution of brightness in the back hemisphere, because only a very limited amount of that plasma will be contributing to the observed spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' It is tempting to try to propose a description of the brightness in a ring above the limb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Unfortunately, we have not found a proper mathematical description for such a ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' One of the difficulties is that, as we assume that it is uncommon for plumes to rise high enough to become visible above the limb, we are expecting con- tributions from at most a small range of polar angles, the rest contributing zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Any orthogonal family of functions trying to describe this paucity of sources requires an excessive number of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' We finally opted for a simplistic description in terms of a small number of discrete sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Each one of these discrete sources over the limb is described by its polar angle χ, its angular distance to the limb θ, and a brightness value Z (see cartoon in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Its polarization is given, as in the case of any other emitting point in the front hemisphere, by the scaled prod- uct of its brightness and the squared sine of the scattering angle, this scattering angle being geometrically related to the distance to the limb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Qoff (v) = � i=0,N Zi sin2 θi cos 2χie−(v−vz)2/σ2, (4) Uoff (v) = � i=0,N Zi sin2 θi sin 2χie−(v−vz)2/σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' (5) The radial velocity of the rising plasma is identically given as a function of brightness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Its redshifted wavelength is analo- gously given by the projection of this velocity onto the line of sight, a projection which is again geometrically dependent on the distance to the limb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Each one of these discrete over-the-limb sources produces a polarization peak in Stokes Q and U that is broadened by a Gaussian profile with a full width at half maxi- mum (FWHM) of 10 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' This FWHM is supposed to encom- pass both the instrumental resolution and various stellar broad- ening mechanisms, that is, thermal, microturbulent, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Cartoon defining the parameters of a discrete source (yellow sphere) in the back hemisphere (in gray) beyond the plane of the sky (bluish plane).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Celestial north is up, in the plane of the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' The image of the front hemisphere corresponds to the inferred brightness distribution of µ Cep in September, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Finally, both sources of polarization, Qdisk, Udisk, and Qoff , Uoff are added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' The last parameter to be determined is the number N of dis- crete sources to be allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' We find it unpractical to leave this number unbound, and prefer to fix it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' We attempted from N = 0 up to N = 5 discrete sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' As expected, having zero sources allows us to recover the basic model, unable to reproduce the anomalous signals on the red wing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' On the other end, we find that beyond four sources we are not learning anything new from the inversion results, and the algorithm becomes unstable, and presents convergence issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' This can be safely understood as an excessive number of new parameters given the available in- formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Between one and four sources is therefore the right number of sources that we can safely infer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Interestingly, we also find that for any individual observation of our long dataset, the inferred value of the polar angle of all the sources is similar, even if the intensity and height of each one of them is different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' This means that the solution found by the inversion algorithm proposes that, at a given polar angle, there are several sources of polarization at different heights and with different intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' That is, the inferred sources clump together on the same region above the limb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' This can be interpreted as one single but ex- tended source over the limb of µ Cep at the time of the observa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' This result appears to justify our intuition that such events of high-rising plasma are not common.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' From this conclusion, one may expect one single source in our model to be sufficient to describe the observed polarization profiles in the red wing, but we find that this is not the case and that we need a minimum of two sources to reproduce the basic spectral features observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' This may be an indication that even if there is a unique object beyond the limb, it has sufficient structure that our description in terms of a Gaussian profile per source is inadequate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Using two or more sources becomes a simple manner of better describing Article number, page 6 of 11 Xi theta SunA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' López Ariste et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' : Raising plumes in µ Cep the extent and structure of the emitting region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Because of this result, we present inversions in this work with just two sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' This has the advantage of capturing the important physical pa- rameter for our work, the main distance of the bright structure to the limb, while easing the constraints on the inversion algo- rithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' The observed structure is often spectrally broader than twice the FWHM of 10 km s−1of every discrete source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' The fit is therefore approximative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' By increasing the number of sources, we improve this fit, but do not bring any further information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' The above developments are illustrated in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Both figures show on top of the observed profiles the solution found by the inversion code as a green continuous line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' This solution is made of three different contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' The basic model describ- ing the front hemisphere as a linear combination of spherical harmonics is plotted in red, and is fully coincident with, and hid- den behind, the full solution between the two dashed lines that limit the contribution of the front hemisphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' This basic model can only be seen as a tail of small signal on the red side of the red velocity boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' This small signal, as mentioned above, is the contribution from the dark sinking plasma, and is insufficient to explain the observed polarization peak in September 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' How- ever, it is almost sufficient to explain the entirety of the redshifted signal in October 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' The two other contributions combined are shown as a black continuous curve, and correspond to two discrete sources above the limb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Again, this black line is only visible when it does not fully coincide with the final solution plotted in green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' As explained, limiting the number of sources to just two results in an approximative fit of the redshifted signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' The full solution profile clearly shows two peaks on the red wing, coinciding with the maxima of the two sources, a feature ab- sent in the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' There is also a clear tail further towards the red in the observations that cannot be captured with just two sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Adding more sources would correct these missed fits, but the parameters of the added sources will not change signifi- cantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' In September 2015, the two sources over the limb bring signal comparable to anything else over the front disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' In Octo- ber 2016, the two sources appear as small contributions that may drop to zero if just the red velocity boundary is shifted a few km s−1 towards the red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' The modified model is therefore able to capture both those cases with important sources over the limb as well as those cases with negligible contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' We used this model, in conjunction with two sources above the limb, to invert the whole available dataset of linearly polar- ized spectra of µ Cep presented in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Imaging from lin- ear spectropolarimetry is subject to a certain number of ambi- guities: Several images, with different distributions of brightness are compatible with the same observed polarized spectra, that is, they are possible alternative solutions of the inversion problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' These latter images are not completely unrelated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' The most com- mon ambiguity appears between two images that are identical but rotated 180 degrees with respect to one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' A compar- ison with images of Betelgeuse made with interferometric tech- niques allows us to determine which of these two rotated images is the one that better corresponds to reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' However, we do not have interferometric images for all dates, and none for µ Cep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Because of this, in the case of Betelgeuse, the best solution for a date with available interferometric images is propagated as the initial solution to the next date, which encourages the inversion code to stay in the group of solutions sharing choices among the possible ambiguities that better compared to interferometric im- ages at one particular date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Similarly, for µ Cep, we inverted the first available date without constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' But for next dates, the so- lution of the previous available date was used as initial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' This ensures a certain time coherence in the series of images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' The inversion code provides values for the polar angle and distance to the limb for the two sources over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' The distance to the limb θ is directly converted into the minimum height h above the stellar surface for this source in the back hemisphere to be visible above the limb: h = 1 − cos θ cos θ R∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' (6) Presented in this manner, the results of our inversions are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' 5 Over the last six years, µ Cep appears to have produced three events in the back hemisphere with plasma being lifted to con- siderable heights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' The first of these events was ongoing when our observations started in September 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' It had completely disappeared when the star was re-observed in late spring 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' The next event started one year later, between January and April 2017, and by January 2018 plasma had reached heights of at least 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='1R∗ and perhaps higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' This plasma appeared at polar angles of 100 degrees and after the winter blind window, the plasma was still at the same position and at even greater heights of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='15R∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Over the spring of 2019, the emitting plasma was seen to be de- scending in height, until it disappeared by the summer of that year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' The beginning of the observations with Neo-Narval at the beginning of 2020 showed µCep to be still quiet, with no partic- ular signals on the red wing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' But this situation changed by the end of the year, with the rapid rise of a new clump of plasma at a polar angle of 0 degrees —and therefore unrelated to the pre- vious one—, which in less than one month reached heights of at least 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='175R∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' The maximum height reached by this event ap- pears to be quite ephemeral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' As fast as it rises, it disappears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' But as it disappears, we are left with a low-lying clump that persists throughout 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Optimistically, we may interpret this as a large event of rising plasma inside of which there is a small clump at high speed reaching even higher heights in a short time before disappearing, perhaps due to a quick cooling, while the rest of the rising plasma is still visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' In all these events, the value of the polar angle of the two sources is quite similar, as can be seen in the right plot of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' As said above, we interpret this result as proof that there is a unique source above the limb but more extended and complex than what our model with two Gaussians can reproduce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Our 8 years of observations of linear polarization of Betel- geuse have not produced any single event sufficiently large to require a modification of the inversion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' In 5 years of ob- servations, µ Cep has produced three such events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' It is possi- ble that this is due to the slightly different stellar parameters of these two stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' The fundamental parameters of µ Cep recently determined by Montargès et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' (2019) show a star similar to Betelgeuse within error bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Rather than invoking fundamental differences between the two stars, we speculate that µ Cep may be at present in a Decin stage (Decin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' 2006), as suggested by Montargès et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' (2019), with common episodes of mass loss, while Betelgeuse may rather be in a quiet stage with rare and separated events of this kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' This is simply speculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' At this point, we lack any clear scenario explaining why and when a red supergiant will enter into a Decin stage, if such episodes exist at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Further observations in time will be needed to see whether or not µ Cep stops producing these events1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' In the fall of 2019, Betelgeuse suffered a large dimming that has been attributed to the formation of a dense dust cloud al- most directly along our line of sight (Montargès et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' 1 It must be said that Decin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' (2006) estimate the duration of such episodes in the tens of years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Article number, page 7 of 11 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' art72_final Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Plots of the height of the two sources visible above the limb and of their polar angle for the observations of µ Cep over the last six years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' For heights below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='05, the source is considered to be absent and the corresponding value of the polar angle is made transparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' López Ariste et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' (2022) suggested that these mass-loss events are triggered from fast-rising plasma in the photosphere reach- ing the escape velocity at a certain height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' The suggestion made by these latter authors stems from the measurement of plasma velocities that are constant with height and sufficiently large to be comparable to escape velocities at the estimated heights of these structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' It is tempting to see this event in Betelgeuse as one example of the more common events in µ Cep of plasma rising sufficiently high to be visible above the limb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' But in the case of Betelgeuse, this event happened in the front hemisphere, rather than in the back hemisphere as in µ Cep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' If we accept that Betelgeuse is at present in a quiet stage of mass loss, un- like µ Cep, events where plasma is ejected from the star appear to still happen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Just by chance, in Betelgeuse, lately, they have not been happening in the regions near the limb, but rather in the front disk, the ultimate example being the one that produced the large dust cloud involved in the great dimming of 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' In µ Cep on the other hand, three such events have taken place in regions around the limb, making it visible to our spectropolari- metric measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Follow-up of a convective plume above the limb Figure 6 shows a time series of spectropolarimetric observations of µ Cep starting on September 15, 2020, and ending on May 1, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' The first five observations during the fall and winter of 2020 show the rapid rise of a convective plume above the celes- tial north limb of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' This plume can easily be identified in the inferred heights shown on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Such behavior can also be seen directly in the profiles as a red peak with negative ampli- tude in Stokes Q that, day after day, shifts to redder and redder wavelengths, meaning that its projection over the line of sight is greater and greater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Our interpretation of this is that plasma be- yond the limb is rising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' First, the parts nearer to the limb become visible above the limb and, as time goes on, plasma farther and farther from the limb becomes visible as it reaches the height at which this is geometrically possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' The plume, which is cen- tered well beyond the limb, is rising over a period of 3 months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' We lose track of the star from January through April 2021, and in the first observation in May the structure has almost completely disappeared: the polarization beyond the red velocity boundary is small and centered very near the limit, as if only the regions closer to the limb were still emitting light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' The plume has disap- peared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' We have chosen this event to illustrate how the rise of the plume can be estimated from direct visual inspection of the pro- files, before the inversion code confirms the interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' The rise of the plume is quite fast, and similarly fast is its disappear- ance, as there is barely any signal of its presence at the opening of the observing window in the following spring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' It may be tempting to say that the plume fell back into the star, but we have no signature of this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' We must recall that, any plasma falling back into the back hemisphere would produce a blueshifted signal that would melt into the signals of rising plasma from the front hemisphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' We have no manner to disen- tangle to two origins of polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' The event of 2018 is better followed during its disappear- ance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' What we observe is that the signal is still highly visible in the red wing, meaning that the emitted plasma is still rising in the back hemisphere, but its height is lower and lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' We interpret this as follows: the upper parts of the plume of plasma, while still rising, stop emitting light in the atomic lines measured here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' This may be because as it cools down, its brightness diminishes, or because atomic lines are no longer excited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Translating our tech- nique to molecular lines, if feasible, would shed light on this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' In both cases, the top of the plume cools down first and stops emit- ting light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' We only measure light from the lower parts, which are still hot enough and still raising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' This process continues until only the lower parts of the plume are emitting measurable sig- nals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Therefore, at the end of these episodes, we do not see the plume falling, but just disappearing from our sensing window of atomic lines in what we interpret as a cooling down phase that starts from the top of the plume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Discussion on the height of the observed structures Looking back at Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' 5 we see that the structure followed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' 6 reached a minimum height of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='175R∗ during those 3 months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' This could be higher, because, using geometry, we can only give the lower bound of this height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Taking 1000 − 1200R⊙ as the radius of the star, this rise requires an average velocity of 15 Article number, page 8 of 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='175 Source 1 Polar angle from celestial North (°) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='150 Source 2 250 Height above limb (R+) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='125 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='100 150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='075 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='050 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='025 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='000 2016/01/01 2017/01/01 2018/01/01 2019/01/01 2020/01/01 2021/01/01 2022/01/01 2016/01/01 2017/01/01 2018/01/01 2019/01/01 2020/01/01 2021/01/01 2022/01/01A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' López Ariste et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' : Raising plumes in µ Cep Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Time series of spectropolarimetric observations of µ Cep corresponding to all dates from September 15, 2020, through May 1, 2021, showing the rise and fall of a convective plume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' The meaning of curve colors and styles is the same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' to 20 km s−1, maintained constant for 3 months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' This velocity fits comfortably with the velocity limit Vp = 70km s−1in µ Cep determined for the basic model of the front hemisphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' These are minimum velocities, as we can only determine minimum heights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' The possibility of detecting this kind of plume over the limb, as offered by µ Cep, is exceptional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' The observation of three such events in over five years may suggest that there is some abnor- mal convective activity in this star, at least when compared with Betelgeuse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Nevertheless, we stick to the assumption that both stars represent different cases of the same physics and that it is just the relatively short span of the observations available that explains the observed differences, and not any fundamental dif- ference between the behaviors of these two RSGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Building on this assumption, the measurement of a geometric height made on these structures is deemed typical of convective plasma fea- tures in RSGs, and we generalise it to all other structures im- aged on such objects with spectropolarimetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' We consider the value of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='1R∗ as the typical height of the plasma in the atmo- spheres of RSGs hot enough to emit atomic spectral lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Mak- ing the link with López Ariste et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' (2022), we consider that this measured geometric height, recovered from spectropolarimetry of the deepest atomic lines in the spectrum, must correspond typ- ically to the height of their uppermost layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Figure 9 of this lat- ter publication must extend therefore up to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='1R∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' It is only by considering that this upper layer is visible for several months at that height that in this latter work it is assumed that the observed structures may well have reached 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='3R∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Conclusion Spectropolarimetric observations of the RSG µ Cep show spec- tral features in linear polarization that were not observed in the better studied Betelgeuse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Such spectral features are much more redshifted than any other signal and are not permanent features: on certain dates, the observed spectra are qualitatively identical to those of Betelgeuse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' We argue that the origin of those unex- pected spectral features is convective plumes in the back hemi- sphere of the star that rise high enough to be visible above the stellar limb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' This hypothesis allows us to conserve the inversion algo- rithms and model that have successfully been applied to Betel- geuse and that have produced images comparable to those from interferometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' However, such a basic model must be extended to allow for temporary sources of polarized light above the stel- lar limb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' We produced an inversion algorithm using this extended model and successfully fitted the observed profiles, including the unexpected new features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' This model assumes the presence of a small number of discrete sources over the limb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Although about four such sources are required to correctly fit the profiles, we re- alize that, at any given date, all those sources appear to be com- bined to describe a unique but extended source on the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' This observation allowed us to reduce the number of sources over the limb to just two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' While the fit with just two sources is not as good as it would with four sources, we still capture the main parameters of the sources and stabilize the convergence of the code, which can be automatically launched to handle the whole dataset available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' The inversion results produce the polar angle position and the height of those sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' This gives us access, for the first Article number, page 9 of 11 2021/01/13 Stokes Q Stokes U off limb sources 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='0 front hemisphere 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='0015 Fit 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='8 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Three events of plasma rising over the limb have been ob- served during the six years of observation of µ Cep with Nar- val and Neo-Narval at the TBL at Pic du Midi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Two of those events were tracked during their rapid rising phase and into their disappearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' The characteristic heights reach 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='1R∗ and even 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='175R∗ in the last observed event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' We consider this a typical value of the heights of the convective structures observed in the photosphere of RSGs, and López Ariste et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' (2022) use this value to calculate a geometric height from the three-dimensional images of Betelgeuse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Thanks to this measurement, these authors demonstrated that the measured velocities in the plasma are very near the escape velocity of Betelgeuse and that this rising plasma is likely a contributor to the mass loss of these stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' This work was supported by the "Programme National de Physique Stellaire" (PNPS) of CNRS/INSU co-funded by CEA and CNES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' acknowledges support under the Erasmus+ EU program for doctoral mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' acknowledges partial support by the Bulgarian NSF project DN 18/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' References Aurière, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=', López Ariste, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=', Mathias, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=', et al.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=', Morin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' 2017, Astronomy and Astrophysics, 603, A129 Article number, page 10 of 11 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' López Ariste et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' : Raising plumes in µ Cep Appendix A: Log of Observations Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Log of Narval and Neo-Narval observations of µ Cep and polarimetric measurements since July 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Date Julian date Stokes Sequence July 10, 2015 7214.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='578 8U+8Q September 05, 2015 7271.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='504 2U+2Q November 10, 2015 7337.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='462 2Q+2U May 16, 2016 7525.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='609 2Q+2U June 08, 2016 7548.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='628 2U June 16, 2016 7556.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='604 2Q June 21, 2016 7561.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='586 2U June 27, 2016 7567.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='569 2Q July 05, 2016 7575.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='566 2Q+2U July 15, 2016 7585.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='442 2Q+2U November 14, 2018 8437.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='322 2Q+2U December 10, 2018 8463.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='313 2Q+2U January 04, 2019 8488.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='239 2Q+2U January 16, 2019 8500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='281 2Q+2U March 21, 2019 8564.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='685 2Q+2U May 05, 2019 8609.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='594 2Q+2U June 01, 2019 8636.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='633 2Q+2U June 18, 2019 8653.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='622 2Q+2U July 18, 2019 8683.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='555 2Q+2U August 02, 2019 8698.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='589 2Q+2U August 15, 2019 8711.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='498 2Q+2U August 30, 2019 8726.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='552 2Q+2U January 06, 2020 8855.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='288 2Q+2U May 17, 2020 8987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='572 1Q+2U June 22, 2020 9023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='621 2Q+2U July 05, 2020 9036.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='583 2Q+2U July 24, 2020 9055.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='576 2Q+2U August 22, 2020 9084.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='518 2Q+2U September 15, 2020 9108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='483 2Q+2U October 16, 2020 9139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='313 2U+2Q November 21, 2020 9175.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='329 2Q+2U December 18, 2020 9202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='256 2Q+2U January 13, 2021 9228.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='271 2Q+2U May 01, 2021 9337.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='647 2Q+2U May 26, 2021 9361.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='603 2Q+2U June 14, 2021 9380.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='536 2Q+2U July 10, 2021 9406.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='619 2Q+2U August 07, 2021 9434.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='455 2Q+2U August 19, 2021 9446.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='59 2Q+2U September 04, 2021 9462.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='43 2Q+2U October 06, 2021 9494.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='441 2Q+2U October 13, 2021 9501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='405 2Q+2U November 09, 2021 9528.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='363 2Q+2U December 13, 2021 9562.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='279 2Q+2U December 22, 2021 9571.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='252 2Q+2U January 11, 2022 9591.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='264 2Q+2U Notes: Columns give the date, the heliocentric Julian date (+2 450 000), and the observed Stokes sequence, that is, how many observations of which Stokes parameter were made at that date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' An observation consists of four exposures with changing polarimetric modulation that, after reduction, produce polariza- tion spectra of either Stokes Q or U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content=' Beyond a 2 year pro- prietary embargo, all data are publicly available at PolarBase (http://polarbase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAzT4oBgHgl3EQfX_ww/content/2301.01326v1.pdf'} +page_content='irap.' 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their ease of interpretation, these weights +are not faithful to the models’ decisions as they +are only one part of an encoder, and other com- +ponents in the encoder layer can have consider- +able impact on information mixing in the out- +put representations. In this work, by expand- +ing the scope of analysis to the whole encoder +block, we propose Value Zeroing, a novel con- +text mixing score customized for Transformers +that provides us with a deeper understanding +of how information is mixed at each encoder +layer. We demonstrate the superiority of our +context mixing score over other analysis meth- +ods through a series of complementary evalu- +ations with different viewpoints based on lin- +guistically informed rationales, probing, and +faithfulness analysis.1 +1 +Introduction +Transformers (Vaswani et al., 2017), with their im- +pressive empirical success, have become a prime +choice of architecture to learn contextualized repre- +sentations across a wide range of modalities, such +as language (Devlin et al., 2019; Brown et al., +2020), vision (Dosovitskiy et al., 2021), vision- +language (Radford et al., 2021; Rombach et al., +2022), and speech (Baevski et al., 2020), mainly +due to their ability to utilize pairwise interactions +between input tokens at every timestep. +To better understand the inner dynamics of +Transformers, we need to trace the information +flow from the input embeddings up to the output +representation (including quantifying the degree +of context mixing, which we will define below). +The attention weights from the multi-head atten- +tion mechanisms offer a straightforward starting +1Code is freely available at https://github.com/ +hmohebbi/ValueZeroing +point for understanding this flow, and these weights +(‘raw attention’) have been used in many studies +(Clark et al., 2019; Kovaleva et al., 2019; Reif et al., +2019; Htut et al., 2019a, inter alia). However, the +reliability and usefulness of raw attention weights +has also been questioned (Jain and Wallace, 2019; +Bibal et al., 2022). In particular, attention weights +tend to concentrate on uninformative tokens in the +context (Voita et al., 2018; Clark et al., 2019), and +removing or altering them may lead to the same +and sometimes even better model performance on +downstream tasks (Jain and Wallace, 2019; Toneva +and Wehbe, 2019; Hassid et al., 2022). These find- +ings suggest that Transformers do not solely rely on +self-attention, and other components in the encoder +block play an essential role in information mixing. +A number of methods have been proposed to +compute some form of ‘effective attention weights’, +with the goal of more faithfully tracing the relative +contributions of different input tokens at various +layers of the Transformer (as we will discuss in +Section 2). These methods show an improvement +over raw attention, but still ignore key components +of the Transformer encoder block. This is a partic- +ularly crucial shortcoming, given that most of the +parameter budget in a Transformer encoder is spent +on position-wise feed-forward networks outside +of the self-attention component, which can have a +considerable impact on the degree of information +mixing in the output representations. +In this paper, we focus on context mixing: the +property of Transformers that in each node, at each +layer, information from the context can be incor- +porated into the representation of the target token. +We propose Value Zeroing, a novel approach to +quantify the contribution each context token has in +determining the final representation of a target to- +ken, at each layer of a Transformer. Value Zeroing +is based on the Explaining-by-Removing intuition +(Covert et al., 2021) shared by many posthoc in- +terpretability methods, but it takes advantage of +arXiv:2301.12971v1 [cs.CL] 30 Jan 2023 + +a specific feature of Transformers: it zeroes only +the value vector of a token t when computing its +importance, but leaves the key and query vectors +(and thus the pattern of information flow) intact. +Based on extensive experiments and three com- +plementary approaches to evaluation, we demon- +strate that importance scores we can obtain with +Value Zeroing provide better interpretations than +other analysis methods. +Firstly, we use a set of grammatical agreement +tasks from the BLiMP corpus (Warstadt et al., +2020) as a case study. Transformer-based mod- +els do extremely well on the task of distinguishing +grammatical from ungrammatical sentences, and +the BLiMP corpus provides information on the cue +words that determine the difference. We find that +Value Zeroing, unlike earlier approaches, indeed re- +veals that Transformers make use of relevant cues. +Secondly, we use information-theoretic probing +(Voita and Titov, 2020; Pimentel et al., 2020) as an +independent approach to track information flow in +Transformer networks. The scores we obtain with +Value Zeroing turn out to be highly correlated with +layer-wise probing performance; that is, probing +accuracy is higher in layers where relevant tokens +are more effectively utilized by the model. +Thirdly, we assess the faithfulness of our method +(Jacovi and Goldberg, 2020); compared to alterna- +tive analysis methods, we show that Value Zeroing +is not only more plausible and human-interpretable, +but also more faithful to models’ decisions. +2 +Related Work +While numerous studies have leveraged the weights +assigned by the self-attention mechanism to gain +intuition about the information mixing process in +Transformers (Clark et al., 2019; Kovaleva et al., +2019; Reif et al., 2019; Lin et al., 2019; Htut et al., +2019b; Raganato and Tiedemann, 2018), it is still +a matter of debate whether attention weights are +suitable for interpreting the model (see Bibal et al. +(2022)’s study for a full discussion). Thus several +post-processing interpretability techniques have +been proposed to convert these weights into scores +that provide a more detailed interpretation of the +inner workings of Transformers. We review the +main approaches below. +Abnar +and +Zuidema +(2020) +propose +the +attention-flow and attention-rollout methods to ap- +proximate information flow in Transformers based +on raw attention weights. The former treats raw +attention weight matrices as a flow network and +returns the maximum flow through each input to- +ken. The latter recursively multiplies the attention +weight matrix at each layer by the preceding ones. +There is, however, an unjustified assumption in +the formulation of these methods that both multi- +head attention and residual connections contribute +equally to the computation of the output. +Kobayashi et al. (2020) propose a method that +incorporates the norm of the transformed value vec- +tors and report a negative correlation between these +norms and raw attention weights on frequent to- +kens, which partially explains the insufficiency of +raw attention weights for context mixing estima- +tion. Kobayashi et al. (2021) extend this method +to the whole self-attention block by incorporating +Residual connections (RES) and Layer Normaliza- +tion (LN) (two components with significant impact +on both model performance and training conver- +gence (Parisotto et al., 2020; Liu et al., 2020)), but +demonstrate that RES and LN components largely +cancel out the mixing process. Kobayashi et al. +(2021)’s method, however, ignores the effect of the +second sublayer in a Transformer’s encoder. +Brunner et al. (2020) and Pascual et al. (2021) +employ a gradient-based approach for analyzing +the interaction of input representations, but the gra- +dient measures the sensitivity between two vectors +and ignores the impact of the input vector. In our +experiments we show that despite their relative suc- +cess in explaining model decisions, gradient-based +approaches are not suitable for layer-wise analysis. +More recently, the effectiveness of combining +these approaches has also been investigated. Us- +ing the rollout method (Abnar and Zuidema, 2020), +Modarressi et al. (2022) aggregated Kobayashi et al. +(2021)’s scores across the layers to provide global +token attributions. +In the same vein, Ferrando +et al. (2022) used rollout to aggregate a variant +of Kobayashi et al. (2021)’s scores: instead of re- +lying on the Euclidean norm of the transformed +vector, they measured Manhattan distance of each +transformed vector to the context vector outputted +from self-attention block. In both studies, however, +the fact that these context vectors might undergo +significant changes after passing through the sec- +ond sublayer in the encoder layer is not taken into +account. We will show (in Section 7) that even at +a global level, when scores are aggregated across +layers of a model, our scores provide better inter- +pretation than prior methods. + +3 +Our Proposed Method +To remedy for the limited scope of the existing +methods, we introduce a new context mixing score +that takes into account all components in a Trans- +former encoder block. +3.1 +Background and Notation +In this section, we set up the notation and briefly +review the internal structure of an encoder layer in +the Transformer architecture. +Each Transformer encoder layer is composed of +two sublayers: a multi-head self-attention mecha- +nism (MHA) and a position-wise fully connected +feed-forward network (FFN), followed by a Resid- +ual connection (RES) and Layer Normalization +(LN) around each of these two sublayers. This +encoder layer produces the next contextualized rep- +resentations (˜x1, ..., ˜xn) for each token in the con- +text, using the output representations from the pre- +vious layer (x1, ..., xn). +MHA. +For each head h ∈ {1, ..., H} in the self- +attention module, each input vector xi is trans- +formed into a query qh +i , a key kh +i , and a value vh +i +vector via separate trainable linear transformations: +qh +i = xiW h +Q + bh +Q +(1) +kh +i = xiW h +K + bh +K +(2) +vh +i = xiW h +V + bh +V +(3) +The context vector zh +i for the ith token of each +attention head is then generated as a weighted sum +over the transformed value vectors: +zh +i = +n +� +j=1 +αh +i,jvh +j +(4) +where αh +i,j is the raw attention weight assigned +to the jth token, and computed as a softmax- +normalized dot product between the corresponding +query and key vectors: +αi,j = softmax +xj∈X +� +qik⊤ +j +√ +d +� +∈ R +(5) +Next, the context vector (zi ∈ Rd) for the ith to- +ken is computed by concatenating all the heads’ +outputs followed by a head-mixing WO projection +and layer normalization: +zi = CONCAT(z1 +i , ..., zH +i )WO +(6) +zi = LNMHA(zi + xi) +(7) +FFN. +Each encoder layer also includes two linear +transformations with a ReLU activation in between, +which is applied to every zi separately and identi- +cally to produce output token representations ˜xi: +˜xi = max(0, ziW1 + b1)W2 + b2 +(8) +˜xi = LNFFN(˜xi + zi) +(9) +3.2 +Value Zeroing +We aim to measure how much a token uses other +context tokens to build its output representation +˜xi at each encoder layer. To this end, we treat +the self-attention mechanism as a fuzzy hash-table, +where we look up the sum of values weighted by +the query-key match in the context. Thus in Eq. 4 +we replace a value vector associated with token +j with a zero vector vh +j ← 0, ∀h ∈ H, where the +context vector for the ith token is being computed. +This provides an alternative output representation +˜x¬j +i +for the ith token that has excluded token j in +the mixing process. By comparing the alternative +output representation ˜x¬j +i +with the original ˜xi, we +can measure how much the output representation +is affected by the exclusion of the jth token: +Ci,j = ˜x¬j +i +∗ ˜xi +(10) +where the operation ∗ can be any pairwise distance +metric that properly considers the characteristics +of the model’s representation space. We opted +for cosine distance throughout our experiments as +its superiority over other dissimilarity metrics has +been supported for textual deep learning models +(Yokoi et al., 2020; Hanawa et al., 2021).2 Com- +puting Eq. 10 for all ˜xi in a given context provides +us with a Value Zeroing matrix score C where the +value of the cell Ci,j ∈ R (ith row, jth column) in +the map denotes the degree to which the ith token +depends on the jth token to form its contextualized +representation. +Note that unlike generic perturbation approaches, +our proposed method does not remove the token +representations xi from the input of an encoder. We +argue that ablating input token representations can- +not be a reliable basis to understand context mix- +ing process since any changes in the input vectors +will lead to changes in the query and key vectors +(Eq. 1 and 2), resulting in a change in the attention +2More details on the choice of distance metric is discussed +in Appendix A.1. + +Phenomenon +UID +Example +Target word +Foil word +Anaphor Number Agreement +ana +Many teenagers were helping [MASK]. +themselves +herself +Determiner-Noun Agreement +dna +Jeffrey has not passed [MASK] museums. +these +this +dnaa +Sara noticed [MASK] white hospitals. +these +this +Subject-Verb Agreement +darn +The pictures of Martha [MASK] not disgust Anne. +do +does +rpsv +Kristen [MASK] fixed this chair. +has +have +Table 1: Examples of the selected tasks with our annotations from the BLiMP benchmark (UIDs are unique +identifiers used in BLiMP). Cue words are underlined. +distribution (cf. Eq. 5). Consequently, there will +be a discrepancy between the alternative attention +weights and those for the original context. Instead, +our method only nullifies the value vector of a spe- +cific token representation. In this way, the token +representation can maintain its identity within the +encoder layer, but it does not contribute to form- +ing other token representations. Moreover, since +our Value Zeroing is computed from the encoder’s +layer outputs, it incorporates all the components in- +side an encoder layer such as multi-head attention, +residual connection, layer normalization, and also +feed-forward networks, resulting in a more reliable +context mixing score than previous methods. +4 +Experimental Setup +4.1 +Data +We used the BLiMP benchmark (Warstadt et al., +2020) which contains a set of pairs of minimally +different sentences that contrast in grammatical ac- +ceptability under a specific linguistic phenomenon. +The benchmark isolates linguistic phenomena such +that only one word determines the true label of +each sentence. We refer to this crucial context to- +ken as the cue word. The nature of this task makes +it especially suitable for evaluating context mix- +ing scores, since it gives us a strong hypothesis on +which context token is the most relevant for the +representation of the masked target word. +From this benchmark, we select five datasets +with three different linguistic phenomena for which +Pre-trained Language Models (PLMs) have shown +high accuracy to ensure that the model captures the +relevant information. We expand contractions such +as doesn’t → does not) and generate dependency +trees using SpaCy (Honnibal and Montani, 2017) +to extract and annotate the position of target and +cue words in a sentence. In Table 1, we provide +an example of each phenomenon in the benchmark +together with our automated annotations. We accu- +mulate examples from the five selected tasks as a +unified dataset for grammatical agreement, result- +ing in 4,276 data points, and divide them equally +into Train and Test sets. The Train set is only used +for the fine-tuning phase; the Test set is used for all +evaluation experiments. +4.2 +Target Model +We conduct our experiments on three Transformer- +based language models: BERT (uncased, Devlin +et al., 2019), RoBERTa (Liu et al., 2019) and +ELECTRA (Clark et al., 2020).3 The results for +the latter two are reported in Appendix A.3. By +replacing the target words with the [MASK] token, +we perform a Masked Language Modeling (MLM) +task using the model’s pre-trained MLM head. For +instance, in the Subject-Verb Agreement example +“The pictures of Martha do not disgust Anne.”, we +replace the verb ‘do’ with the [MASK] token and +feed the example to the model. +We perform our experiments on both pre-trained +and fine-tuned versions of each model. Including +a fine-tuned model in our analysis study gives us +a complementary insight into the importance of +the cue words, since fine-tuning allows the model +to concentrate on the most helpful words for the +downstream task of choice (i.e., agreement) and +makes sure that target word representations take the +cue word into account. We use prompt fine-tuning +(Schick and Schütze, 2021a,b; Karimi Mahabadi +et al., 2022) and compute Cross Entropy loss only +over the output logits corresponding to the target +and foil classes. Accuracy is 0.96 for pre-trained +and 0.99 for fine-tuned BERT. +4.3 +Baselines +Here we describe the existing context mixing meth- +ods which we include in our experiments. For each +method, we select the mth row of its context mixing +map where m is the position of the [MASK] token, +3Base, with 12 layers and 12 attention heads, obtained +from the Transformers library (Wolf et al., 2020). + +resulting in a 1-D array of scores for each context +token. We normalize the scores for all tokens in +a sentence so that they are all positive values and +sum to one. For completeness, we also include +a few gradient-based attribution methods in our +comparisons. +Rand: random scores generated from a uniform +distribution for each sentence in the dataset. +Attn: raw attention scores αm from Eq. 5. +Attn-rollout & Attn-flow: two methods for ap- +proximating the attention flow based on raw atten- +tion weights (Abnar and Zuidema, 2020). +Attn-Norm: norm-based method of Kobayashi +et al. (2020) that also incorporates the norm of +the transformed input vectors to compute context +mixing scores. +Attn-Norm + RES + LN: the extended norm- +based method of Kobayashi et al. (2021) in which +they also incorporates Residual connection (RES) +and Layer Normalization (LN) located only in the +first sublayer of a Transformer’s encoder. +ALTI: Aggregation of Layer-wise Token-to-token +Interactions, proposed by Ferrando et al. (2022). +GradXinput, IG & DL: feature attribution scores +that also make use of top-down information +from the classification layer on top of the Trans- +former. +We consider three popular variants of +gradient-based attribution scores: Gradient×Input +(GradXinput) (Samek et al., 2019; Yuan et al., +2019), Integrated Gradients (IG) (Sundararajan +et al., 2017), and DeepLift (DL) (Shrikumar et al., +2017). For gradient-based methods, we use the +pre-trained MLM head which has been trained dur- +ing the pre-training of the BERT to compute the +gradient of the true label with respect to the token +representations at each layer. +5 +Evaluation 1: Cue Alignment +As cue words are the only indicators of the true la- +bels in our dataset, we expect that when the model +performs well, it overwhelmingly depends on these +words to form the representation of a [MASK] to- +ken in a given context. To quantify the alignment +between a context mixing score and the cue word, +we first define a binary cue vector ξ according to +the following condition: +ξi = +� +1, +the ith token ∈ Cue words +0, +otherwise +(11) +Then we compare the cue vector and the prediction +of a context mixing score S in two different ways: +Dot Product. +We quantify cue alignment as S·ξ, +which measures the total score mass the model +assigns to cue words to form the representation of +the target token. +Average Precision. +We quantify cue alignment +as the average precision between the two vectors, +which is a weighted mean of precision at each recall +level: +AP = +� +n +(Rn − Rn−1)Pn +(12) +where Pn and Rn are the precision and recall at the +nth threshold. This metric relies on the ranking of +tokens rather than the magnitude of their weights.4 +Figures 1 and 2 show the alignment between +the cue vector and different analysis methods us- +ing dot product and average precision for the pre- +trained and fine-tuned model, respectively. In com- +parison with the other context mixing methods, +Value Zeroing shows a higher degree of the target +model incorporating cue words into the representa- +tion of the [MASK] token across all layers. +As can be seen from the first two columns in all +graphs, raw self-attention weights (Attn) always +perform worse than even random scores in high- +lighting cue words. This is in line with previous +studies showing that raw attention weights often +pay attention to uninformative tokens (Voita et al., +2018; Clark et al., 2019) and do not reflect the +appropriate context (Kim et al., 2019). The same +pattern holds for their aggregated versions (Attn- +rollout and Attn-flow), which are based solely on +attention weights. However, we can see a signif- +icant improvement in the results for Attn-norm +where the norm of transformed value vectors are +also taken into account, confirming that value vec- +tors play an essential role in the context mixing pro- +cess. The method Attn-norm + RES + LN, which +expands Attn-norm to the whole self-attention +block by adding Residual connection and Layer +Normalization, would seem to show that the model +is incapable of utilizing the cue words. However, +incorporating also the second part of the encoder +layer via our method shows the model does indeed +use the cue words. +4We also employed Probes-needed (Zhong et al., 2019) +metric in our evaluation which intuitively counts the number +of non-cue tokens we need to probe to find cue words based on +a given score. As its motivation is similar to Average Precision +and the results show the same pattern, we relegate the results +with this metric to the Appendix A.2. + +Rand +Attn +Attn-rollout +Attn-flow +Attn-norm +Attn-norm + RES +Attn-norm + RES + LN +ALTI +GradXinput +IG +DL +Value Zeroing +1 +2 +3 +4 +5 +6 +7 +8 +9 10 11 12 +Layer +0.12 0.06 0.03 0.06 0.11 0.07 0.04 0.13 0.19 0.12 0.19 0.10 +0.12 0.08 0.05 0.10 0.13 0.08 0.04 0.12 0.18 0.12 0.18 0.14 +0.12 0.09 0.06 0.12 0.15 0.08 0.05 0.12 0.17 0.09 0.16 0.19 +0.12 0.10 0.06 0.12 0.18 0.10 0.07 0.12 0.16 0.08 0.14 0.21 +0.12 0.10 0.06 0.12 0.19 0.10 0.08 0.12 0.15 0.06 0.13 0.24 +0.12 0.08 0.05 0.12 0.16 0.08 0.07 0.12 0.13 0.07 0.12 0.17 +0.12 0.09 0.05 0.12 0.17 0.09 0.07 0.12 0.14 0.09 0.11 0.17 +0.12 0.12 0.05 0.12 0.19 0.09 0.09 0.12 0.12 0.08 0.10 0.21 +0.12 0.12 0.04 0.12 0.22 0.10 0.09 0.12 0.10 0.06 0.09 0.28 +0.12 0.09 0.04 0.12 0.18 0.06 0.06 0.12 0.09 0.05 0.07 0.21 +0.13 0.09 0.04 0.12 0.19 0.06 0.06 0.12 0.05 0.03 0.03 0.26 +0.13 0.04 0.04 0.12 0.17 0.03 0.02 0.12 0.00 0.00 0.00 0.21 +Dot Product +Rand +Attn +Attn-rollout +Attn-flow +Attn-norm +Attn-norm + RES +Attn-norm + RES + LN +ALTI +GradXinput +IG +DL +Value Zeroing +1 +2 +3 +4 +5 +6 +7 +8 +9 10 11 12 +0.32 0.16 0.16 0.16 0.34 0.23 0.22 0.42 0.54 0.29 0.56 0.32 +0.32 0.18 0.17 0.20 0.39 0.25 0.24 0.15 0.52 0.26 0.53 0.36 +0.33 0.22 0.18 0.22 0.39 0.25 0.24 0.16 0.49 0.23 0.51 0.37 +0.31 0.21 0.16 0.22 0.51 0.31 0.31 0.16 0.45 0.20 0.36 0.48 +0.33 0.27 0.16 0.22 0.53 0.33 0.33 0.16 0.40 0.18 0.30 0.54 +0.32 0.25 0.16 0.22 0.41 0.28 0.28 0.16 0.34 0.18 0.27 0.41 +0.32 0.27 0.16 0.22 0.43 0.29 0.29 0.16 0.34 0.19 0.25 0.43 +0.32 0.29 0.16 0.22 0.46 0.30 0.30 0.16 0.28 0.19 0.23 0.47 +0.32 0.32 0.16 0.22 0.57 0.34 0.34 0.16 0.26 0.17 0.22 0.57 +0.31 0.28 0.16 0.22 0.45 0.29 0.29 0.16 0.29 0.17 0.21 0.45 +0.33 0.29 0.16 0.22 0.54 0.32 0.31 0.16 0.23 0.16 0.18 0.54 +0.33 0.24 0.16 0.22 0.43 0.27 0.27 0.16 0.12 0.12 0.12 0.44 +Average Precision +Figure 1: Layer-wise alignment between the cue vector and different analysis methods averaged over Test set +examples for the pre-trained model. Higher value (darker color) is better. +Rand +Attn +Attn-rollout +Attn-flow +Attn-norm +Attn-norm + RES +Attn-norm + RES + LN +ALTI +GradXinput +IG +DL +Value Zeroing +1 +2 +3 +4 +5 +6 +7 +8 +9 10 11 12 +Layer +0.12 0.06 0.03 0.06 0.11 0.07 0.04 0.13 0.18 0.12 0.18 0.10 +0.12 0.08 0.05 0.10 0.13 0.08 0.04 0.12 0.18 0.12 0.17 0.14 +0.12 0.08 0.06 0.12 0.15 0.08 0.05 0.12 0.17 0.09 0.16 0.19 +0.12 0.09 0.06 0.12 0.18 0.10 0.07 0.12 0.16 0.08 0.14 0.21 +0.12 0.09 0.06 0.12 0.19 0.10 0.08 0.12 0.15 0.06 0.13 0.25 +0.12 0.09 0.05 0.12 0.16 0.09 0.07 0.12 0.13 0.08 0.12 0.18 +0.12 0.10 0.05 0.12 0.18 0.09 0.08 0.12 0.14 0.10 0.12 0.19 +0.12 0.13 0.05 0.12 0.20 0.10 0.09 0.12 0.13 0.11 0.11 0.24 +0.12 0.13 0.04 0.12 0.25 0.12 0.11 0.12 0.10 0.07 0.09 0.35 +0.12 0.09 0.04 0.12 0.18 0.06 0.06 0.12 0.09 0.05 0.06 0.21 +0.13 0.07 0.04 0.12 0.19 0.05 0.05 0.12 0.05 0.03 0.04 0.25 +0.13 0.04 0.04 0.12 0.17 0.03 0.02 0.12 0.00 0.00 0.00 0.22 +Dot Product +Rand +Attn +Attn-rollout +Attn-flow +Attn-norm +Attn-norm + RES +Attn-norm + RES + LN +ALTI +GradXinput +IG +DL +Value Zeroing +1 +2 +3 +4 +5 +6 +7 +8 +9 10 11 12 +0.32 0.16 0.16 0.16 0.34 0.23 0.23 0.41 0.53 0.29 0.55 0.33 +0.32 0.18 0.17 0.20 0.39 0.25 0.24 0.15 0.51 0.27 0.54 0.36 +0.33 0.22 0.18 0.22 0.38 0.25 0.24 0.16 0.49 0.26 0.52 0.37 +0.31 0.21 0.16 0.22 0.50 0.31 0.31 0.16 0.45 0.23 0.35 0.48 +0.33 0.28 0.16 0.22 0.56 0.34 0.34 0.16 0.41 0.20 0.30 0.58 +0.32 0.25 0.16 0.22 0.42 0.28 0.28 0.16 0.34 0.20 0.28 0.43 +0.32 0.29 0.16 0.22 0.47 0.30 0.30 0.16 0.33 0.26 0.26 0.46 +0.32 0.30 0.16 0.22 0.52 0.32 0.32 0.16 0.30 0.25 0.25 0.53 +0.32 0.35 0.16 0.22 0.68 0.39 0.39 0.16 0.25 0.19 0.23 0.68 +0.31 0.29 0.16 0.22 0.45 0.29 0.29 0.16 0.31 0.18 0.22 0.45 +0.33 0.28 0.16 0.22 0.53 0.32 0.32 0.16 0.25 0.19 0.18 0.53 +0.33 0.24 0.16 0.22 0.45 0.28 0.28 0.16 0.12 0.12 0.12 0.46 +Average Precision +Figure 2: Layer-wise alignment between the cue vector and different analysis methods averaged over Test set +examples for the fine-tuned model. Higher value (darker color) is better. +The gradient-based scores, in contrast to the +other methods, highlight the cue words only in +the earlier layers of the model. In the next section, +by using a layer-wise probing experiment, we will +show that these scores are not reliable for identify- +ing the relevant context in individual layers. +6 +Evaluation 2: Context Mixing versus +Probing +In this section, we investigate the relationship be- +tween cue word alignment and probing perfor- +mance across layers. We hypothesize that if a layer +aligns better with the cue word according to a reli- +able context mixing score, then the representation +of the masked token on that layer can be used more +effectively by a probing classifier to decode number +agreement with the cue word. +To verify our hypothesis, we obtain the represen- +tation of masked tokens in test examples across all +layers. Since all examples in our dataset share +the same number agreement property, we asso- +ciate each masked representation with a Singular +or Plural label. Next, we perform an information- +theoretic probing analysis using Minimum De- +scription Length (MDL) to measures the degree to +which representations encode number agreement. +We chose MDL as our probe since it is theoreti- +cally justified and has been shown to provide more +reliable results than conventional probes (Voita and +Titov, 2020; Fayyaz et al., 2021). +To compute MDL, we employed the online cod- +ing of Voita and Titov (2020). Since MDL can be +affected by the number of data points (N), we mea- +sure compression as our evaluation metric which is + +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +layer +1.6 +2 +2.4 +2.8 +compression +finetuned +False +True +Figure 3: Layer-wise compression of probing classi- +fiers using pre-trained and fine-tuned representations. +defined as follows: +Compression = N · log2(K) +MDL +(13) +where K refers to the number of classes (2 in our +case). This metric is equal to 1 (no compression) +for a random guessing classifier. A higher value for +Compression indicates more accurate label predic- +tion for the probing classifier. +Figure 3 reports compression of probing classi- +fiers based on representations obtained from both +pre-trained and fine-tuned models across all lay- +ers. We also include the results for the embedding +layer of the model (layer 0) which can serve as a +non-contextualized baseline. We can see a jump +in probing performance at layers 4 (1.52 → 1.72) +and 9 (2.03 → 2.45) in the fine-tuned setup, the +same layers for which we found a higher alignment +with cue words in Figures 1 and 2. +Table 2 presents the correlation between layer- +wise Compression scores and layer-wise cue align- +ment scores from Section 5 for different analysis +methods. As we can see, alignment according to +Value Zeroing is highly positively correlated with +the probing performance. This suggests that when +Value Zeroing indicates that the model uses cue +words to form representations of the masked to- +kens in a particular layer, these representations are +in fact better at encoding number agreement. +Recall that based on Figures 1 and 2, according +to the gradient-based methods the masked tokens +pay more attention to the cue words only in earlier +layers. However, we can see a highly negative cor- +relation with probing results for these scores. Due +to the nature of the task the probing score goes up +monotonically along the layers. At the same time, +the gradient attribution score goes up monotoni- +cally as you get closer to the bottom embedding +layers, suggesting that gradient-based methods are +unreliable for layer-wise analysis and identifying +important tokens in the context mixing process. +Method +ρPT +ρFT +Rand +-0.07 +-0.02 +Attn +0.10 +0.08 +Attn-norm +0.52 +0.56 +Attn-norm + RES +-0.35 +-0.24 +Attn-norm + RES + LN +0.12 +0.17 +ALTI +-0.01 +-0.12 +GradXinput +-0.96 +-0.99 +IG +-0.86 +-0.77 +DL +-0.97 +-1.00 +Value Zeroing +0.65 +0.64 +Table 2: Spearman’s ρ correlation between layer-wise +probing performance (Comp.) +and layer-wise cue +alignment scores based on Dot Product. PT and FT +refer to pre-trained and fine-tuned conditions, respec- +tively. +7 +Evaluation 3: Faithfulness Analysis +Our experimental results in Sections 5 and 6 show +that the Value Zeroing score matches our prior +linguistically-informed expectations. However, it +is not always clear whether a plausible context mix- +ing score that matches human expectations is also +faithful to the model and reflects its decision mak- +ing process (Herman, 2017; Wiegreffe and Pinter, +2019; Jacovi and Goldberg, 2020). +In this section we employ the notion of input +ablation (Covert et al., 2021) to evaluate the faith- +fulness of our context mixing score. The influence +of a target token on a model’s decision is often +estimated as the drop in the model’s predicted prob- +ability of the correct class after blanking out the +target token from the input. A higher drop for an +ablated token indicates that the token is more in- +fluential on the model’s decision (DeYoung et al., +2020; Abnar and Zuidema, 2020; Atanasova et al., +2020; Wang et al., 2022). We use this blank-out +approach as a base for analyzing and comparing the +faithfulness of the existing context mixing scores +and our proposed one. +To estimate the blank-out scores in BERT, we +calculate the probability of its output y using a +softmax function normalized over only the corre- +sponding logit values of target t and foil words +(cf. Table 1), and compute blank-out scores for a +given input token i as p(yt|e) − p(yt|e\ei), where +ei refers to the input embedding of input token +i. We compare these blank-out scores with con- +text mixing scores, aggregated across all layers +of the model. For gradient-based scores, calculat- +ing them with respect to the tokens in the input + +Method +ρPT +ρFT +Rand +-0.01 +0.00 +Attn +-0.10 +-0.07 +Attn-norm +0.19 +0.14 +Attn-norm + RES +0.03 +-0.05 +Attn-norm + RES + LN +-0.08 +-0.17 +ALTI +-0.14 +-0.10 +GradXinput +0.11 +0.16 +IG +0.07 +0.21 +DL +0.20 +0.29 +Value Zeroing +0.26 +0.31 +Table 3: Spearman’s ρ correlation between the blank- +out scores and different aggregated context mixing and +attribution scores. PT and FT refer to pre-trained and +fine-tuned conditions, respectively. +embedding layer (ℓ = 0) provides us with aggre- +gated scores since the backpropagation of gradients +passes through all layers to the beginning of the +model. For other scores, we use the rollout (Abnar +and Zuidema, 2020) aggregation method. +Table 3 shows Spearman’s rank correlation be- +tween the blank-out scores and different aggregated +context mixing scores. The highest correlation for +our method indicates that Value Zeroing is more +faithful in explaining the model behaviour com- +pared to other analysis methods. +Qualitative Analysis. +We also take a closer look +at the aggregated scores for a qualitative compar- +ison. In Figure 4, we illustrate different scores +obtained from a fine-tuned BERT model for a cor- +rectly classified example, where the model is asked +to fill the masked token with one of the verbs were +or is as target and foil classes, respectively. Ac- +cording to Value Zeroing scores, the model mainly +relies on the main subject (pictures) as a cue +word to form a contextualized representation of +the [MASK] token, while the word pictures is also +important for the model’s final decision based on +the blank-out scores. In this example, the blank- +out score for the cue word is 0.99, meaning the +model fully loses its confidence in the target class +when the cue word is replaced with an [UNK] +token. Surprisingly, gradient-based methods tend +to highlight the word hat which is an agreement +attractor, and attention-based scores tend to focus +on the [CLS] token which has been idle during +fine-tuning process. +Overall, our faithfulness evaluation and qualita- +tive analysis suggest that Value Zeroing can explain +model decisions at a global level when it is aggre- +blank-out: +[CLS] the pictures of some hat [MASK] scar ##ing marcus . [SEP] +Attn: +[CLS] the pictures of some hat [MASK] scar ##ing marcus . [SEP] +Attn-norm: +[CLS] the pictures of some hat [MASK] scar ##ing marcus . [SEP] +Attn-norm+RES: +[CLS] the pictures of some hat [MASK] scar ##ing marcus . [SEP] +Attn-norm+RES+LN: [CLS] the pictures of some hat [MASK] scar ##ing marcus . [SEP] +ALTI: +[CLS] the pictures of some hat [MASK] scar ##ing marcus . [SEP] +GradXinput: +[CLS] the pictures of some hat [MASK] scar ##ing marcus . [SEP] +IG: +[CLS] the pictures of some hat [MASK] scar ##ing marcus . [SEP] +DL: +[CLS] the pictures of some hat [MASK] scar ##ing marcus . [SEP] +Value Zeroing: +[CLS] the pictures of some hat [MASK] scar ##ing marcus . [SEP] +Figure 4: Most influential tokens on the target repre- +sentation in a fine-tuned BERT model according to dif- +ferent aggregated context mixing scores compared to +blank-out scores. +gated across layers. The context mixing maps per +layer are provided in Appendix A.4, where some +more meaningful patterns can be found in Value +Zeroing scores (in both layer-wise and aggregated +setups) in contrast to other context mixing scores. +8 +Discussion +Although some desiderata such as plausibility (Lei +et al., 2016; Strout et al., 2019) and faithfulness +(Lakkaraju et al., 2019; Jacovi and Goldberg, 2020) +are taken into account when developing explana- +tion and analysis methods, evaluating them is still +a challenge due to lack of a standard ground truth. +Evaluating context mixing scores, where token-to- +token interactions in a context are also considered, +is even more challenging. Several studies have used +gradient-based scores as an anchor of faithfulness, +and measure how strongly context mixing scores +correlate with them (Jain and Wallace, 2019; Ab- +nar and Zuidema, 2020; Modarressi et al., 2022). +However, the reliability of gradient-based scores +can be questioned, especially when different vari- +ations of them show considerable disagreement +(Neely et al., 2022; Pruthi et al., 2022; Krishna +et al., 2022). Thus, we suggest using controlled +tasks for which we have strong prior expectations +for evaluating these methods. In our study, we use +a set of number agreement tasks to provide such +priors, since the cue words are the only sources of +information in the context for performing well in +the task. +Another point worth discussing is the concern +raised by Kobayashi et al. (2021) that BERT tends +to preserve token representations rather than mix- +ing them at each layer. We argue that their ob- +servation is due to the context-mixing ratio they + +defined by comparing the norm of residual effects +against other token representations. In our view, +this ratio is dominated by residuals and neglects the +fact that a token representation carried by residual +connections is indeed a contextualized representa- +tion outputted from previous layers. We keep the +residuals intact within the encoder layer by zeroing +only the value vectors and focusing on the context +mixing performed by all tokens. +9 +Conclusion +In this paper, we propose Value Zeroing as a novel +approach for quantifying the information mixing +process in Transformers to address the shortcom- +ings of previous methods. We performed exten- +sive complementary experiments and showed that +our method outperforms others in three different +evaluation setups. Since our approach requires no +supervision, it could be an interesting option for +improving model efficiency by removing token rep- +resentations across layers. +10 +Limitations +As is the case for most attempts at interpreting +Deep Learning models, our evaluation of our (and +others’) proposed methods are not definite since +we have no gold standard of what happens inside +a model, although we try to remedy for that by +conducting independent and complementary evalu- +ation schemes. +Our proposed method is customized for deep +neural models based on the Transformer archi- +tecture and cannot be easily generalized to other +(mathematically different) modeling architectures. +Our evaluations were based on encoder-based mod- +els, and focused on the Text modality. 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Timkey and van +Schijndel (2021) questioned the informativity of +standard representational distance measures such +as cosine and Euclidean by observing that only a +small subset of rogue dimensions contribute to the +anisotropy of a contextualized representation space. +They proposed using simple post-processing tech- +niques to correct for such these rough dimensions. +We followed their suggestion and normalized the +representations before computing distances, but we +did not observe any noticeable difference in our +scores compared to using non-normalized repre- +sentations (Fig. A.1). We also repeated our ex- +periment with Spearman’s and Euclidean distance +metrics and observed the same pattern in the results +(Fig. A.1). +We believe that in anisotropy studies that use +clustering methods, the choice of distance metrics +is crucial. However, we compute each token’s dis- +tance from itself (not from other tokens) and com- +pare them relatively. This might explain why we +observe the same pattern for different distance met- +rics. +A.2 +More metrics +Figure A.2 reports the cue alignment evaluation +for BERT model based on Probes-needed (Zhong +et al., 2019) metric. +A.3 +More PLMs +We replicated our experiment for the cue alignment +for two more PLMs; RoBERTa (Liu et al., 2019) +and ELECTRA (generator, Clark et al., 2020). As +we can see in Figures A.3 and A.4, our method +consistently outperforms other methods on all mod- +els in both pre-trained and fine-tuned setups. Due +to the fact that our scores are based on zeroing +value vectors, our method can be easily applied to +any Transformer-based models even with different +modalities. +A.4 +Qualitative Analysis: Layer-wise +Context Mixing Maps +This section illustrates different context mix- +ing maps obtained from a fine-tuned BERT +model for the correctly classified example of +“The pictures of some hat [MASK] scaring Marcus.” +VZ (cosine) +VZ (cosine, normalized) +VZ (spearmanr) +VZ (spearmanr, normalized) +VZ (euclidean) +VZ (euclidean, normalized) +1 +2 +3 +4 +5 +6 +7 +8 +9 10 11 12 +Layer +0.10 0.10 0.10 0.10 0.10 0.10 +0.14 0.15 0.14 0.14 0.13 0.13 +0.19 0.19 0.19 0.19 0.15 0.15 +0.21 0.21 0.21 0.21 0.18 0.18 +0.24 0.24 0.23 0.23 0.19 0.19 +0.17 0.17 0.17 0.17 0.16 0.16 +0.17 0.17 0.17 0.17 0.17 0.17 +0.21 0.21 0.21 0.21 0.19 0.19 +0.28 0.28 0.28 0.28 0.22 0.22 +0.21 0.21 0.21 0.21 0.18 0.18 +0.26 0.26 0.26 0.26 0.19 0.19 +0.21 0.21 0.21 0.21 0.17 0.17 +Dot Product +VZ (cosine) +VZ (cosine, normalized) +VZ (spearmanr) +VZ (spearmanr, normalized) +VZ (euclidean) +VZ (euclidean, normalized) +1 +2 +3 +4 +5 +6 +7 +8 +9 10 11 12 +0.32 0.33 0.33 0.32 0.32 0.33 +0.36 0.37 0.36 0.37 0.36 0.37 +0.37 0.37 0.38 0.37 0.37 0.37 +0.48 0.48 0.48 0.48 0.48 0.48 +0.54 0.54 0.54 0.54 0.54 0.54 +0.41 0.41 0.41 0.41 0.41 0.41 +0.43 0.42 0.42 0.42 0.43 0.42 +0.47 0.47 0.47 0.47 0.47 0.47 +0.57 0.57 0.57 0.57 0.57 0.57 +0.45 0.45 0.46 0.46 0.45 0.45 +0.54 0.54 0.54 0.54 0.54 0.54 +0.44 0.44 0.44 0.44 0.44 0.44 +Average Precision +(a) Pre-trained BERT +VZ (cosine) +VZ (cosine, normalized) +VZ (spearmanr) +VZ (spearmanr, normalized) +VZ (euclidean) +VZ (euclidean, normalized) +1 +2 +3 +4 +5 +6 +7 +8 +9 10 11 12 +Layer +0.10 0.10 0.10 0.10 0.11 0.11 +0.14 0.15 0.14 0.14 0.13 0.13 +0.19 0.19 0.19 0.19 0.15 0.15 +0.21 0.21 0.21 0.21 0.18 0.18 +0.25 0.25 0.25 0.25 0.20 0.20 +0.18 0.18 0.17 0.17 0.17 0.17 +0.19 0.19 0.19 0.19 0.18 0.18 +0.24 0.24 0.24 0.24 0.21 0.21 +0.35 0.35 0.34 0.34 0.25 0.25 +0.21 0.21 0.20 0.21 0.18 0.18 +0.25 0.25 0.24 0.24 0.19 0.19 +0.22 0.22 0.22 0.21 0.17 0.17 +Dot Product +VZ (cosine) +VZ (cosine, normalized) +VZ (spearmanr) +VZ (spearmanr, normalized) +VZ (euclidean) +VZ (euclidean, normalized) +1 +2 +3 +4 +5 +6 +7 +8 +9 10 11 12 +0.33 0.33 0.33 0.33 0.33 0.33 +0.36 0.37 0.36 0.37 0.36 0.37 +0.37 0.37 0.37 0.37 0.37 0.37 +0.48 0.48 0.48 0.47 0.48 0.48 +0.58 0.58 0.57 0.57 0.58 0.58 +0.43 0.43 0.43 0.43 0.43 0.43 +0.46 0.46 0.46 0.46 0.46 0.46 +0.53 0.53 0.53 0.53 0.53 0.53 +0.68 0.68 0.68 0.68 0.68 0.68 +0.45 0.45 0.45 0.45 0.45 0.45 +0.53 0.53 0.53 0.53 0.53 0.53 +0.46 0.45 0.45 0.45 0.45 0.45 +Average Precision +(b) Fine-tuned BERT +Figure A.1: Layer-wise alignment between the cue vec- +tor and different Value Zeroing (VZ) scores computed +based on 1) different distance metric and 2) whether +representations are normalized. + +Rand +Attn +Attn-rollout +Attn-flow +Attn-norm +Attn-norm + RES +Attn-norm + RES + LN +ALTI +GradXinput +IG +DL +Value Zeroing +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +Layer +4.15 6.52 6.52 6.52 5.18 5.55 5.61 3.21 1.46 3.44 1.35 5.35 +4.05 5.89 6.08 4.84 3.82 4.76 4.98 6.47 1.62 3.70 1.51 3.96 +4.08 5.76 5.86 4.04 3.66 4.48 4.72 6.30 1.79 4.32 1.72 4.04 +4.19 4.67 5.91 3.94 2.10 3.08 3.14 6.30 2.14 4.73 2.77 2.19 +4.04 3.62 5.86 3.94 1.72 2.61 2.62 6.30 2.72 5.05 3.37 1.75 +4.08 4.20 5.92 3.94 2.52 3.39 3.40 6.30 3.25 5.06 3.75 2.53 +4.12 3.49 5.89 3.94 2.28 3.24 3.25 6.30 3.29 4.99 3.93 2.31 +4.12 3.41 5.90 3.94 2.29 3.20 3.20 6.30 3.65 5.07 4.01 2.30 +4.20 2.88 5.88 3.94 1.59 2.55 2.56 6.30 4.13 5.78 4.38 1.59 +4.18 3.64 5.90 3.94 2.59 3.45 3.47 6.30 3.84 5.83 4.79 2.60 +4.07 3.58 5.92 3.94 2.59 3.45 3.53 6.30 5.04 6.08 5.72 2.58 +4.02 4.45 5.94 3.94 3.40 4.27 4.36 6.30 4.36 4.36 4.36 3.35 +Probes-needed +2 +3 +4 +5 +6 +(a) Pre-trained BERT +Rand +Attn +Attn-rollout +Attn-flow +Attn-norm +Attn-norm + RES +Attn-norm + RES + LN +ALTI +GradXinput +IG +DL +Value Zeroing +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +Layer +4.15 6.50 6.50 6.50 5.13 5.52 5.58 3.31 1.52 3.30 1.34 5.31 +4.05 5.89 6.05 4.83 3.82 4.75 4.96 6.60 1.64 3.48 1.46 3.97 +4.08 5.77 5.84 3.93 3.89 4.69 4.87 6.42 1.76 3.69 1.62 4.21 +4.19 4.72 5.88 3.83 2.07 3.05 3.11 6.44 2.05 4.06 2.66 2.16 +4.04 3.42 5.83 3.83 1.49 2.40 2.41 6.44 2.50 4.59 3.14 1.47 +4.08 4.06 5.87 3.83 2.36 3.26 3.26 6.44 3.07 4.43 3.49 2.37 +4.12 3.18 5.84 3.83 1.95 2.93 2.95 6.44 3.07 3.57 3.66 1.99 +4.12 3.11 5.84 3.83 1.95 2.87 2.87 6.44 3.21 3.96 3.60 1.95 +4.20 2.40 5.84 3.83 1.05 2.03 2.03 6.44 4.00 5.36 4.06 1.05 +4.18 3.24 5.88 3.83 2.31 3.18 3.21 6.44 3.40 5.29 4.51 2.31 +4.07 3.54 5.92 3.83 2.45 3.31 3.34 6.44 4.55 4.97 5.44 2.46 +4.02 4.25 5.94 3.83 3.10 3.95 4.11 6.44 4.36 4.36 4.36 3.06 +Probes-needed +2 +3 +4 +5 +6 +(b) Fine-tuned BERT +Figure A.2: The layer-wise alignment based on Probes-needed metric between the cue vector and different analysis +methods averaged over Test set examples. Lower value (darker blue) is better. + +Rand +Attn +Attn-rollout +Attn-flow +Attn-norm +Attn-norm + RES +Attn-norm + RES + LN +ALTI +GradXinput +IG +DL +Value Zeroing +1 +2 +3 +4 +5 +6 +7 +8 +9 10 11 12 +Layer +0.08 0.08 0.04 0.08 0.09 0.06 0.05 0.08 0.08 0.09 0.10 0.12 +0.08 0.08 0.06 0.08 0.13 0.06 0.05 0.08 0.07 0.10 0.10 0.21 +0.08 0.07 0.06 0.08 0.11 0.06 0.05 0.08 0.07 0.09 0.09 0.15 +0.08 0.05 0.05 0.08 0.09 0.04 0.03 0.08 0.07 0.09 0.10 0.09 +0.08 0.06 0.04 0.08 0.09 0.04 0.03 0.08 0.07 0.09 0.10 0.10 +0.08 0.07 0.04 0.08 0.11 0.05 0.04 0.08 0.07 0.09 0.09 0.13 +0.08 0.06 0.04 0.08 0.09 0.04 0.03 0.08 0.07 0.09 0.09 0.11 +0.08 0.08 0.04 0.08 0.12 0.05 0.04 0.08 0.08 0.08 0.09 0.16 +0.08 0.07 0.03 0.08 0.12 0.05 0.04 0.08 0.07 0.07 0.08 0.14 +0.08 0.07 0.03 0.08 0.12 0.05 0.04 0.08 0.06 0.05 0.07 0.16 +0.08 0.07 0.03 0.08 0.12 0.04 0.03 0.08 0.05 0.04 0.04 0.16 +0.08 0.04 0.03 0.08 0.10 0.03 0.02 0.08 0.00 0.00 0.00 0.12 +Dot Product +Rand +Attn +Attn-rollout +Attn-flow +Attn-norm +Attn-norm + RES +Attn-norm + RES + LN +ALTI +GradXinput +IG +DL +Value Zeroing +1 +2 +3 +4 +5 +6 +7 +8 +9 10 11 12 +0.25 0.22 0.19 0.22 0.33 0.21 0.20 0.24 0.25 0.27 0.29 0.36 +0.25 0.25 0.18 0.20 0.41 0.25 0.24 0.12 0.20 0.29 0.28 0.42 +0.25 0.23 0.19 0.18 0.42 0.25 0.25 0.11 0.18 0.27 0.27 0.42 +0.25 0.17 0.18 0.18 0.27 0.20 0.20 0.11 0.20 0.25 0.28 0.27 +0.25 0.16 0.16 0.15 0.31 0.20 0.20 0.11 0.21 0.25 0.28 0.30 +0.25 0.20 0.15 0.13 0.38 0.25 0.25 0.11 0.19 0.24 0.25 0.38 +0.25 0.16 0.15 0.13 0.30 0.21 0.20 0.11 0.18 0.22 0.23 0.31 +0.26 0.20 0.15 0.13 0.39 0.25 0.25 0.11 0.21 0.23 0.23 0.41 +0.25 0.21 0.15 0.13 0.34 0.23 0.23 0.11 0.21 0.24 0.24 0.35 +0.26 0.23 0.15 0.13 0.33 0.22 0.21 0.11 0.19 0.24 0.23 0.34 +0.25 0.20 0.15 0.13 0.35 0.22 0.22 0.11 0.18 0.23 0.22 0.36 +0.25 0.18 0.15 0.13 0.31 0.20 0.20 0.11 0.08 0.08 0.08 0.31 +Average Precision +(a) Pre-trained RoBERTa +Rand +Attn +Attn-rollout +Attn-flow +Attn-norm +Attn-norm + RES +Attn-norm + RES + LN +ALTI +GradXinput +IG +DL +Value Zeroing +1 +2 +3 +4 +5 +6 +7 +8 +9 10 11 12 +Layer +0.08 0.08 0.04 0.08 0.09 0.06 0.05 0.08 0.09 0.09 0.11 0.12 +0.08 0.08 0.06 0.08 0.12 0.06 0.05 0.08 0.08 0.10 0.10 0.20 +0.08 0.06 0.06 0.08 0.11 0.06 0.05 0.08 0.07 0.10 0.10 0.15 +0.08 0.03 0.05 0.08 0.07 0.03 0.02 0.08 0.08 0.10 0.09 0.05 +0.08 0.06 0.04 0.08 0.08 0.04 0.03 0.08 0.08 0.09 0.09 0.09 +0.08 0.06 0.04 0.08 0.10 0.05 0.04 0.08 0.07 0.09 0.08 0.09 +0.08 0.05 0.04 0.08 0.09 0.04 0.03 0.08 0.07 0.09 0.07 0.11 +0.08 0.09 0.04 0.08 0.11 0.05 0.04 0.08 0.06 0.07 0.07 0.14 +0.08 0.09 0.03 0.08 0.14 0.06 0.05 0.08 0.07 0.06 0.06 0.19 +0.08 0.07 0.03 0.08 0.13 0.05 0.04 0.08 0.05 0.05 0.05 0.16 +0.08 0.06 0.03 0.08 0.11 0.04 0.04 0.08 0.03 0.02 0.02 0.13 +0.08 0.02 0.03 0.08 0.06 0.01 0.01 0.08 0.00 0.00 0.00 0.04 +Dot Product +Rand +Attn +Attn-rollout +Attn-flow +Attn-norm +Attn-norm + RES +Attn-norm + RES + LN +ALTI +GradXinput +IG +DL +Value Zeroing +1 +2 +3 +4 +5 +6 +7 +8 +9 10 11 12 +0.25 0.22 0.19 0.22 0.33 0.21 0.20 0.23 0.28 0.25 0.33 0.36 +0.25 0.25 0.18 0.20 0.40 0.24 0.24 0.12 0.22 0.28 0.33 0.42 +0.25 0.22 0.18 0.18 0.42 0.25 0.25 0.11 0.21 0.27 0.31 0.40 +0.25 0.14 0.18 0.18 0.20 0.16 0.16 0.11 0.21 0.28 0.28 0.19 +0.25 0.16 0.16 0.15 0.27 0.19 0.19 0.11 0.21 0.26 0.25 0.27 +0.25 0.18 0.15 0.14 0.31 0.21 0.21 0.11 0.20 0.27 0.20 0.31 +0.25 0.18 0.15 0.16 0.28 0.19 0.19 0.11 0.18 0.24 0.20 0.30 +0.26 0.23 0.15 0.13 0.36 0.24 0.24 0.11 0.17 0.22 0.19 0.37 +0.25 0.24 0.15 0.13 0.44 0.28 0.28 0.11 0.19 0.21 0.19 0.44 +0.26 0.20 0.15 0.13 0.36 0.24 0.24 0.11 0.15 0.19 0.17 0.38 +0.25 0.20 0.15 0.13 0.34 0.22 0.22 0.11 0.15 0.16 0.16 0.34 +0.25 0.15 0.15 0.13 0.17 0.14 0.14 0.11 0.08 0.08 0.08 0.17 +Average Precision +(b) Fine-tuned RoBERTa +Figure A.3: Layer-wise alignment between the cue vector and different analysis methods averaged over Test set +examples for RoBERTa. Higher value (darker color) is better. + +Rand +Attn +Attn-rollout +Attn-flow +Attn-norm +Attn-norm + RES +Attn-norm + RES + LN +ALTI +GradXinput +IG +DL +Value Zeroing +1 +2 +3 +4 +5 +6 +7 +8 +9 10 11 12 +Layer +0.10 0.11 0.05 0.11 0.11 0.05 0.04 0.11 0.15 0.11 0.15 0.14 +0.10 0.11 0.08 0.10 0.14 0.07 0.05 0.10 0.15 0.11 0.15 0.19 +0.10 0.07 0.08 0.11 0.15 0.07 0.05 0.10 0.14 0.12 0.14 0.22 +0.10 0.11 0.07 0.11 0.20 0.09 0.06 0.10 0.14 0.11 0.14 0.34 +0.10 0.05 0.06 0.11 0.09 0.03 0.02 0.10 0.13 0.11 0.13 0.11 +0.10 0.09 0.06 0.11 0.19 0.08 0.06 0.10 0.12 0.10 0.11 0.22 +0.10 0.10 0.05 0.11 0.17 0.07 0.06 0.10 0.12 0.08 0.11 0.20 +0.10 0.14 0.04 0.11 0.25 0.10 0.08 0.10 0.12 0.07 0.10 0.36 +0.10 0.13 0.04 0.11 0.19 0.08 0.07 0.10 0.11 0.06 0.09 0.25 +0.10 0.11 0.04 0.11 0.18 0.06 0.06 0.10 0.08 0.03 0.05 0.22 +0.10 0.04 0.04 0.11 0.13 0.04 0.03 0.10 0.05 0.01 0.03 0.15 +0.11 0.06 0.04 0.11 0.12 0.03 0.03 0.10 0.00 0.00 0.00 0.12 +Dot Product +Rand +Attn +Attn-rollout +Attn-flow +Attn-norm +Attn-norm + RES +Attn-norm + RES + LN +ALTI +GradXinput +IG +DL +Value Zeroing +1 +2 +3 +4 +5 +6 +7 +8 +9 10 11 12 +0.29 0.36 0.22 0.36 0.36 0.22 0.21 0.39 0.53 0.34 0.55 0.37 +0.29 0.34 0.24 0.27 0.44 0.27 0.27 0.15 0.51 0.30 0.51 0.44 +0.29 0.23 0.19 0.22 0.48 0.29 0.28 0.13 0.46 0.29 0.45 0.51 +0.29 0.27 0.18 0.23 0.63 0.35 0.35 0.14 0.41 0.28 0.40 0.63 +0.28 0.17 0.18 0.23 0.27 0.19 0.19 0.14 0.40 0.27 0.36 0.26 +0.29 0.22 0.17 0.23 0.42 0.27 0.27 0.14 0.34 0.27 0.27 0.44 +0.29 0.29 0.17 0.23 0.45 0.28 0.28 0.14 0.33 0.23 0.25 0.45 +0.29 0.31 0.17 0.23 0.63 0.36 0.36 0.14 0.32 0.22 0.26 0.64 +0.30 0.39 0.17 0.23 0.51 0.31 0.31 0.14 0.30 0.22 0.24 0.51 +0.30 0.28 0.16 0.22 0.49 0.31 0.30 0.14 0.26 0.17 0.17 0.50 +0.30 0.17 0.16 0.22 0.38 0.25 0.25 0.14 0.22 0.15 0.18 0.38 +0.31 0.18 0.16 0.22 0.33 0.22 0.22 0.14 0.10 0.10 0.10 0.33 +Average Precision +(a) Pre-trained ELECTRA +Rand +Attn +Attn-rollout +Attn-flow +Attn-norm +Attn-norm + RES +Attn-norm + RES + LN +ALTI +GradXinput +IG +DL +Value Zeroing +1 +2 +3 +4 +5 +6 +7 +8 +9 10 11 12 +Layer +0.10 0.10 0.05 0.10 0.11 0.05 0.04 0.11 0.16 0.13 0.17 0.14 +0.10 0.11 0.08 0.10 0.15 0.07 0.05 0.10 0.16 0.12 0.16 0.20 +0.10 0.06 0.07 0.10 0.15 0.06 0.05 0.10 0.16 0.14 0.16 0.21 +0.10 0.10 0.07 0.11 0.22 0.09 0.07 0.10 0.15 0.13 0.15 0.40 +0.10 0.05 0.06 0.11 0.10 0.03 0.02 0.10 0.15 0.12 0.14 0.12 +0.10 0.09 0.05 0.11 0.21 0.08 0.06 0.10 0.13 0.11 0.13 0.26 +0.10 0.10 0.05 0.11 0.20 0.08 0.07 0.10 0.14 0.09 0.13 0.28 +0.10 0.22 0.04 0.11 0.35 0.15 0.12 0.10 0.11 0.07 0.10 0.54 +0.10 0.16 0.04 0.11 0.22 0.10 0.09 0.10 0.09 0.05 0.08 0.33 +0.10 0.10 0.04 0.11 0.18 0.07 0.06 0.10 0.06 0.02 0.04 0.21 +0.10 0.03 0.04 0.11 0.11 0.03 0.03 0.10 0.02 0.00 0.01 0.11 +0.11 0.02 0.03 0.11 0.06 0.01 0.01 0.10 0.00 0.00 0.00 0.04 +Dot Product +Rand +Attn +Attn-rollout +Attn-flow +Attn-norm +Attn-norm + RES +Attn-norm + RES + LN +ALTI +GradXinput +IG +DL +Value Zeroing +1 +2 +3 +4 +5 +6 +7 +8 +9 10 11 12 +0.29 0.36 0.22 0.36 0.36 0.22 0.21 0.40 0.60 0.41 0.62 0.37 +0.29 0.33 0.23 0.25 0.45 0.28 0.27 0.15 0.57 0.38 0.58 0.45 +0.29 0.22 0.19 0.22 0.45 0.27 0.27 0.14 0.53 0.40 0.54 0.49 +0.29 0.24 0.18 0.23 0.63 0.35 0.34 0.14 0.47 0.34 0.46 0.63 +0.28 0.17 0.18 0.22 0.29 0.20 0.20 0.14 0.46 0.31 0.41 0.28 +0.29 0.21 0.17 0.22 0.47 0.29 0.29 0.14 0.38 0.32 0.31 0.49 +0.29 0.26 0.17 0.23 0.59 0.34 0.34 0.14 0.38 0.27 0.28 0.60 +0.29 0.57 0.17 0.23 0.80 0.43 0.42 0.14 0.30 0.23 0.27 0.78 +0.30 0.56 0.17 0.23 0.65 0.36 0.36 0.14 0.26 0.21 0.22 0.64 +0.30 0.25 0.16 0.22 0.49 0.30 0.30 0.14 0.23 0.20 0.19 0.49 +0.30 0.20 0.16 0.23 0.31 0.21 0.21 0.14 0.16 0.15 0.16 0.32 +0.31 0.17 0.16 0.23 0.20 0.16 0.15 0.14 0.10 0.10 0.10 0.20 +Average Precision +(b) Fine-tuned ELECTRA +Figure A.4: Layer-wise alignment between the cue vector and different analysis methods averaged over Test set +examples for ELECTRA. Higher value (darker color) is better. + +Attn +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 1 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 2 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 3 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 4 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 5 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus 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+Figure A.5: Raw attention scores (Attn) averaged over all attention heads at each different layer. +Attn + rollout +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 1 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 2 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 3 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 4 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 5 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. 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(2020)’s scores (Attn-norm) across layers. +Attn-norm + rollout +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 1 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 2 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 3 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 4 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 5 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 6 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 7 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 8 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 9 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 10 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 11 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 12 +Figure A.8: Kobayashi et al. (2020)’s scores (Attn-norm) aggregated by rollout method across layers. + +Attn-norm + RES +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 1 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 2 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 3 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 4 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 5 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 6 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 7 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 8 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 9 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 10 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 11 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 12 +Figure A.9: Kobayashi et al. (2021)’s scores (Attn-norm + RES) across layers. +Attn-norm + RES + rollout +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 1 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 2 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 3 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 4 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 5 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 6 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 7 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 8 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 9 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 10 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 11 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 12 +Figure A.10: Kobayashi et al. (2021)’s scores (Attn-norm + RES) aggregated by rollout method across layers. + +Attn-norm + RES + LN +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 1 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 2 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 3 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 4 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 5 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 6 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 7 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 8 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 9 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 10 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 11 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 12 +Figure A.11: Kobayashi et al. (2021)’s scores (Attn-norm + RES + LN) across layers. +Attn-norm + RES + LN + rollout +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 1 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 2 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 3 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 4 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 5 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 6 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 7 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 8 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 9 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 10 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 11 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 12 +Figure A.12: Kobayashi et al. (2021)’s scores (Attn-norm + RES + LN) aggregated by rollout method across layers. + +ALTI without rollout +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 1 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 2 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 3 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 4 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 5 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 6 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 7 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 8 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 9 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 10 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 11 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 12 +Figure A.13: Ferrando et al. (2022)’s scores (ALTI) without aggregation (rollout) across layers. +ALTI (with rollout) +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 1 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 2 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 3 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 4 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 5 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 6 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 7 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 8 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 9 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 10 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 11 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 12 +Figure A.14: Ferrando et al. (2022)’s scores (ALTI) across layers (the rollout method is inherently incorporated in +ALTI). + +Value Zeroing +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 1 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 2 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 3 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 4 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 5 +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] 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+[SEP] +[CLS] +the +pictures +of +some +hat +[MASK] +scar +##ing +marcus +. +[SEP] +Layer: 12 +Figure A.16: Our global scores (Value Zeroing) aggregated by rollout method across layers. + diff --git a/YdFOT4oBgHgl3EQf9zRP/content/tmp_files/load_file.txt b/YdFOT4oBgHgl3EQf9zRP/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a87f88c167d46045cb35917dd17f456368e87a07 --- /dev/null +++ b/YdFOT4oBgHgl3EQf9zRP/content/tmp_files/load_file.txt @@ -0,0 +1,3375 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf,len=3374 +page_content='Quantifying Context Mixing in Transformers Hosein Mohebbi1 Willem Zuidema2 Grzegorz Chrupała1 Afra Alishahi1 1 CSAI, Tilburg University 2 ILLC, University of Amsterdam {h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='mohebbi, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='alishahi}@tilburguniversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='edu w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='zuidema@uva.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='nl grzegorz@chrupala.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='me Abstract Self-attention weights and their transformed variants have been the main source of infor- mation for analyzing token-to-token interac- tions in Transformer-based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' But de- spite their ease of interpretation, these weights are not faithful to the models’ decisions as they are only one part of an encoder, and other com- ponents in the encoder layer can have consider- able impact on information mixing in the out- put representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' In this work, by expand- ing the scope of analysis to the whole encoder block, we propose Value Zeroing, a novel con- text mixing score customized for Transformers that provides us with a deeper understanding of how information is mixed at each encoder layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' We demonstrate the superiority of our context mixing score over other analysis meth- ods through a series of complementary evalu- ations with different viewpoints based on lin- guistically informed rationales, probing, and faithfulness analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='1 1 Introduction Transformers (Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=', 2017), with their im- pressive empirical success, have become a prime choice of architecture to learn contextualized repre- sentations across a wide range of modalities, such as language (Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=', 2020), vision (Dosovitskiy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=', 2021), vision- language (Radford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Rombach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=', 2022), and speech (Baevski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=', 2020), mainly due to their ability to utilize pairwise interactions between input tokens at every timestep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' To better understand the inner dynamics of Transformers, we need to trace the information flow from the input embeddings up to the output representation (including quantifying the degree of context mixing, which we will define below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' The attention weights from the multi-head atten- tion mechanisms offer a straightforward starting 1Code is freely available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='com/ hmohebbi/ValueZeroing point for understanding this flow, and these weights (‘raw attention’) have been used in many studies (Clark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Kovaleva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Reif et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Htut et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=', 2019a, inter alia).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' However, the reliability and usefulness of raw attention weights has also been questioned (Jain and Wallace, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Bibal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' In particular, attention weights tend to concentrate on uninformative tokens in the context (Voita et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Clark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=', 2019), and removing or altering them may lead to the same and sometimes even better model performance on downstream tasks (Jain and Wallace, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Toneva and Wehbe, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Hassid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' These find- ings suggest that Transformers do not solely rely on self-attention, and other components in the encoder block play an essential role in information mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' A number of methods have been proposed to compute some form of ‘effective attention weights’, with the goal of more faithfully tracing the relative contributions of different input tokens at various layers of the Transformer (as we will discuss in Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' These methods show an improvement over raw attention, but still ignore key components of the Transformer encoder block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' This is a partic- ularly crucial shortcoming, given that most of the parameter budget in a Transformer encoder is spent on position-wise feed-forward networks outside of the self-attention component, which can have a considerable impact on the degree of information mixing in the output representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' In this paper, we focus on context mixing: the property of Transformers that in each node, at each layer, information from the context can be incor- porated into the representation of the target token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' We propose Value Zeroing, a novel approach to quantify the contribution each context token has in determining the final representation of a target to- ken, at each layer of a Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Value Zeroing is based on the Explaining-by-Removing intuition (Covert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=', 2021) shared by many posthoc in- terpretability methods, but it takes advantage of arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='12971v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='CL] 30 Jan 2023 a specific feature of Transformers: it zeroes only the value vector of a token t when computing its importance, but leaves the key and query vectors (and thus the pattern of information flow) intact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Based on extensive experiments and three com- plementary approaches to evaluation, we demon- strate that importance scores we can obtain with Value Zeroing provide better interpretations than other analysis methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Firstly, we use a set of grammatical agreement tasks from the BLiMP corpus (Warstadt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=', 2020) as a case study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Transformer-based mod- els do extremely well on the task of distinguishing grammatical from ungrammatical sentences, and the BLiMP corpus provides information on the cue words that determine the difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' We find that Value Zeroing, unlike earlier approaches, indeed re- veals that Transformers make use of relevant cues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Secondly, we use information-theoretic probing (Voita and Titov, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Pimentel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=', 2020) as an independent approach to track information flow in Transformer networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' The scores we obtain with Value Zeroing turn out to be highly correlated with layer-wise probing performance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' that is, probing accuracy is higher in layers where relevant tokens are more effectively utilized by the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Thirdly, we assess the faithfulness of our method (Jacovi and Goldberg, 2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' compared to alterna- tive analysis methods, we show that Value Zeroing is not only more plausible and human-interpretable, but also more faithful to models’ decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' 2 Related Work While numerous studies have leveraged the weights assigned by the self-attention mechanism to gain intuition about the information mixing process in Transformers (Clark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Kovaleva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Reif et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Htut et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=', 2019b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Raganato and Tiedemann, 2018), it is still a matter of debate whether attention weights are suitable for interpreting the model (see Bibal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' (2022)’s study for a full discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Thus several post-processing interpretability techniques have been proposed to convert these weights into scores that provide a more detailed interpretation of the inner workings of Transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' We review the main approaches below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Abnar and Zuidema (2020) propose the attention-flow and attention-rollout methods to ap- proximate information flow in Transformers based on raw attention weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' The former treats raw attention weight matrices as a flow network and returns the maximum flow through each input to- ken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' The latter recursively multiplies the attention weight matrix at each layer by the preceding ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' There is, however, an unjustified assumption in the formulation of these methods that both multi- head attention and residual connections contribute equally to the computation of the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Kobayashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' (2020) propose a method that incorporates the norm of the transformed value vec- tors and report a negative correlation between these norms and raw attention weights on frequent to- kens, which partially explains the insufficiency of raw attention weights for context mixing estima- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Kobayashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' (2021) extend this method to the whole self-attention block by incorporating Residual connections (RES) and Layer Normaliza- tion (LN) (two components with significant impact on both model performance and training conver- gence (Parisotto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=', 2020)), but demonstrate that RES and LN components largely cancel out the mixing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Kobayashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' (2021)’s method, however, ignores the effect of the second sublayer in a Transformer’s encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Brunner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' (2020) and Pascual et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' (2021) employ a gradient-based approach for analyzing the interaction of input representations, but the gra- dient measures the sensitivity between two vectors and ignores the impact of the input vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' In our experiments we show that despite their relative suc- cess in explaining model decisions, gradient-based approaches are not suitable for layer-wise analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' More recently, the effectiveness of combining these approaches has also been investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Us- ing the rollout method (Abnar and Zuidema, 2020), Modarressi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' (2022) aggregated Kobayashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' (2021)’s scores across the layers to provide global token attributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' In the same vein, Ferrando et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' (2022) used rollout to aggregate a variant of Kobayashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' (2021)’s scores: instead of re- lying on the Euclidean norm of the transformed vector, they measured Manhattan distance of each transformed vector to the context vector outputted from self-attention block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' In both studies, however, the fact that these context vectors might undergo significant changes after passing through the sec- ond sublayer in the encoder layer is not taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' We will show (in Section 7) that even at a global level, when scores are aggregated across layers of a model, our scores provide better inter- pretation than prior methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' 3 Our Proposed Method To remedy for the limited scope of the existing methods, we introduce a new context mixing score that takes into account all components in a Trans- former encoder block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='1 Background and Notation In this section, we set up the notation and briefly review the internal structure of an encoder layer in the Transformer architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Each Transformer encoder layer is composed of two sublayers: a multi-head self-attention mecha- nism (MHA) and a position-wise fully connected feed-forward network (FFN), followed by a Resid- ual connection (RES) and Layer Normalization (LN) around each of these two sublayers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' This encoder layer produces the next contextualized rep- resentations (˜x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=', ˜xn) for each token in the con- text, using the output representations from the pre- vious layer (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=', xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' MHA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' For each head h ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' H} in the self- attention module,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' each input vector xi is trans- formed into a query qh i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' a key kh i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' and a value vh i vector via separate trainable linear transformations: qh i = xiW h Q + bh Q (1) kh i = xiW h K + bh K (2) vh i = xiW h V + bh V (3) The context vector zh i for the ith token of each attention head is then generated as a weighted sum over the transformed value vectors: zh i = n � j=1 αh i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='jvh j (4) where αh i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='j is the raw attention weight assigned to the jth token,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' and computed as a softmax- normalized dot product between the corresponding query and key vectors: αi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='j = softmax xj∈X � qik⊤ j √ d � ∈ R (5) Next,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' the context vector (zi ∈ Rd) for the ith to- ken is computed by concatenating all the heads’ outputs followed by a head-mixing WO projection and layer normalization: zi = CONCAT(z1 i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=', zH i )WO (6) zi = LNMHA(zi + xi) (7) FFN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Each encoder layer also includes two linear transformations with a ReLU activation in between, which is applied to every zi separately and identi- cally to produce output token representations ˜xi: ˜xi = max(0, ziW1 + b1)W2 + b2 (8) ˜xi = LNFFN(˜xi + zi) (9) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='2 Value Zeroing We aim to measure how much a token uses other context tokens to build its output representation ˜xi at each encoder layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' To this end, we treat the self-attention mechanism as a fuzzy hash-table, where we look up the sum of values weighted by the query-key match in the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Thus in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' 4 we replace a value vector associated with token j with a zero vector vh j ← 0, ∀h ∈ H, where the context vector for the ith token is being computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' This provides an alternative output representation ˜x¬j i for the ith token that has excluded token j in the mixing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' By comparing the alternative output representation ˜x¬j i with the original ˜xi, we can measure how much the output representation is affected by the exclusion of the jth token: Ci,j = ˜x¬j i ∗ ˜xi (10) where the operation ∗ can be any pairwise distance metric that properly considers the characteristics of the model’s representation space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' We opted for cosine distance throughout our experiments as its superiority over other dissimilarity metrics has been supported for textual deep learning models (Yokoi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Hanawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='2 Com- puting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' 10 for all ˜xi in a given context provides us with a Value Zeroing matrix score C where the value of the cell Ci,j ∈ R (ith row, jth column) in the map denotes the degree to which the ith token depends on the jth token to form its contextualized representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Note that unlike generic perturbation approaches, our proposed method does not remove the token representations xi from the input of an encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' We argue that ablating input token representations can- not be a reliable basis to understand context mix- ing process since any changes in the input vectors will lead to changes in the query and key vectors (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' 1 and 2), resulting in a change in the attention 2More details on the choice of distance metric is discussed in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Phenomenon UID Example Target word Foil word Anaphor Number Agreement ana Many teenagers were helping [MASK].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' themselves herself Determiner-Noun Agreement dna Jeffrey has not passed [MASK] museums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' these this dnaa Sara noticed [MASK] white hospitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' these this Subject-Verb Agreement darn The pictures of Martha [MASK] not disgust Anne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' do does rpsv Kristen [MASK] fixed this chair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' has have Table 1: Examples of the selected tasks with our annotations from the BLiMP benchmark (UIDs are unique identifiers used in BLiMP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Cue words are underlined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' distribution (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Consequently, there will be a discrepancy between the alternative attention weights and those for the original context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Instead, our method only nullifies the value vector of a spe- cific token representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' In this way, the token representation can maintain its identity within the encoder layer, but it does not contribute to form- ing other token representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Moreover, since our Value Zeroing is computed from the encoder’s layer outputs, it incorporates all the components in- side an encoder layer such as multi-head attention, residual connection, layer normalization, and also feed-forward networks, resulting in a more reliable context mixing score than previous methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' 4 Experimental Setup 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='1 Data We used the BLiMP benchmark (Warstadt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=', 2020) which contains a set of pairs of minimally different sentences that contrast in grammatical ac- ceptability under a specific linguistic phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' The benchmark isolates linguistic phenomena such that only one word determines the true label of each sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' We refer to this crucial context to- ken as the cue word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' The nature of this task makes it especially suitable for evaluating context mix- ing scores, since it gives us a strong hypothesis on which context token is the most relevant for the representation of the masked target word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' From this benchmark, we select five datasets with three different linguistic phenomena for which Pre-trained Language Models (PLMs) have shown high accuracy to ensure that the model captures the relevant information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' We expand contractions such as doesn’t → does not) and generate dependency trees using SpaCy (Honnibal and Montani, 2017) to extract and annotate the position of target and cue words in a sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' In Table 1, we provide an example of each phenomenon in the benchmark together with our automated annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' We accu- mulate examples from the five selected tasks as a unified dataset for grammatical agreement, result- ing in 4,276 data points, and divide them equally into Train and Test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' The Train set is only used for the fine-tuning phase;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' the Test set is used for all evaluation experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='2 Target Model We conduct our experiments on three Transformer- based language models: BERT (uncased, Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=', 2019), RoBERTa (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=', 2019) and ELECTRA (Clark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='3 The results for the latter two are reported in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' By replacing the target words with the [MASK] token, we perform a Masked Language Modeling (MLM) task using the model’s pre-trained MLM head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' For instance, in the Subject-Verb Agreement example “The pictures of Martha do not disgust Anne.”, we replace the verb ‘do’ with the [MASK] token and feed the example to the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' We perform our experiments on both pre-trained and fine-tuned versions of each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Including a fine-tuned model in our analysis study gives us a complementary insight into the importance of the cue words, since fine-tuning allows the model to concentrate on the most helpful words for the downstream task of choice (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=', agreement) and makes sure that target word representations take the cue word into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' We use prompt fine-tuning (Schick and Schütze, 2021a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Karimi Mahabadi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=', 2022) and compute Cross Entropy loss only over the output logits corresponding to the target and foil classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Accuracy is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='96 for pre-trained and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='99 for fine-tuned BERT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='3 Baselines Here we describe the existing context mixing meth- ods which we include in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' For each method, we select the mth row of its context mixing map where m is the position of the [MASK] token, 3Base, with 12 layers and 12 attention heads, obtained from the Transformers library (Wolf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' resulting in a 1-D array of scores for each context token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' We normalize the scores for all tokens in a sentence so that they are all positive values and sum to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' For completeness, we also include a few gradient-based attribution methods in our comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Rand: random scores generated from a uniform distribution for each sentence in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Attn: raw attention scores αm from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Attn-rollout & Attn-flow: two methods for ap- proximating the attention flow based on raw atten- tion weights (Abnar and Zuidema, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Attn-Norm: norm-based method of Kobayashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' (2020) that also incorporates the norm of the transformed input vectors to compute context mixing scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Attn-Norm + RES + LN: the extended norm- based method of Kobayashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' (2021) in which they also incorporates Residual connection (RES) and Layer Normalization (LN) located only in the first sublayer of a Transformer’s encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' ALTI: Aggregation of Layer-wise Token-to-token Interactions, proposed by Ferrando et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' GradXinput, IG & DL: feature attribution scores that also make use of top-down information from the classification layer on top of the Trans- former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' We consider three popular variants of gradient-based attribution scores: Gradient×Input (GradXinput) (Samek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=', 2019), Integrated Gradients (IG) (Sundararajan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=', 2017), and DeepLift (DL) (Shrikumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' For gradient-based methods, we use the pre-trained MLM head which has been trained dur- ing the pre-training of the BERT to compute the gradient of the true label with respect to the token representations at each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' 5 Evaluation 1: Cue Alignment As cue words are the only indicators of the true la- bels in our dataset, we expect that when the model performs well, it overwhelmingly depends on these words to form the representation of a [MASK] to- ken in a given context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' To quantify the alignment between a context mixing score and the cue word, we first define a binary cue vector ξ according to the following condition: ξi = � 1, the ith token ∈ Cue words 0, otherwise (11) Then we compare the cue vector and the prediction of a context mixing score S in two different ways: Dot Product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' We quantify cue alignment as S·ξ, which measures the total score mass the model assigns to cue words to form the representation of the target token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Average Precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' We quantify cue alignment as the average precision between the two vectors, which is a weighted mean of precision at each recall level: AP = � n (Rn − Rn−1)Pn (12) where Pn and Rn are the precision and recall at the nth threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' This metric relies on the ranking of tokens rather than the magnitude of their weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='4 Figures 1 and 2 show the alignment between the cue vector and different analysis methods us- ing dot product and average precision for the pre- trained and fine-tuned model, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' In com- parison with the other context mixing methods, Value Zeroing shows a higher degree of the target model incorporating cue words into the representa- tion of the [MASK] token across all layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' As can be seen from the first two columns in all graphs, raw self-attention weights (Attn) always perform worse than even random scores in high- lighting cue words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' This is in line with previous studies showing that raw attention weights often pay attention to uninformative tokens (Voita et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Clark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=', 2019) and do not reflect the appropriate context (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' The same pattern holds for their aggregated versions (Attn- rollout and Attn-flow), which are based solely on attention weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' However, we can see a signif- icant improvement in the results for Attn-norm where the norm of transformed value vectors are also taken into account, confirming that value vec- tors play an essential role in the context mixing pro- cess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' The method Attn-norm + RES + LN, which expands Attn-norm to the whole self-attention block by adding Residual connection and Layer Normalization, would seem to show that the model is incapable of utilizing the cue words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' However, incorporating also the second part of the encoder layer via our method shows the model does indeed use the cue words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' 4We also employed Probes-needed (Zhong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=', 2019) metric in our evaluation which intuitively counts the number of non-cue tokens we need to probe to find cue words based on a given score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' As its motivation is similar to Average Precision and the results show the same pattern, we relegate the results with this metric to the Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Rand Attn Attn-rollout Attn-flow Attn-norm Attn-norm + RES Attn-norm + RES + LN ALTI GradXinput IG DL Value Zeroing 1 2 3 4 5 6 7 8 9 10 11 12 Layer 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='06 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='46 Average Precision Figure 2: Layer-wise alignment between the cue vector and different analysis methods averaged over Test set examples for the fine-tuned model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Higher value (darker color) is better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' The gradient-based scores, in contrast to the other methods, highlight the cue words only in the earlier layers of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' In the next section, by using a layer-wise probing experiment, we will show that these scores are not reliable for identify- ing the relevant context in individual layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' 6 Evaluation 2: Context Mixing versus Probing In this section, we investigate the relationship be- tween cue word alignment and probing perfor- mance across layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' We hypothesize that if a layer aligns better with the cue word according to a reli- able context mixing score, then the representation of the masked token on that layer can be used more effectively by a probing classifier to decode number agreement with the cue word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' To verify our hypothesis, we obtain the represen- tation of masked tokens in test examples across all layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Since all examples in our dataset share the same number agreement property, we asso- ciate each masked representation with a Singular or Plural label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Next, we perform an information- theoretic probing analysis using Minimum De- scription Length (MDL) to measures the degree to which representations encode number agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' We chose MDL as our probe since it is theoreti- cally justified and has been shown to provide more reliable results than conventional probes (Voita and Titov, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Fayyaz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' To compute MDL, we employed the online cod- ing of Voita and Titov (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Since MDL can be affected by the number of data points (N), we mea- sure compression as our evaluation metric which is 1 2 3 4 5 6 7 8 9 10 11 12 layer 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='6 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='8 compression finetuned False True Figure 3: Layer-wise compression of probing classi- fiers using pre-trained and fine-tuned representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' defined as follows: Compression = N · log2(K) MDL (13) where K refers to the number of classes (2 in our case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' This metric is equal to 1 (no compression) for a random guessing classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' A higher value for Compression indicates more accurate label predic- tion for the probing classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Figure 3 reports compression of probing classi- fiers based on representations obtained from both pre-trained and fine-tuned models across all lay- ers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' We also include the results for the embedding layer of the model (layer 0) which can serve as a non-contextualized baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' We can see a jump in probing performance at layers 4 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='52 → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='72) and 9 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='03 → 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='45) in the fine-tuned setup, the same layers for which we found a higher alignment with cue words in Figures 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Table 2 presents the correlation between layer- wise Compression scores and layer-wise cue align- ment scores from Section 5 for different analysis methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' As we can see, alignment according to Value Zeroing is highly positively correlated with the probing performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' This suggests that when Value Zeroing indicates that the model uses cue words to form representations of the masked to- kens in a particular layer, these representations are in fact better at encoding number agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Recall that based on Figures 1 and 2, according to the gradient-based methods the masked tokens pay more attention to the cue words only in earlier layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' However, we can see a highly negative cor- relation with probing results for these scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Due to the nature of the task the probing score goes up monotonically along the layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' At the same time, the gradient attribution score goes up monotoni- cally as you get closer to the bottom embedding layers, suggesting that gradient-based methods are unreliable for layer-wise analysis and identifying important tokens in the context mixing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Method ρPT ρFT Rand 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='02 Attn 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='08 Attn-norm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='56 Attn-norm + RES 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='24 Attn-norm + RES + LN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='17 ALTI 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='12 GradXinput 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='99 IG 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='77 DL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='97 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='00 Value Zeroing 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='64 Table 2: Spearman’s ρ correlation between layer-wise probing performance (Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=') and layer-wise cue alignment scores based on Dot Product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' PT and FT refer to pre-trained and fine-tuned conditions, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' 7 Evaluation 3: Faithfulness Analysis Our experimental results in Sections 5 and 6 show that the Value Zeroing score matches our prior linguistically-informed expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' However, it is not always clear whether a plausible context mix- ing score that matches human expectations is also faithful to the model and reflects its decision mak- ing process (Herman, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Wiegreffe and Pinter, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Jacovi and Goldberg, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' In this section we employ the notion of input ablation (Covert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=', 2021) to evaluate the faith- fulness of our context mixing score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' The influence of a target token on a model’s decision is often estimated as the drop in the model’s predicted prob- ability of the correct class after blanking out the target token from the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' A higher drop for an ablated token indicates that the token is more in- fluential on the model’s decision (DeYoung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Abnar and Zuidema, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Atanasova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' We use this blank-out approach as a base for analyzing and comparing the faithfulness of the existing context mixing scores and our proposed one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' To estimate the blank-out scores in BERT, we calculate the probability of its output y using a softmax function normalized over only the corre- sponding logit values of target t and foil words (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Table 1), and compute blank-out scores for a given input token i as p(yt|e) − p(yt|e\\ei), where ei refers to the input embedding of input token i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' We compare these blank-out scores with con- text mixing scores, aggregated across all layers of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' For gradient-based scores, calculat- ing them with respect to the tokens in the input Method ρPT ρFT Rand 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='00 Attn 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='07 Attn-norm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='14 Attn-norm + RES 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='05 Attn-norm + RES + LN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='17 ALTI 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='10 GradXinput 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='16 IG 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='21 DL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='29 Value Zeroing 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='31 Table 3: Spearman’s ρ correlation between the blank- out scores and different aggregated context mixing and attribution scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' PT and FT refer to pre-trained and fine-tuned conditions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' embedding layer (ℓ = 0) provides us with aggre- gated scores since the backpropagation of gradients passes through all layers to the beginning of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' For other scores, we use the rollout (Abnar and Zuidema, 2020) aggregation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Table 3 shows Spearman’s rank correlation be- tween the blank-out scores and different aggregated context mixing scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' The highest correlation for our method indicates that Value Zeroing is more faithful in explaining the model behaviour com- pared to other analysis methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Qualitative Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' We also take a closer look at the aggregated scores for a qualitative compar- ison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' In Figure 4, we illustrate different scores obtained from a fine-tuned BERT model for a cor- rectly classified example, where the model is asked to fill the masked token with one of the verbs were or is as target and foil classes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Ac- cording to Value Zeroing scores, the model mainly relies on the main subject (pictures) as a cue word to form a contextualized representation of the [MASK] token, while the word pictures is also important for the model’s final decision based on the blank-out scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' In this example, the blank- out score for the cue word is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='99, meaning the model fully loses its confidence in the target class when the cue word is replaced with an [UNK] token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Surprisingly, gradient-based methods tend to highlight the word hat which is an agreement attractor, and attention-based scores tend to focus on the [CLS] token which has been idle during fine-tuning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Overall, our faithfulness evaluation and qualita- tive analysis suggest that Value Zeroing can explain model decisions at a global level when it is aggre- blank-out: [CLS] the pictures of some hat [MASK] scar ##ing marcus .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' [SEP] Attn: [CLS] the pictures of some hat [MASK] scar ##ing marcus .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' [SEP] Attn-norm: [CLS] the pictures of some hat [MASK] scar ##ing marcus .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' [SEP] Attn-norm+RES: [CLS] the pictures of some hat [MASK] scar ##ing marcus .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' [SEP] Attn-norm+RES+LN: [CLS] the pictures of some hat [MASK] scar ##ing marcus .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' [SEP] ALTI: [CLS] the pictures of some hat [MASK] scar ##ing marcus .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' [SEP] GradXinput: [CLS] the pictures of some hat [MASK] scar ##ing marcus .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' [SEP] IG: [CLS] the pictures of some hat [MASK] scar ##ing marcus .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' [SEP] DL: [CLS] the pictures of some hat [MASK] scar ##ing marcus .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' [SEP] Value Zeroing: [CLS] the pictures of some hat [MASK] scar ##ing marcus .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' [SEP] Figure 4: Most influential tokens on the target repre- sentation in a fine-tuned BERT model according to dif- ferent aggregated context mixing scores compared to blank-out scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' gated across layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' The context mixing maps per layer are provided in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='4, where some more meaningful patterns can be found in Value Zeroing scores (in both layer-wise and aggregated setups) in contrast to other context mixing scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' 8 Discussion Although some desiderata such as plausibility (Lei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Strout et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=', 2019) and faithfulness (Lakkaraju et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Jacovi and Goldberg, 2020) are taken into account when developing explana- tion and analysis methods, evaluating them is still a challenge due to lack of a standard ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Evaluating context mixing scores, where token-to- token interactions in a context are also considered, is even more challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Several studies have used gradient-based scores as an anchor of faithfulness, and measure how strongly context mixing scores correlate with them (Jain and Wallace, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Ab- nar and Zuidema, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Modarressi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' However, the reliability of gradient-based scores can be questioned, especially when different vari- ations of them show considerable disagreement (Neely et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Pruthi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Krishna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Thus, we suggest using controlled tasks for which we have strong prior expectations for evaluating these methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' In our study, we use a set of number agreement tasks to provide such priors, since the cue words are the only sources of information in the context for performing well in the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Another point worth discussing is the concern raised by Kobayashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' (2021) that BERT tends to preserve token representations rather than mix- ing them at each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' We argue that their ob- servation is due to the context-mixing ratio they defined by comparing the norm of residual effects against other token representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' In our view, this ratio is dominated by residuals and neglects the fact that a token representation carried by residual connections is indeed a contextualized representa- tion outputted from previous layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' We keep the residuals intact within the encoder layer by zeroing only the value vectors and focusing on the context mixing performed by all tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' 9 Conclusion In this paper, we propose Value Zeroing as a novel approach for quantifying the information mixing process in Transformers to address the shortcom- ings of previous methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' We performed exten- sive complementary experiments and showed that our method outperforms others in three different evaluation setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Since our approach requires no supervision, it could be an interesting option for improving model efficiency by removing token rep- resentations across layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' 10 Limitations As is the case for most attempts at interpreting Deep Learning models, our evaluation of our (and others’) proposed methods are not definite since we have no gold standard of what happens inside a model, although we try to remedy for that by conducting independent and complementary evalu- ation schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Our proposed method is customized for deep neural models based on the Transformer archi- tecture and cannot be easily generalized to other (mathematically different) modeling architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Our evaluations were based on encoder-based mod- els, and focused on the Text modality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' In the future, we will extend our experiments to more modalities, such as speech and vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Acknowledgments This publication is part of the project InDeep: Inter- preting Deep Learning Models for Text and Sound (with project number NWA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='1292.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='399) of Na- tional Research Agenda (NWA-ORC) programme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Funding by the Dutch Research Council (NWO) is gratefully acknowledged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' References Samira Abnar and Willem Zuidema.' metadata={'source': 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vector norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' In Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing (EMNLP), pages 7057–7075, Online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Association for Computa- tional Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Goro Kobayashi, Tatsuki Kuribayashi, Sho Yokoi, and Kentaro Inui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Incorporating Residual and Normalization Layers into Analysis of Masked 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Methods in Natural Language Processing, pages 4527–4546, Online and Punta Cana, Dominican Re- public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Mariya Toneva and Leila Wehbe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Interpret- ing and improving natural-language processing (in machines) with natural language-processing (in the brain).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' In Advances in Neural 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Curran Associates, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Elena Voita, Pavel Serdyukov, Rico Sennrich, and Ivan Titov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Context-aware neural machine trans- lation learns anaphora resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' In Proceedings of the 56th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pages 1264–1274, Melbourne, Australia.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Association for Computa- tional Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Lijie Wang, Yaozong Shen, Shu ping Peng, Shuai Zhang, Xinyan Xiao, Hao Liu, Hongxuan Tang, Ying Chen, Hua Wu, and Haifeng Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' A fine-grained interpretability evaluation benchmark for neural nlp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' ArXiv, abs/2205.' metadata={'source': 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Natural Language Processing (EMNLP- IJCNLP), pages 11–20, Hong Kong, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Associ- ation for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Thomas Wolf, Lysandre Debut, Victor Sanh, Julien Chaumond, Clement Delangue, Anthony Moi, Pier- ric Cistac, Tim Rault, Remi Louf, Morgan Funtow- icz, Joe Davison, Sam Shleifer, Patrick von Platen, Clara Ma, Yacine Jernite, Julien Plu, Canwen Xu, Teven Le Scao, Sylvain Gugger, Mariama Drame, Quentin Lhoest, and Alexander Rush.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Trans- formers: State-of-the-art natural language process- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' In Proceedings of the 2020 Conference on Em- pirical Methods in Natural Language Processing: System Demonstrations, pages 38–45, Online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Asso- ciation for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Sho Yokoi, Ryo Takahashi, Reina Akama, Jun Suzuki, and Kentaro Inui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Word rotator’s distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' In Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing (EMNLP), pages 2944–2960, Online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Association for Computa- tional Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Hao Yuan, Yongjun Chen, Xia Hu, and Shuiwang Ji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Interpreting deep models for text analysis via optimization and regularization methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' In Pro- ceedings of the Thirty-Third AAAI Conference on Ar- tificial Intelligence and Thirty-First Innovative Ap- plications of Artificial Intelligence Conference and Ninth AAAI Symposium on Educational Advances in Artificial Intelligence, AAAI’19/IAAI’19/EAAI’19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' AAAI Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Ruiqi Zhong, Steven Shao, and Kathleen McKeown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Fine-grained sentiment analysis with faithful attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' ArXiv, abs/1908.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='06870.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' A Appendices A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='1 On the choice of distance function The purpose of this section is to inspect the impact of selecting different distance metrics when com- puting Value Zeroing in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Timkey and van Schijndel (2021) questioned the informativity of standard representational distance measures such as cosine and Euclidean by observing that only a small subset of rogue dimensions contribute to the anisotropy of a contextualized representation space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' They proposed using simple post-processing tech- niques to correct for such these rough dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' We followed their suggestion and normalized the representations before computing distances, but we did not observe any noticeable difference in our scores compared to using non-normalized repre- sentations (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' We also repeated our ex- periment with Spearman’s and Euclidean distance metrics and observed the same pattern in the results (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' We believe that in anisotropy studies that use clustering methods, the choice of distance metrics is crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' However, we compute each token’s dis- tance from itself (not from other tokens) and com- pare them relatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' This might explain why we observe the same pattern for different distance met- rics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='2 More metrics Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='2 reports the cue alignment evaluation for BERT model based on Probes-needed (Zhong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=', 2019) metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='3 More PLMs We replicated our experiment for the cue alignment for two more PLMs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' RoBERTa (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=', 2019) and ELECTRA (generator, Clark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' As we can see in Figures A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='3 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content='4, our method consistently outperforms other methods on all mod- els in both pre-trained and fine-tuned setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' Due to the fact that our scores are based on zeroing value vectors, our method can be easily applied to any Transformer-based models even with different modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} +page_content=' A.' 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global scores (Value Zeroing) aggregated by rollout method across layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFOT4oBgHgl3EQf9zRP/content/2301.12971v1.pdf'} diff --git a/ZtFRT4oBgHgl3EQfPze-/content/tmp_files/2301.13519v1.pdf.txt b/ZtFRT4oBgHgl3EQfPze-/content/tmp_files/2301.13519v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..5236d1ccf9fb29f068cbf5fbdd1e15ca5f21f3b5 --- /dev/null +++ b/ZtFRT4oBgHgl3EQfPze-/content/tmp_files/2301.13519v1.pdf.txt @@ -0,0 +1,5416 @@ +Restricted distance-type Gaussian +estimators based on density power +divergence and their aplications in +hypothesis testing +A. Felipe, M. Jaenada, P. Miranda and L. Pardo +Department of Statistics and O.R., Complutense University of Madrid, Spain +Abstract +Zhang (2019) presented a general estimation approach based on the +Gaussian distribution for general parametric models where the likelihood +of the data is difficult to obtain or unknown, but the mean and variance- +covariance matrix are known. Castilla and Zografos (2021) extended the +method to density power divergence-based estimators, which are more ro- +bust than the likelihood-based Gaussian estimator against data contami- +nation. Here, we present the restricted minimum density power divergence +Gaussian estimator (MDPDGE) and study it asymptotic and robustness +properties through it asymptotic distribution and influence function, re- +spectively. Restricted estimators are required in many practical situations +and provide here constrained estimators to inherent restrictions of the un- +derlying distribution. Further, we derive robust Rao-type test statistics +based on the MDPDGE for testing simple a composite null hypothesis and +we deduce explicit expression for some main important distributions. Fi- +nally, we empirically evaluate the efficiency and robustness of the method +through a simulation study. +AMS 2001 Subject Classification: 62F35, 62J12 +Keywords and phrases: Gaussian estimator, Minimum density power diver- +gence Gaussian estimator, Robustness, Influence function, Restricted Minimum +density power divergence Gaussian estimator, Rao-type tests, Elliptical family +of distributions +1 +Introduction +Let Y 1, ..., Y n be independent and identically distributed observations from a +m-dimensional random vector Y with probability density function fθ(y), θ ∈ +Θ ⊂ Rd. We denote, +Eθ [Y ] = µ (θ) and Covθ [Y ] = Σ (θ) . +(1) +1 +arXiv:2301.13519v1 [math.ST] 31 Jan 2023 + +The log-likelihood function, in order to get the maximum likelihood estimator +(MLE), is given by, +l (θ) = +n� +i=1 +log fθ(yi) +being y1, ..., yn realizations of the m-dimensional random vectors Y1, ..., Yn. +If could be the case that the computation of fθ(y) is difficult or it is unknown +but the computation of Eθ [Y ] and Covθ [Y ] is not. In this case, Zhang (2019) +proposed to assume that fθ(y) follows a m−dimensional normal distribution +with vector mean µ (θ) and variance-covariance matrix Σ (θ) as given in (1). +From a statistical point of view this procedure can be justified on the basis +of the maximum-entropy principle, see Kapur (1989), because the distribution +which is consistent +with the given information, vector mean and variance- +covariance matrix, and at the same time has maximum uncertainty, in terms +of Shannon entropy, is the multidimensional normal population with vector +mean and variance-covariance given. Hence Zhang (2019) defines the Gaussian +likelihood function of θ, under y1, ..., yn, by +lG (θ) = −nm +2 log 2π − n +2 log |Σ (θ)| − 1 +2 +n� +i=1 +(yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) +(2) +and the Gaussian estimator of θ by, +�θG = arg max +θ∈Θ lG (θ) . +This idea consists of on assuming that the observations y1, ..., yn are from a +m-variate normal population with vector mean µ (θ) and variance-covariance +matrix Σ (θ) , the vector mean and variance-covariance matrix associated to the +population Y . +The Gaussian estimator is related with the Kullback-Leibler divergence and +work well in the sense of the asymptotic efficiency but it has, as the classi- +cal MLE, serious robustness problems. For this reason Castilla and Zografos +(2022) extended the Guassian estimator on the basis of the density power di- +vergence (DPD) defining the minimum density power divergence Gaussian esti- +mator (MDPDGE), by +�θ +τ +G = arg +max +θ∈Θ⊂Rd Hτ +n (θ) +(3) +being +Hτ +n (θ) += +τ + 1 +τ (2π)mτ/2 |Σ (θ)|τ/2 +1 +n +(4) +� n� +i=1 +exp +� +−τ +2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) +� +− +τ +(1 + τ)(m/2)+1 +� +− 1 +τ += +a |Σ (θ)|− τ +2 1 +n +� n� +i=1 +exp +� +−τ +2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) +� +− b +� +− 1 +τ , +2 + +where, +a = +τ + 1 +τ (2π)mτ/2 +and b = +τ +(1 + τ)(m/2)+1 . +(5) +It is immediate to see that the Gaussian estimator, �θG, of Zhang (2019) can be +defined by, +�θG = arg +max +θ∈Θ⊂Rd H0 +n (θ) +with +H0 +n (θ) = lim +τ→0 Hτ +n (θ) = −n +2 log |Σ (θ)|−1 +2 +n� +i=1 +(yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) . +Hence, the MDPDGE is based in the DPD measure, defined by Basu et al (1998), +while the Gaussian estimator is based on the Kullback- Leibler divergence. All +details about the MDPDGE, �θ +τ +G, can be seen in Castilla and Zografos (2022). In +particular, in the cited paper, the consistency and the asymptotic distribution of +the MDPDGE, �θ +τ +G, was studied. Given Y 1, ..., Y n independent and identically +distributed vectors from the m-dimensional random vector Y , the MDPDGE, +�θ +τ +G, defined in (3) satisfies, +√n +� +�θ +τ +G − θ +� +L +−→ +n−→∞ N(0d, Jτ(θ)−1Kτ(θ)Jτ(θ)−1) +(6) +being +Jτ(θ) = +� +Jij +τ (θ) +� +i,j=1,..,,d and Kτ(θ) = +� +Kij +τ (θ) +� +i,j=1,..,,d . +The elements Jij +τ (θ) and Kij +τ (θ) of the matrices Jτ (θ) and Kτ (θ) are given +by, +Jij +τ (θ) += +� +1 +(2π)m/2 |Σ (θ)|1/2 +�τ +1 +(1 + τ)(m/2)+2 +(7) +� +(τ + 1) trace +� +Σ (θ)−1 ∂µ (θ) +∂θ +�∂µ (θ) +∂θ +�T � ++∆i +τ∆j +τ + 1 +2trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +Σ (θ)−1 ∂Σ (θ) +∂θi +�� +and +Kij +τ (θ) += +� +1 +(2π)m/2 |Σ (θ)|1/2 +�2τ +1 +(1 + 2τ)(m/2)+2 +� +∆i +2τ∆j +2τ +(8) ++ (1 + 2τ) trace +� +Σ (θ)−1 ∂µ (θ) +∂θ +�∂µ (θ) +∂θ +�T � ++1 +2trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +Σ (θ)−1 ∂Σ (θ) +∂θi +�� +− +� +1 +(2π)m/2 |Σ (θ)|1/2 +�2τ +1 +(1 + τ)m+2 ∆i +τ∆j +τ, +3 + +with ∆i +τ = τ +2trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +� +. +On certain occasions we have an additional knowledge about the true param- +eter because it satisfies certain restrictions, i.e., the parameter space is restricted +in the way, +{θ ∈ Θ/ g(θ) = 0r} , +(9) +where g : Rd → Rr is a vector-valued function mapping such that the d × r +matrix +G (θ) = ∂gT (θ) +∂θ +(10) +exists and is continuous in θ and rank(G (θ)) = r; also 0r denotes the null vector +of dimension r. In the following the parameter space given in (9) will be denoted +by Θ0 because it will represent, in most of the situations, a composite null +hypothesis. The superscript T in (10) represents the transpose of the matrix. +The most popular estimator of θ under the restriction given in (9) is the +restricted MLE (RMLE) that is an estimator which maximizes the loglikelihood +function subject to the restrictions g(θ) = 0r (see Silvey, 1975). The RMLE +has the same robustness problems that the MLE. Later some restricted estima- +tors have been considered in the statistical literature to overcome the problem +of robustness of the RMLE. We will only mention the restricted estimators +based on divergence measures: In Pardo et al (2002), the restricted minimum +Phi-divergence estimator was introduced and its properties studied. Basu et +al (2018) presented the restricted minimum density power divergence estima- +tors (RMDPDE) and studied some applications of them in testing hypothesis. +In Ghosh (2015) the theoretical robustness properties of the RMDPDE were +studied. More recently Jaenada et al (2022) considered the restricted R´enyi +pseudodistance estimator as well as its use in defining Rao-type tests based on +it. But if it is not easy to get the fθ(y) or it is unknown the previous estimators +can not be obtained and for this reason in this paper we are going to introduce +and to study the restricted minimum density power divergence Gaussian esti- +mator (RMDPDGE). In Section 2 we introduce the RMDPDGE and we obtain +its asymptotic distribution. The rest of the paper goes as follows: Section 3 is +devoted to get the influence function of the RMDPDGE. Some statistical appli- +cations for testing are presented in Section 4. A simulation study is carried out +in Section 5. Finally, we present an Appendix in which we have included the +proofs of some of the results appearing in the paper. +2 +Restricted minimum density power divergence +Gaussian estimators +We shall begin giving the definition of the RMDPDGE. +Definition 1 Let Y 1, ..., Y n be independent and identically distributed ob- +servations from a m-dimensional random vector Y with Eθ [Y ] = µ (θ) and +4 + +Covθ [Y ] = Σ (θ), θ ∈ Θ ⊂ Rd. The RMDPDGE, �θ +τ +G, is defined by +�θ +τ +G = arg +max +θ∈Θ/ g(θ)=0r +Hτ +n (θ) , +where Hτ +n (θ) was given in (4). +The main purpose in this Section is to get the asymptotic distribution of �θ +τ +G. +Before presenting the theorem with the asymptotic distribution we shall give +some previous results that are necessary in order to proof the theorem. +Proposition 2 Let Y 1, ..., Y n be independent and identically distributed ob- +servations from a m-dimensional random vector Y with Eθ [Y ] = µ (θ) and +Covθ [Y ] = Σ (θ), θ ∈ Θ ⊂ Rd. Then, +√n +� +1 +τ + 1 +∂Hτ +n (θ) +∂θ +� +L +−→ +n−→∞ N(0d, Kτ(θ)), +where Kτ(θ) was defined in (8). +Proof. See Appendix A. +Proposition 3 Let Y 1, ..., Y n be independent and identically distributed ob- +servations from a m-dimensional random vector Y with with Eθ [Y ] = µ (θ) +and Covθ [Y ] = Σ (θ), θ ∈ Θ ⊂ Rd, we have, +∂2Hτ +n (θ) +∂θ∂θT +P +−→ +n−→∞ − (τ + 1) Jτ(θ), +where Jτ(θ) was defined in (7). +Proof. See Appendix B. +In the next theorem we present the asymptotic distribution of the RMD- +PDGE, �θ +τ +G. +Theorem 4 Let Y 1, ..., Y n be independent and identically distributed observa- +tions from a m-dimensional random vector Y with Eθ [Y ] = µ (θ) and Covθ [Y ] = +Σ (θ), θ ∈ Θ ⊂ Rd. Suppose the true distribution of Y belongs to the model and +θ ∈ Θ0 is the true parameter. Then the RMDPDGE �θ +τ +G of θ obtained under +the constraints g(θ) = 0r has the asymptotic distribution, +n1/2(�θ +τ +G − θ) +L +−→ +n−→∞ N(0d, M τ(θ)) +(11) +where +M τ(θ) = P ∗ +τ(θ)Kτ (θ) P ∗ +τ(θ)T , +P ∗ +τ(θ) = Jτ(θ)−1 − Qτ(θ)G(θ)T Jτ(θ)−1, +(12) +Qτ(θ) = J−1 +τ (θ)G(θ) +� +G(θ)T Jτ(θ)−1G(θ) +�−1 , +(13) +and Jτ(θ) and Kτ (θ) were defined in (7) and (8), respectively. +5 + +Proof. The estimating equations for the RMDPDGE are given by +� +∂ +∂θHτ +n(θ) + G(θ)λn = 0d, +g(�θ +τ +G) = 0r, +(14) +where λn is a vector of Lagrangian multipliers. Now we consider θn = θ0 + +mn−1/2, where ||m|| < k, for 0 < k < ∞. We have, +∂ +∂θ Hτ +n(θ)|θ=θn = ∂ +∂θ Hτ +n(θ) + +∂2 +∂θT ∂θ +Hτ +n(θ)|θ=θ∗ (θn − θ) + o(||θn − θ||2) +and +n1/2 ∂ +∂θ Hτ +n(θ) +���� +θ=θn += +n1/2 ∂ +∂θ Hτ +n(θ) + +∂2 +∂θT ∂θ +Hτ +n(θ)|θ=θ∗ n1/2 (15) +(θn − θ) + o(n1/2||θn − θ||2) +where θ∗ belongs to the segment joining θ and θ∗ +However, +o(n1/2||θn − θ||2) = o(n1/2||m||2/n) = o(n−1/2||m||2) = o(Op(1)) = op(1). +Since +lim +n→∞ +∂2 +∂θT ∂θ +Hτ +n(θ) = − (τ + 1) Jτ(θ) +we obtain +n1/2 ∂ +∂θ Hτ +n(θ) +���� +θ=θn += n1/2 ∂ +∂θ Hτ +n(θ)−(τ + 1) n1/2Jτ(θ)(θn−θ)+op(1). (16) +Taking into account that G(θ) is continuous in θ +n1/2g(θn) = G(θ)T n1/2(θn − θ) + op(1). +(17) +The RMDPDGE �θ +τ +G must satisfy the conditions in (14), and in view of (16) +and (17) we have +n1/2 ∂ +∂θ Hτ +n(θ) − (τ + 1) Jτ(θ)n1/2(�θ +τ +G − θ) + G(θ)n1/2λn + op(1) = 0p. (18) +From (17) it follows that +GT (θ)n1/2(�θ +τ +G − θ) + op(1) = 0r. +(19) +Now we can express equations (18) and (19) in the matrix form as +� (τ + 1) Jτ(θ) +−G(θ) +−GT (θ) +0 +� � +n1/2(�θ +τ +G − θ0) +n1/2λn +� += +� +n1/2 ∂ +∂θHτ +n(θ) +0 +� ++ op(1). +6 + +Therefore +� +n1/2(�θ +τ +G − θ) +n1/2λn +� += +� (τ + 1) Jτ(θ) +−G(θ) +−GT (θ) +0 +�−1 � +−n1/2 ∂ +∂θHτ +n(θ) +0r +� ++op(1). +But +� (τ + 1) Jτ(θ) +−G(θ) +−GT (θ) +0 +�−1 += +� +L∗ +τ(θ) +Qτ(θ) +Qτ(θ0)T +Rτ(θ) +� +, +where +L∗ +τ(θ) += +1 +τ + 1Jτ(θ)−1 − Qτ(θ)G(θ)T Jτ(θ)−1 += +1 +τ + 1P ∗ +τ(θ) +Qτ(θ) += +J−1 +τ (θ)G(θ) +� +G(θ)T Jτ(θ)−1G(θ) +�−1 +Rτ(θ) += +G(θ)T Jτ(θ)−1G(θ) +P ∗ +τ(θ0) and Qτ(θ0) are as given in (12) and (13) respectively. Then, +n1/2(�θ +τ +G − θ) = (τ + 1)−1 P ∗ +τ(θ)n1/2 ∂ +∂θ Hτ +n(θ) + op(1), +(20) +and we know by Proposition 2 that +n1/2 (τ + 1)−1 ∂ +∂θ Hτ +n(θ) +L +−→ +n−→∞ N (0, Kτ (θ)) . +(21) +Now by (20) and (21) we have the desired result presented in (11). +Remark 5 Notice that the result in (6) is a special case of the previous theorem +when there is no restriction on the parametric space, in the sense that G, defined +in (10), is the null matrix. In this case the matrix P ∗ +τ(θ) given in (12) becomes +P ∗ +τ(θ) = Jτ(θ)−1. Therefore, the asymptotic variance-covariance matrix of the +unrestricted estimator may be reconstructed from the previous theorem. +3 +Influence function for the RMDPDGE +Based on +• ∂ |Σ (θ)|−τ/2 +∂θ += − τ +2 |Σ (θ)|−τ/2 trace +� +Σ (θ)−1 ∂Σ (θ) +∂θ +� +• +∂ +∂θ (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) = −2 +� +∂µ(θ) +∂θ +�T +Σ (θ)−1 (yi − µ (θ))− +(yi − µ (θ))T +� +Σ (θ)−1 ∂Σ (θ) +∂θ +Σ (θ)−1 +� +(yi − µ (θ)) , +7 + +we have +∂ +∂θ Hτ +n(θ) += +1 +n +n� +i=1 +� +−a τ +2 |Σ (θ)|−τ/2 trace +� +Σ (θ)−1 ∂Σ (θ) +∂θ +� +exp +� +−τ +2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) +� ++ ba τ +2 |Σ (θ)|−τ/2 +trace +� +Σ (θ)−1 ∂Σ (θ) +∂θ +� ++ a τ +2 |Σ (θ)|−τ/2 exp +� +−τ +2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) +� +� +2 +�∂µ (θ) +∂θ +�T +Σ (θ)−1 (yi − µ (θ)) ++ (yi − µ (θ))T +� +Σ (θ)−1 ∂Σ (θ) +∂θ +Σ (θ)−1 +� +(yi − µ (θ)) +�� += +1 +n +n� +i=1 +Ψτ(yi; θ), +where +Ψτ(yi; θ) += +a τ +2 |Σ (θ)|−τ/2 +� +−trace +� +Σ (θ)−1 ∂Σ (θ) +∂θ +� +(22) +exp +� +−τ +2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) +� ++ b trace +� +Σ (θ)−1 ∂Σ (θ) +∂θ +� ++ exp +� +−τ +2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) +� +� +2 +�∂µ (θ) +∂θ +�T +Σ (θ)−1 (yi − µ (θ)) ++ (yi − µ (θ))T +� +Σ (θ)−1 ∂Σ (θ) +∂θ +Σ (θ)−1 +� +(yi − µ (θ)) +�� +. +Therefore the estimating equations are, +n� +i=1 +Ψτ(yi; θ) = 0d, +(23) +and the MDPDGE is an M-estimator. +Based on the general theory of M- +estimators we know that +√n +� +�θ +τ +G − θ +� +L +−→ +n−→∞ N +� +0d, S−1MS−1� +being +S = −E +�∂2Hτ +n (θ) +∂θ∂θT +� +and M = Cov +�√n ∂ +∂θ Hτ +n(θ) +� +. +Based on Propositions 2 and 3 we have, +S = (τ + 1) Jτ (θ) and M = (τ + 1)2 Kτ (θ) +8 + +and we get the result given in (6). +To analyze the robustness of an estimator, Hampel et al. (1986) introduced +the concept of Influence Function (IF). Since then, the IF have been widely +used in the statistical literature to measure robustness in different statistical +contexts. Intuitively, the IF describes the effect of an infinitesimal contamina- +tion of the model on the estimate. Then, IFs associated to locally robust (B- +robust) estimators should be bounded. Let us now obtain the IF of RMDPDE. +We consider the contaminated model gε(y) = (1 − ε)fθ(y) + ε∆y, with ∆y the +indicator function in y, and we denote �θ +τ +G,ε = �Tτ(Gε), being Gε the distribution +function associated to gε. By definition �θ +τ +G,εis the minimizer of Hτ +n(θ) subject +to g(�θ +τ +G,ε) = 0. Taking into account that the MDPDGE is an M-estimator we +have that the influence function of the MDPDGE is given by +IF(y, �Tτ, θ) = Jτ(θ)−1Ψτ(y; θ), +(24) +where Jτ(θ) was defined in (7) and Ψτ(y; θ) in (22). The influence function +of the RMDPDGE will be obtained with the additional condition g(�θ +τ +G,ε) = 0. +Differentiating this last equation gives, at ε = 0, +G (θ)T IF(y, �Tτ, θ) = 0. +(25) +Based on (24) and (25) we have +� Jτ(θ) +G (θ)T +� +IF(y, �Tτ, θ) = +� +Ψτ(y; θ) +0 +� +. +Therefore, +� +Jτ(θ)T , G (θ) +� � Jτ(θ) +G (θ)T +� +IF(y, �Tτ, θ) = Jτ(θ)T Ψτ(y; θ) +and the influence function of the RMDPDGE, �θ +τ +G, is given by +� +Jτ(θ)T Jτ(θ)) + G (θ) G (θ)T �−1 +Jτ(θ)T Ψτ(y; θ). +(26) +We can observe that the influence function of the �θ +τ +G, obtained in (26) will be +bounded if the influence function of the MDPDGE, �θ +τ +G, given in (24) is bounded. +In general it is not easy to see if it is bounded or not but in particular situations +is not difficult. On the other hand if there are not restrictions, G (θ) = 0, and +therefore (26) coincides with (24). +In Section 4.1 we shall present the expression of Jτ(θ) and ψτ(y, θ) for the +exponential and Poisson models. Based on that results we present in Figure 1 +the influence function of the MDPDGE, �θ +τ +G, for θ = 4 and τ = 0, 0, 2 and 0.8 +for the exponential model. We can see that for τ = 0, the influence function +is not bounded and for τ = 0, 2 and 0.8 is bounded. This fact points out the +robustness of the MDPDGE, �θ +τ +G, for τ > 0. +9 + +Figure 1: Influence function of the MDPDGE for the exponential model with +τ = 0 (red), τ = 0.2 (black) and τ = 0.8 (green) +4 +Rao-type tests based on RMDPDGE +In the last years many robust test statistics have been introduced in the statis- +tical literature based on minimum distance estimators. We pay special atten- +tion to the procedures based on density power divergence (MDPDE) as well as +the procedures based on Renyi’s pseudodistance estimator (MRPE). The test +statistics are essentially of two types: Wald-type tests and Rao-type tests. Some +references are the following: Basu et al. (2013, 2016, 2017, 2018a, 2018b, 2022a, +2022b), Castilla et al (2016, 2021), Jaenada et al (2022a, 2022b), Men´endez et +al (1995) and references therein. +In this section we are going to introduce and study the Rao-type tests based +on RMDPDGE. We analyze the case of simple null hypothesis because for com- +posite null hypothesis it is necessary a separated paper. Note that, +1 +√n +n� +i=1 +1 +τ + 1Ψτ(yi; θ) = √n +1 +τ + 1 +∂ +∂θ Hτ +n(θ), +and hence by Proposition 2, +1 +√n +n� +i=1 +1 +τ + 1Ψτ(yi; θ) +L→ +n→∞ N (0p, Kτ (θ)) . +Definition 6 Let Y 1, ..., Y n be independent and identically distributed ob- +servations from a m-dimensional random vector Y with Eθ [Y ] = µ (θ) and +Covθ [Y ] = Σ (θ), θ ∈ Θ ⊂ Rd. For testing +H0 : θ = θ0 versus H1 : θ ̸= θ0. +(27) +10 + +IF(y) +10 +8 +6 +4 +2 +0 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +14 +15 +ythe Rao-type test statistic, based on RMDPDGE, is defined by +Rτ (θ0) = 1 +nU τ +n(θ0)T Kτ (θ0)−1 U τ +n(θ0) +where +U τ +n(θ) = +� +1 +τ + 1 +n� +i=1 +Ψ1 +τ(yi; θ), ..., +1 +τ + 1 +�n +i=1 Ψd +τ(yi; θ) +�T +and Ψτ(yi; θ) = +� +Ψ1 +τ(yi; θ), ...., Ψd +τ(yi; θ) +� +. +In the following we shall call Rτ (θ0), ”Rao-type test based on RMDPDGE”. +Theorem 7 Let Y 1, ..., Y n be independent and identically distributed obser- +vations from a m- dimensional random vector Y with Eθ [Y ] = µ (θ) and +Covθ [Y ] = Σ (θ), θ ∈ Θ ⊂ Rd. Under the null hypothesis given in (27) we +have, +Rτ (θ0) +L→ +n→∞ χ2 +d. +Proof. It is clear that +1 +√nU τ +n(θ) = +1 +√n +n� +i=1 +1 +τ + 1Ψτ(yi; θ) = √n +1 +τ + 1 +∂ +∂θ Hτ +n(θ) +L +−→ +n−→∞ N (0, Kτ (θ)) . +Now the result follows. +Remark 8 Based on previous theorem, if the sample size is large enough, one +can use the 100(1 − α) percentile, χ2 +d,α, of the chi-square with d degrees of +freedom defined by the equation Pr +� +χ2 +d > χ2 +d,α +� += α, to propose the decision +rule: ”Reject H0, with a significant level α, if Rτ (θ0) > χ2 +d,α”. +Example 9 (Elliptical distributions). +The m-dimensional random vector Y +has an elliptical distribution if its characteristic function has the form +ϕY (t) = exp +� +it2µ +� +ψ +�1 +2t2Σt +� +being µ a m−dimensional column vector, Σ a positive definite matrix and ψ(t) +the so-called characteristic generator function. The function ψ may depend on +the dimension of random vector Y . In general, it does not follow that Y has a +joint density function, fY (y), but this density exists, it is given by +fY (y) = cm |Σ|− 1 +2 gm +�1 +2 (y − µ)T Σ−1 (y − µ) +� +for some density generator function gm which could depend on the dimension +m. The elliptical family of distributions is denoted by Em (µ, Σ,gm) . In the case +the density exists cm is given explicitly by +cm = (2π)− m +2 Γ +�m +2 +� �� +x +m +2 −1gm (x) dx +�−1 +. +11 + +For more details about the family Em (µ, Σ,gm) see for instance Fang et al +(1987), Gupta and Varga (1993), Cambanis et al (1981), Fang and Zhang +(1990).and references therein. In Fang et al (1987), for instance, can be seen +that +E [Y ] = µ and Cov [Y ] = cY Σ +where cY = −2ψ′(0). +In this case, the parameter to be estimated is θ = +� +µT , Σ +� +whose dimension +is s = m + m(m+1) +2 +. In the following we shall denote µ(θ) instead of µ and +Σ (θ) instead of Σ, in order to be consistent with our previous notation. +If we are interested in testing +H0 : (µ(θ), Σ (θ)) = (µ0, Σ0) versus H1 : (µ(θ), Σ (θ)) ̸= (µ0, Σ0) +(28) +where µ0 and Σ0 completely known. Hence, We shall reject the null hypothesis +given in (28) if +Rτ (µ0, Σ0) = 1 +nU τ +n(µ0, Σ0)T Kτ (µ0, Σ0)−1 U τ +n(µ0, Σ0) > χ2 +m+ m(m+1) +2 +,α +where +U τ +n(µ0, Σ0 ) = +n� +i=1 +1 +τ + 1Ψτ(yi; µ0, Σ0) +with Ψτ(yi; µ0, Σ0) is obtained from (22) replacing Σ (θ) by cY Σ and µ (θ) by +µ and Kτ (µ0, Σ0) is obtained from (8) replacing Σ (θ) by cY Σ and µ (θ) by +µ . +Let us now establish the consistency of the Score-type test based on RMD- +PDGE. In the following we shall denote, +Y τ(θ) += +a τ +2 |Σ (θ)|−τ/2 +� +−trace +� +Σ (θ)−1 ∂Σ (θ) +∂θ +� +(29) +exp +� +−τ +2 (Y − µ (θ))T Σ (θ)−1 (Y − µ (θ)) +� ++ b trace +� +Σ (θ)−1 ∂Σ (θ) +∂θ +� ++ exp +� +−τ +2 (Y − µ (θ))T Σ (θ)−1 (Y − µ (θ)) +� +� +2 +�∂µ (θ) +∂θ +�T +Σ (θ)−1 (Y − µ (θ)) ++ (Y − µ (θ))T +� +Σ (θ)−1 ∂Σ (θ) +∂θ +Σ (θ)−1 +� +(Y − µ (θ)) +�� +, +where a and b were defined in (5). We can observe that +∂ +∂θHn(θ) is the sample +mean of a random sample of size n from the m-dimensional population Y τ(θ). +Theorem 10 Let Y 1, ..., Y n be independent and identically distributed obser- +vations from a m- dimensional random vector Y with Eθ [Y ] = µ (θ) and +12 + +Covθ [Y ] = Σ (θ), θ ∈ Θ ⊂ Rd. Let θ ∈ Θ with θ ̸= θ0, with θ0 defined +in (27), and let us assume that Eθ [Y τ(θ0)] ̸= 0d. Then, +lim +n→∞ P +� +Rτ (θ0) > χ2 +d,α +� += 1. +Proof. We have, +1 +nU τ +n(θ0) = 1 +n +n� +i=1 +1 +τ + 1Ψτ(Y i; θ) = +1 +τ + 1 +∂ +∂θ Hτ +n(θ) +P→ +n→∞ +1 +τ + 1E [Y τ(θ0)] , +where Y τ(θ0) was defined in (29). Therefore +P +� +Rτ (θ0) > χ2 +d,α +� += P +� 1 +nRτ (θ0) > 1 +nχ2 +d,α +� +−→ +n→∞ I +� +1 +(τ + 1)2 Eθ [Y τ(θ0)] K−1 +τ +(θ) ET +θ [Y τ(θ0)] > 0 +� += 1, +where I(·) is the indicator function. +Now let us derive the asymptotic distribution of Rτ (θ0) under local Pitman- +type alternative hypotheses of the form H1,n : θ = θn, where θn = θ0 +n−1/2d. +Such results help to determine the asymptotic contiguous power of the Score- +type test based on RMDPDGE. +Theorem 11 Let Y 1, ..., Y n be independent and identically distributed ob- +servations from a m-dimensional random vector Y with Eθ [Y ] = µ (θ) and +Covθ [Y ] = Σ (θ), θ ∈ Θ ⊂ Rd. Under the contiguous alternative hypothesis +H1,n : θn = θ0 + n−1/2d, the asymptotic distribution of the Rao-type test based +on RMDPDGE, Rτ (θ0) , is a non-central chi-square distribution with d degrees +of freedom and non-centrality parameter given by +δτ(θ0, d) = dT Jτ (θ0) K−1 +τ +(θ0) Jτ (θ0) d +Proof. Consider the Taylor series expansion +1 +√nU τ +n(θn) = +1 +√nU τ +n(θ0) + 1 +n +∂U τ +n(θ) +∂θT +���� +θ=θ∗ +n +d, +where θ∗ +n belongs to the line segment joining θ0 and θ0 + +1 +√nd. Now, by propo- +sition 3 +But +1 +n +∂U τ +n(θ) +∂θT += +1 +τ + 1 +∂2Hτ +n (θ) +∂θ ∂θT +P +−→ +n−→∞ −Jτ(θ) +Therefore, +1 +√nU τ +n(θ) +���� +θ=θ0+n−1/2d +L +−→ +n→∞ N (−Jτ (θ0) d, Kτ (θ0)) , +and +Rτ (θ0) +L +−→ +n→∞ χ2 +p (δτ(θ0, d)) , +13 + +with δτ(θ0, d) given by +δτ(θ0, d) = dT Jτ (θ0) K−1 +τ +(θ0) Jτ (θ0) d. +Remark 12 The family of Score-type tests, Rτ (θ0) , presented in this Section +for simple null hypothesis can be extended to composite null hypothesis. If we +are interested in testing H0 : θ ∈ Θ0 = {θ ∈ Θ/ g(θ) = 0r} we can consider +the family of Score-type tests given by +Rτ +� +�θ +τ +G +� += 1 +nU τ +n(�θ +τ +G)T Qτ(�θ +τ +G) +� +Qτ(�θ +τ +G)Kτ +� +�θ +τ +G +� +Qτ(�θ +τ +G) +�−1 +Qτ(�θ +τ +G)T U τ +n(�θ +τ +G). +(30) +However, the analysis of the family of test statistics presented in (30) deserves +a new paper, +that will be finished soon, in the line of the paper of Basu et al +(2022a). +4.1 +Rao-type test based on MDPDGE for univariate dis- +tributions and θ ∈ Θ +Let Y1, ...., Yn a random sample from the population Y, with +E [Y ] = µ (θ) and V ar [Y ] = σ2 (θ) . +Based on (23) the estimating equation is given by +n� +i=1 +Ψτ (yi, θ) = 0 +with +Ψτ (yi, θ) += +(τ + 1) +� +σ2 (θ) +�−τ/2 +2 (2π)τ/2 +�� +−∂ log σ2 (θ) +∂θ ++ ∂ log σ2 (θ) +∂θ +�yi − µ (θ) +σ (θ) +�2 +(31) ++2∂µ (θ) +∂θ +(yi − µ (θ)) +1 +σ2 (θ) +� +exp +� +− +τ +2σ2 (θ) (yi − µ (θ))2 +� ++ +τ +(1 + τ)3/2 +∂ log σ2 (θ) +∂θ +� +. +The expressions of Jτ (θ) and Kτ (θ) are given by +Jτ (θ) += +1 +� +2πσ (θ)2� τ +2 +1 +(1 + τ)5/2 +� +(τ + 1) σ−2 (θ) +�∂µ (θ) +∂θ +�2 ++ τ 2 +4 +�∂ log σ2 (θ) +∂θ +�2 ++1 +2 +�∂ log σ2 (θ) +∂θ +�2� +14 + +and +Kτ (θ) += +� +1 +(2π)1/2 σ (θ) +�2τ � +1 +(1 + 2τ)5/2 +� +τ 2 +�∂ log σ2 (θ) +∂θ +�2 +(32) ++ (1 + 2τ) σ−2 (θ) +�∂µ (θ) +∂θ +�2 ++ 1 +2 +�∂ log σ2 (θ) +∂θ +�2� +− +τ 2 +4 (1 + τ)3 +�∂ log σ2 (θ) +∂θ +�2� +. +Therefore, if we are interesting in testing +H0 : θ = θ0 versus H1 : θ ̸= θ0, +on the basis of the Rao-type tests based on RMDPDGE, Rτ (θ0) , we shall reject +the null hypothesis if +Rτ (θ0) = 1 +nU τ +n (θ0)2Kτ (θ0)−1 > χ2 +1,α +where +U τ +n (θ0) = +1 +τ + 1 +n� +i=1 +Ψτ (yi, θ) +with Ψτ (yi, θ) is defined in (31) and Kτ (θ) is given in (32). Let us consider +some special cases. +4.1.1 +Poisson Model +We shall assume that the random variable Y is Poisson with parameter θ. In +this case E [Y ] = V ar [Y ] = θ. The RMDPDGE, for τ > 0, is given by, +�θ +τ +G = arg max +θ +� +τ + 1 +τ (2πθ) +τ +2 +� +1 +n +n� +i=1 +exp +� +− τ +2θ (yi − θ)2� +− +τ +(1 + τ)3/2 +� +− 1 +τ +� +, +and for τ → 0, we have. +�θG = arg max +θ +� +−1 +2 log 2π − 1 +2 log θ − 1 +n +n� +i=1 +1 +2θ (yi − θ)2 +� +. +On the other hand, +Ψτ (yi, θ) = +τ + 1 +2 (2πθ) +τ +2 θ2 +� +� +−2θ2 + y2 +i +� +exp +� +− τ +2θ (yi − θ)2� ++ +τθ +(1 + τ) +3 +2 +� +and for τ = 0, we obtain the result presented by Zhang (2019) +Ψ0 (yi, θ) = +1 +2θ2 +� +−2θ2 + y2 +i +� +. +15 + +In relation to Kτ (θ) , we get, +Kτ (θ) = +� 1 +2π +�τ +1 +2θ2+τ +� +1 +(1 + 2τ)5/2 +� +� +2τ 2 + 2θ + 4θτ + 1 +� +− +τ 2 +2 (1 + τ)3 +�� +and hence the Rao-type test based on RMDPDGE, for τ > 0, Rτ (θ0) , for +testing, +H0 : θ = θ0 versus H1 : θ ̸= θ0, +is given by +Rτ (θ0) = 1 +n +1 +� +2 (2πθ) +τ +2 θ2�2 +� +n� +i=1 +� +� +−2θ2 +0 + y2 +i +� +exp +� +− τ +2θ0 +(yi − θ0)2 +� ++ +τθ +(1 + τ) +3 +2 +��2 +Kτ (θ0)−1 . +For τ = 0 we get +R0 (θ0) = 1 +4n +� n� +i=1 +�−2θ2 +0 + y2 +i +θ2 +0 +��2 +K0 (θ0)−1 , +with +K0 (θ0) = 2θ0 + 1 +2θ2 +0 +. +We shall reject the null hypothesis if +Rτ (θ0) > χ2 +1,α. +4.1.2 +Exponential model +Assume now that the random variable Y is exponential +fθ(x) = 1 +θ exp +� +−x +θ +� +, x > 0. +(33) +In this case E [Y ] = θ and V ar [Y ] = θ2. The RMDPDGE, for τ > 0, is +given by +�θ +τ +G = arg max +θ +� +τ + 1 +τ +� +1 +θ +√ +2π +�τ � +1 +n +n� +i=1 +exp +� +−τ +2 +�yi − θ +θ +�2� +− +τ +(1 + τ)3/2 +� +− 1 +τ +� +, +and for τ → 0, we have. +�θG = arg max +θ +� +−1 +2 log 2π − log θ − 1 +n +n� +i=1 +1 +2 +�yi − θ +θ +�2� +. +On the other hand, +Ψτ (yi, θ) = +(τ + 1) +θτ+3 �√ +2π +�τ +� +� +y2 +i − yiθ − θ2� +exp +� +−τ +2 +�yi − θ +θ +�2� ++ +τθ2 +(1 + τ) +3 +2 +� +, +16 + +and for τ = 0, we get +Ψ0 (yi, θ) = 1 +θ3 +� +y2 +i − yiθ + θ2� +. +The matrix Kτ (θ) has the expression +Kτ (θ) = +1 +(2π)τ θ2(τ+1) +� +1 +(1 + 2τ)5/2 +� +4τ 2 + 2τ + 3 +� +− +τ 2 +(1 + τ)3 +� +and +K0 (θ) = 2 +θ2 . +Hence the Rao-type test based on RMDPDGE, for τ > 0, for testing +H0 : θ = θ0 versus H1 : θ ̸= θ0, +is given by +Rτ (θ0) += +1 +n +1 +θ2τ+6 +0 +(2π)τ +� +n� +i=1 +� +� +y2 +i − yiθ0 − θ2 +0 +� +exp +� +−τ +2 +�yi − θ0 +θ0 +�2� ++ +τθ2 +0 +(1 + τ) +3 +2 +��2 +(34) +×Kτ (θ0)−1 . +and by +R0 (θ0) = 1 +2n +1 +θ4 +0 +� n� +i=1 +�� +y2 +i − yiθ0 − θ2 +0 +���2 +(35) +for τ = 0. +5 +Simulation study +We analyze here the performance of the Rao-type tests based on the MDPDGE, +Rτ (θ0) , in terms of robustness and efficiency. We compare the proposed general +method assuming Gaussian distribution with Rao-type test statistics based on +the true parametric distribution underlying the data. +We consider the exponential model with density function fθ0(x) given in (33). +For the exponential model, the Rao-type test statistic based on MDPDGE is, +for τ > 0, as given in (34) and for τ = 0 as given in (35). To evaluate the +robustness of the tests we generate samples from an exponential mixture, +f ε +θ0(x) = (1 − ε)fθ0(x) + εf2θ0(x), +where θ0 denotes the parameter of the exponential distribution and ε is the con- +tamination proportion. The uncontaminated model is thus obtained by setting +ε = 0. +For comparison purposes we have also considered the robust Rao-type tests +based on the restricted MDPDE, introduced and studied in Basu et al (2021). +17 + +The efficiency loss caused by the Gaussian assumption should be advertised by +the poorer performance of the Rao-type tests based on the restricted MDPDGE +with respect to their analogous based on the restricted MDPDE. For the ex- +ponential model, the famility Rao-type test statistics based on the restricted +MDPDE is given, for β > 0, as +Sβ +n(θ0) = +� 4β2 + 1 +(2β + 1)3 − +β2 +(β + 1)4 +�−1 1 +n +� +1 +θ0 +n +� +i=1 +(yi − θ0) exp +� +−βyi +θ0 +� ++ +nβ +(β + 1)2 +�2 +. +For β = 0, the above test reduces to the classical Rao test given by +Sn (θ0) = Sβ=0,n (θ0) = +�√n +¯Xn − θ0 +θ0 +�2 +. +We consider the testing problem +H0 : θ0 = 2 vs H1 : θ ̸= 2. +and we empirically examine the level and power of both Rao-type test statistics, +the usual test based on the parametric model and the Gaussian-based test by +setting the true value of the parameter θ0 = 2 and θ0 = 1, respectively. Different +sample sizes were considered, namely n = 10, 20, 30, 40, 50, 60, 70, 80, 90, 100 +and 200, but simulation results were quite similar and so, for brevity, we only +report here results for n = 20 and n = 40. +The empirical level of the test is computed +�αn (ε) = Number of times +� +Rτ +n (θ0) (or Sβ +n (θ0)) > χ2 +1,0.05 = 3.84146 +� +Number of simulated samples +. +We set ε = 0%, 5%, 10% and 20% of contamination proportions and perform +the Monte-Carlo study over R = 10000 replications. The tuning parameters τ +and β are fixed from a grid of values, namely {0, 0.1, ..., 0.7}. +Simulation results are presented in Tables 1 and 2 for n = 20 and n = 40, +respectively. +The robustness advantage in terms of level of both Rao-type tests +considered, Rτ (θ0) and Sβ +n(θ0) with positive values of the tuning with respect +to the test statistics with τ = 0 and β = 0 is clearly shown, as their simulated +levels are closer to the nominal in the presence of contamination. +Regarding the power of the tests, uncontaminated scenarios there are values +at least so good than the corresponding to τ = 0 and β = 0 and for contaminated +data the power corresponding to τ > 0 and β > 0 are higher. +The loss of efficiency caused by the Guassian assumption can be measured by +the discrepancy of the estimated levels and powers between the family of Rao- +type tests based on the restricted MDPDGE and the MDPDE. As expected, +empirical levels of the test statistics based on the MDPDGE are quite higher +than the corresponding levels of the test based in the MDPDE. However, the test +statistic based on the parametric model, Sβ +n(θ0), is quite conservative and so the +corresponding powers are higher than those of the proposed test, Rτ (θ0) . Based +18 + +Table 1: +Simulated levels for different contamination proportions and different +tuning parametersτ, β = 0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7 for the Rao-type tests +Rτ (θ0) and Sβ +n(θ0) with for n = 20. +τ +�αn (0) +�αn (0.05) +�αn (0.10) +�αn (0.20) +�πn (0) +�πn (0.1) +�πn (0.15) +�πn (0.20) +0.0 +0.2601 +0.3093 +0.3453 +0.4661 +0.9278 +0.6791 +0.6887 +0.5088 +0.1 +0.1895 +0.1748 +0.1561 +0.1989 +0.9544 +0.7213 +0.7301 +0.0595 +0.2 +0.2120 +0.1776 +0.1417 +0.1174 +0.9747 +0.8398 +0.8430 +0.6395 +0.3 +0.2532 +0.2113 +0.1660 +0.1275 +0.9826 +0.8963 +0.8961 +0.7301 +0.4 +0.2963 +0.2447 +0.1986 +0.1471 +0.9863 +0.9228 +0.9257 +0.7893 +0.5 +0.3243 +0.2773 +0.2307 +0.1695 +0.9875 +0.9363 +0.9386 +0.8254 +0.6 +0.3512 +0.3055 +0.2599 +0.1899 +0.9885 +0.9441 +0.9437 +0.8434 +0.7 +0.3751 +0.3258 +0.2762 +0.2060 +0.9884 +0.9466 +0.9469 +0.8541 +β +0.0 +0.0453 +0.0682 +0.1048 +0.1909 +0.7200 +0.4365 +0.4384 +0.2323 +0.1 +0.0476 +0.0602 +0.0780 +0.1417 +0.7799 +0.5223 +0.5267 +0.3029 +0.2 +0.0498 +0.0552 +0.0667 +0.1103 +0.7922 +0.5751 +0.5780 +0.3558 +0.3 +0.0494 +0.0517 +0.0584 +0.0897 +0.7882 +0.5997 +0.6024 +0.3878 +0.4 +0.0489 +0.0505 +0.0535 +0.0773 +0.7779 +0.6067 +0.6058 +0.4106 +0.5 +0.0494 +0.0498 +0.0504 +0.0692 +0.7634 +0.6048 +0.6037 +0.4221 +0.6 +0.0491 +0.0504 +0.0497 +0.0647 +0.7492 +0.6008 +0.5986 +0.4265 +0.7 +0.0502 +0.0495 +0.0494 +0.0613 +0.7348 +0.5932 +0.5919 +0.4259 +Table 2: +Simulated levels for different contamination proportions and different +tuning parametersτ, β = 0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7 for the Rao-type tests +Rτ (θ0) and Sβ +n(θ0) with for n = 40. +τ +�αn (0) +�αn (0.05) +�αn (0.10) +�αn (0.20) +�πn (0) +�πn (0.1) +�πn (0.15) +�πn (0.20) +0.0 +0.3014 +0.3588 +0.4407 +0.5919 +0.9948 +0.8064 +0.7591 +0.5957 +0.1 +0.2393 +0.1934 +0.1757 +0.2032 +0.9991 +0.9540 +0.9229 +0.7712 +0.2 +0.7712 +0.2559 +0.1970 +0.1317 +0.9995 +0.9916 +0.9846 +0.9204 +0.3 +0.4257 +0.3485 +0.2782 +0.1753 +0.9997 +0.9997 +0.9953 +0.9694 +0.4 +0.5021 +0.4294 +0.3572 +0.2388 +0.9999 +0.9989 +0.9978 +0.9851 +0.5 +0.5642 +0.4920 +0.4253 +0.2993 +0.9999 +0.9992 +0.9986 +0.9908 +0.6 +0.6084 +0.5415 +0.4742 +0.3491 +1.0000 +0.9992 +0.9994 +0.9935 +0.7 +0.6416 +0.5755 +0.5081 +0.3831 +1.0000 +0.9994 +0.9994 +0.9948 +β +0.0 +0.0467 +0.0758 +0.1309 +0.2728 +0.9838 +0.8093 +0.7483 +0.4905 +0.1 +0.0469 +0.0623 +0.0959 +0.1987 +0.9870 +0.8770 +0.8317 +0.6072 +0.2 +0.0464 +0.0554 +0.0800 +0.1526 +0.9862 +0.9010 +0.8687 +0.6778 +0.3 +0.0481 +0.0529 +0.0704 +0.1220 +0.9846 +0.9084 +0.8804 +0.7169 +0.4 +0.0483 +0.0518 +0.0649 +0.1036 +0.9808 +0.9059 +0.8809 +0.7316 +0.5 +0.0500 +0.0519 +0.0618 +0.0929 +0.9756 +0.9008 +0.8742 +0.7338 +0.6 +0.0500 +0.0501 +0.0577 +0.0858 +0.9689 +0.8914 +0.8662 +0.7317 +0.7 +0.0504 +0.0519 +0.0562 +0.0801 +0.9634 +0.8813 +0.8562 +0.7258 +19 + +on the presented results, it seems that the proposed Rao-type test, Rτ (θ0) , +performs reasonably well and offers an appealing alternative for situations where +the probability density function of the true model is unknown or it is very +complicated to work with it. +References +[1] Basu, A.; Mandal, A.; Martin, N. and Pardo, L. 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Journal of Multivariate +Analysis, 169, 234-247. +6 +Appendix +In the different Sections of the Appendix will be important the following results: +1. Results in relation to the derivatives +(a) ∂Σ (θ) +∂θi += |Σ (θ)| trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +� +. +(b) +∂trace(Σ(θ)) +∂θi += trace +�∂Σ (θ) +∂θi +� +. +(c) ∂Σ (θ) +∂θi +−1 += −Σ (θ)−1 ∂Σ (θ) +∂θi +Σ (θ)−1 . +2. Let Y be a normal population with vector mean µ and variance-covariance +Σ we have, +(a) E +� +(Y − µ)T A (Y − µ) +� += Trace (AΣ) . +(b) E +� +(Y − µ)T A (Y − µ) (Y − µ)T B (Y − µ) +� += Trace +� +AΣ +� +B + BT � +Σ +� ++ +Trace (AΣ) Trace (BΣ) . +(c) E +� +(Y − µ)T A (Y − µ) (Y − µ) +� += 0. +6.1 +Appendix A (Proof of Proposition 2) +The expresion of Hτ +n(θ) is given by, +Hτ +n(θ) = a |Σ (θ)|−τ/2 +� 1 +n +n� +i=1 +exp +� +−τ +2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) +� +− b +� +−1 +τ +and we consider the d-dimensional random vector Y τ(θ) defined in (29). Ap- +plying Central Limit Theorem we have, +√n ∂ +∂θ Hτ +n(θ) = +1 +√n +n� +i=1 +Ψτ(yi; θ) +L +−→ +n−→∞ N(0m, Sτ(θ0)) +with +Sτ(θ0) = Cov [Y τ(θ)] = E +� +Y τ(θ)T Y τ(θ) +� +22 + +because +E [Y τ(θ)] = 0d. +We are going to see that E [Y τ(θ)] = 0d. We have, +E [Y τ(θ)] += +a τ +2 |Σ (θ)|−τ/2 E +� +−trace +� +Σ (θ)−1 ∂Σ (θ) +∂θ +� +exp +� +−τ +2 (Y − µ (θ))T Σ (θ)−1 (Y − µ (θ)) +� ++ b trace +� +Σ (θ)−1 ∂Σ (θ) +∂θ +� ++ exp +� +−τ +2 (Y − µ (θ))T Σ (θ)−1 (Y − µ (θ)) +� +� +−2 +�∂µ (θ) +∂θ +�T +Σ (θ)−1 (Y − µ (θ)) ++ (Y − µ (θ))T +� +Σ (θ)−1 ∂Σ (θ) +∂θ +Σ (θ)−1 +� +(Y − µ (θ)) +�� += +a τ +2 |Σ (θ)|−τ/2 +� +−trace +� +Σ (θ)−1 ∂Σ (θ) +∂θ +� +1 +(τ + 1)m/2 ++ +τ +(τ + 1) +m +2 +1 trace +� +Σ (θ)−1 ∂Σ (θ) +∂θ +� ++ +1 +(τ + 1) +m +2 +1 trace +� +Σ (θ)−1 ∂Σ (θ) +∂θ +�� += +0d. +We can observe that Y τ(θ) is a d-dimensional vector whose j-th component is +Y j +τ (θ) += +a τ +2 |Σ (θ)|−τ/2 +� +−trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +� +exp +� +−τ +2 (y − µ (θ))T Σ (θ)−1 (y − µ (θ)) +� ++ b trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +� ++ exp +� +−τ +2 (y − µ (θ))T Σ (θ)−1 (y − µ (θ)) +� +� +−2 +�∂µ (θ) +∂θj +�T +Σ (θ)−1 (y − µ (θ)) ++ (y − µ (θ))T +� +Σ (θ)−1 ∂Σ (θ) +∂θj +Σ (θ)−1 +� +(y − µ (θ)) +�� +, j = 1, ..., d. +Therefore the element (i, j) of the matrix Sτ(θ0) is given by +E +� +Y i +τ (θ)Y j +τ (θ) +� +. +23 + +We are going to get Y i +τ (θ)Y j +τ (θ). We have, +Y i +τ (θ)Y j +τ (θ) += +� +a τ +2 |Σ (θ)|−τ/2 +� +−trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +� +exp +� +−τ +2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) +� ++ b trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +� ++ exp +� +−τ +2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) +� +� +2 +�∂µ (θ) +∂θi +�T +Σ (θ)−1 (yi − µ (θ)) ++ (yi − µ (θ))T +� +Σ (θ)−1 ∂Σ (θ) +∂θj +Σ (θ)−1 +� +(yi − µ (θ)) +��� +. +� +a τ +2 |Σ (θ)|−τ/2 +� +−trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +� +exp +� +−τ +2 (y − µ (θ))T Σ (θ)−1 (y − µ (θ)) +� ++ b trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +� ++ exp +� +−τ +2 (y − µ (θ))T Σ (θ)−1 (y − µ (θ)) +� +� +2 +�∂µ (θ) +∂θj +�T +Σ (θ)−1 (y − µ (θ)) ++ (y − µ (θ))T +� +Σ (θ)−1 ∂Σ (θ) +∂θj +Σ (θ)−1 +� +(y − µ (θ)) +��� +. +24 + +Therefore, Y i +τ (θ)Y j +τ (θ) is given by +a2 �τ +2 +�2 +|Σ (θ)|−τ +� +trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +� +trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +� +exp +� +−2τ +2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) +� +−trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +� +trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +� +b +exp +� +−2τ +2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) +� +−trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +� +exp +� +−2τ +2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) +� +� +2 +�∂µ (θ) +∂θi +�T +Σ (θ)−1 (yi − µ (θ)) + (yi − µ (θ))T +� +Σ (θ)−1 ∂Σ (θ) +∂θj +Σ (θ)−1 +� +(yi − µ (θ)) +� +−b trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +� +trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +� +exp +� +−τ +2 (y − µ (θ))T Σ (θ)−1 (y − µ (θ)) +� ++b2trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +� +trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +� ++b trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +� +exp +� +−τ +2 (y − µ (θ))T Σ (θ)−1 (y − µ (θ)) +� +� +2 +�∂µ (θ) +∂θj +�T +Σ (θ)−1 (y − µ (θ)) + (y − µ (θ))T +� +Σ (θ)−1 ∂Σ (θ) +∂θj +Σ (θ)−1 +� +(y − µ (θ)) +� +−trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +� +exp +� +−2τ +2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) +� +� +2 +�∂µ (θ) +∂θj +�T +Σ (θ)−1 (y − µ (θ)) + (y − µ (θ))T +� +Σ (θ)−1 ∂Σ (θ) +∂θj +Σ (θ)−1 +� +(y − µ (θ)) +� ++b trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +� +exp +� +−2τ +2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) +� +� +2 +�∂µ (θ) +∂θj +�T +Σ (θ)−1 (y − µ (θ)) + (y − µ (θ))T +� +Σ (θ)−1 ∂Σ (θ) +∂θj +Σ (θ)−1 +� +(y − µ (θ)) +� ++ exp +� +−2τ +2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) +� +� +2 +�∂µ (θ) +∂θi +�T +Σ (θ)−1 (yi − µ (θ)) + (yi − µ (θ))T +� +Σ (θ)−1 ∂Σ (θ) +∂θi +Σ (θ)−1 +� +(yi − µ (θ)) +� +� +2 +�∂µ (θ) +∂θj +�T +Σ (θ)−1 (y − µ (θ)) + (y − µ (θ))T +� +Σ (θ)−1 ∂Σ (θ) +∂θj +Σ (θ)−1 +� +(y − µ (θ)) +�� +. +25 + +We can write Y i +τ (θ)Y j +τ (θ) by +Y i +τ (θ)Y j +τ (θ) = a2 �τ +2 +�2 +|Σ (θ)|−τ {C1 + C2 + C3 + C4 + C5 + C6 + C7 + C8 + C9} +and +E [Yi(θ)Yj(θ)] = a2 �τ +2 +�2 +|Σ (θ)|−τ E [C1 + C2 + C3 + C4 + C5 + C6 + C7 + C8 + C9] . +(36) +Now we are going to calculate the different expectations appearing in (36). +We have, +C1 = trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +� +trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +� +exp +� +−2τ +2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) +� +. +Therefore, +E [C1] += +trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +� +trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +� +� +1 +(2π)m/2 |Σ (θ)|1/2 exp +� +−1 +2 (y − µ (θ))T Σ (θ)−1 (y − µ (θ)) +� +exp +� +−1 +2 (y − µ (θ))T +�Σ (θ) +2τ +�−1 +(y − µ (θ)) +� +dy += +trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +� +trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +� +� +1 +2τ + 1 +� m +2 � +1 +(2π)m/2 ��� Σ(θ) +2τ+1 +��� +1/2 exp +� +−1 +2 (y − µ (θ))T +� Σ (θ) +2τ + 1 +�−1 +(y − µ (θ)) +� +dy += +trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +� +trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +� � +1 +2τ + 1 +� m +2 +The expression of C2 is given by +C2 += +−trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +� +trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +� +b +exp +� +−2τ +2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) +� +26 + +and +E [C2] += +− +τ +(1 + τ) +m +2 +1 trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +� +trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +� +� +1 +(2π)m/2 |Σ (θ)|1/2 exp +� +−1 +2 (y − µ (θ))T Σ (θ)−1 (y − µ (θ)) +� +exp +� +−1 +2 (y − µ (θ))T +�Σ (θ) +τ +�−1 +(y − µ (θ)) +� +dy += +− +τ +(1 + τ) +m +2 +1 trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +� +trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +� +1 +(1 + τ) +m +2 +� +1 +(2π)m/2 ��� Σ(θ) +τ+1 +��� +1/2 exp +� +−1 +2 (y − µ (θ))T +�Σ (θ) +τ + 1 +�−1 +(y − µ (θ)) +� +dy += +− +τ +(1 + τ)m+1 trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +� +trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +� +. +The expression of C3 is given by +C3 += +−trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +� +exp +� +−2τ +2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) +� +� +2 +�∂µ (θ) +∂θi +�T +Σ (θ)−1 (yi − µ (θ)) + (yi − µ (θ))T +� +Σ (θ)−1 ∂Σ (θ) +∂θj +Σ (θ)−1 +� +(yi − µ (θ)) +� +. +Then +E [C3] += +−trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +� +1 +(1 + 2τ)m/2 +� +1 +(2π)m/2 ��� Σ(θ) +2τ+1 +��� +1/2 exp +� +−1 +2 (y − µ (θ))T Σ (θ)−1 (y − µ (θ)) +� +(y − µ (θ))T +� +Σ (θ)−1 ∂Σ (θ) +∂θj +Σ (θ)−1 +� +(y − µ (θ)) dy += +−trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +� +trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +� +1 +(1 + 2τ) +m +2 +1 . +In relation to C4 we have, +C4 = −b trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +� +trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +� +exp +� +−τ +2 (y − µ (θ))T Σ (θ)−1 (y − µ (θ)) +� +27 + +and +E [C4] += +− +τ +(1 + τ) +m +2 +1 trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +� +trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +� +� +1 +(2π)m/2 |Σ (θ)|1/2 exp +� +−1 +2 (y − µ (θ))T Σ (θ)−1 (y − µ (θ)) +� +exp +� +−τ +2 (y − µ (θ))T (Σ (θ))−1 (y − µ (θ)) +� +dy += +− +τ +(1 + τ) +m +2 +1 trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +� +trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +� +1 +(1 + τ) +m +2 +� +1 +(2π)m/2 ��� Σ(θ) +τ+1 +��� +1/2 exp +� +−1 +2 (y − µ (θ))T +�Σ (θ) +τ + 1 +�−1 +(y − µ (θ)) +� +dy += +− +τ +(1 + τ)m+1 trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +� +trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +� +. +In relation with C5 we have, +C5 = b2 trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +� +trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +� +and +E [C5] = +τ 2 +(1 + τ)m+2 trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +� +trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +� +. +The expression of C6 is given by, +C6 += +b trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +� +exp +� +−τ +2 (y − µ (θ))T Σ (θ)−1 (y − µ (θ)) +� +� +2 +�∂µ (θ) +∂θj +�T +Σ (θ)−1 (y − µ (θ)) + (y − µ (θ))T +� +Σ (θ)−1 ∂Σ (θ) +∂θj +Σ (θ)−1 +� +(y − µ (θ)) +� +and +E [C6] += +τ +(1 + τ) +m +2 +1 trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +� +trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +� +1 +(1 + τ) +m +2 +1 += +τ +(1 + τ)m+2 trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +� +trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +� +. +The expression of C7 is +C7 += +−trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +� +exp +� +−2τ +2 (y − µ (θ))T Σ (θ)−1 (y − µ (θ)) +� +� +2 +�∂µ (θ) +∂θj +�T +Σ (θ)−1 (y − µ (θ)) + (y − µ (θ))T +� +Σ (θ)−1 ∂Σ (θ) +∂θj +Σ (θ)−1 +� +(y − µ (θ)) +� +28 + +and +E [C7] = − +τ +(1 + 2τ) +m +2 +1 trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +� +trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +� +. +In relation to C7, +C8 += +b trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +� +exp +� +−τ +2 (y − µ (θ))T Σ (θ)−1 (y − µ (θ)) +� +� +2 +�∂µ (θ) +∂θj +�T +Σ (θ)−1 (y − µ (θ)) + (y − µ (θ))T +� +Σ (θ)−1 ∂Σ (θ) +∂θj +Σ (θ)−1 +� +(y − µ (θ)) +� +and +E [C8] += +τ +(1 + τ) +m +2 +1 trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +� +trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +� +1 +(1 + τ) +m +2 +1 += +τ +(1 + τ)m+2 trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +� +trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +� +. +Finally, +C9 += +exp +� +−2τ +2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) +� +� +2 +�∂µ (θ) +∂θi +�T +Σ (θ)−1 (yi − µ (θ)) + (yi − µ (θ))T +� +Σ (θ)−1 ∂Σ (θ) +∂θi +Σ (θ)−1 +� +(yi − µ (θ)) +� +� +2 +�∂µ (θ) +∂θj +�T +Σ (θ)−1 (yi − µ (θ)) + (yi − µ (θ))T +� +Σ (θ)−1 ∂Σ (θ) +∂θj +Σ (θ)−1 +� +(yi − µ (θ)) +� +. +Therefore, +C9 += +exp +� +−2τ +2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) +� � +4 (yi − µ (θ))T Σ (θ)−1 +�∂µ (θ) +∂θi +�T +∂µ (θ) +∂θj +Σ (θ)−1 (yi − µ (θ)) ++2 +�∂µ (θ) +∂θi +�T +Σ (θ)−1 (yi − µ (θ)) (y − µ (θ))T +� +Σ (θ)−1 ∂Σ (θ) +∂θj +Σ (θ)−1 +� +(yi − µ (θ)) ++2 (yi − µ (θ))T +� +Σ (θ)−1 ∂Σ (θ) +∂θi +Σ (θ)−1 +� +(yi − µ (θ)) +�∂µ (θ) +∂θj +�T +Σ (θ)−1 (yi − µ (θ)) ++ (yi − µ (θ))T +� +Σ (θ)−1 ∂Σ (θ) +∂θi +Σ (θ)−1 +� +(yi − µ (θ)) +� += +A1 + A2 + A3 + A4 +and +E [C9] = E [A1] + E [A4] +29 + +because E [A2] = E [A3] = 0. We have, +E [A1] += +E +� +exp −2τ +2 (Y − µ (θ))T Σ (θ)−1 (Y − µ (θ)) 4 (Y − µ (θ))T Σ (θ)−1 +�∂µ (θ) +∂θi +�T ∂µ (θ) +∂θj +Σ (θ)−1 (Y − µ (θ)) +� += +4 +� +1 +(2π)m/2 |Σ (θ)|1/2 exp +� +−1 +2 (y − µ (θ))T +� Σ (θ) +2τ + 1 +�−1 +(y − µ (θ)) +� +(y − µ (θ))T +Σ (θ)−1 +�∂µ (θ) +∂θi +�T ∂µ (θ) +∂θj +Σ (θ)−1 (y − µ (θ)) dy += +4 +1 +(2τ + 1)m/2 +� +1 +(2π)m/2 ��� Σ(θ) +2τ+1 +��� +1/2 exp +� +−1 +2 (y − µ (θ))T +� Σ (θ) +2τ + 1 +�−1 +(y − µ (θ)) +� +(y − µ (θ))T Σ (θ)−1 +�∂µ (θ) +∂θi +�T ∂µ (θ) +∂θj +Σ (θ)−1 (y − µ (θ)) dy += +4 +1 +(2τ + 1)m/2 trace +� +Σ (θ)−1 +�∂µ (θ) +∂θi +�T ∂µ (θ) +∂θj +Σ (θ)−1 Σ (θ) +2τ + 1 +� += +4 +(2τ + 1)m/2+1 trace +� +Σ (θ)−1 +�∂µ (θ) +∂θi +�T ∂µ (θ) +∂θj +� +. +30 + +Now we are going to get E [A4] . +E [A4] += +E +� +exp +� +−2τ +2 (Y − µ (θ))T Σ (θ)−1 (Y − µ (θ)) +� +(Y − µ (θ))T +� +Σ (θ)−1 ∂Σ (θ) +∂θi +Σ (θ)−1 +� +(Y − µ (θ)) (Y − µ (θ))T +� +Σ (θ)−1 ∂Σ (θ) +∂θj +Σ (θ)−1 +� +(Y − µ (θ)) +� += +� +1 +(2π)m/2 |Σ (θ)|1/2 exp +� +−1 +2 (y − µ (θ))T +� Σ (θ) +2τ + 1 +�−1 +(y − µ (θ)) +� +(y − µ (θ))T +� +Σ (θ)−1 ∂Σ (θ) +∂θi +Σ (θ)−1 +� +(y − µ (θ)) (y − µ (θ))T +� +Σ (θ)−1 ∂Σ (θ) +∂θj +Σ (θ)−1 +� +(y − µ (θ)) dy += +1 +(2τ + 1)m/2 +� +trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +Σ (θ)−1 Σ (θ) +2τ + 1 +� +� +Σ (θ)−1 ∂Σ (θ) +∂θi +Σ (θ)−1 + Σ (θ)−1 ∂Σ (θ) +∂θj +Σ (θ)−1 +� Σ (θ) +2τ + 1 +� ++ +1 +(2τ + 1)m/2 trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +Σ (θ)−1 Σ (θ) +2τ + 1 +� +trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +Σ (θ)−1 Σ (θ) +2τ + 1 +� += +1 +(2τ + 1) +m +2 +2 trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +Σ (θ)−1 ∂Σ (θ) +∂θj +� ++ +1 +(2τ + 1) +m +2 +2 trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +Σ (θ)−1 +�∂Σ (θ) +∂θj +�T � ++ +1 +(2τ + 1) +m +2 +2 trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +� +trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +� += +2 +1 +(2τ + 1) +m +2 +2 trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +Σ (θ)−1 ∂Σ (θ) +∂θj +� ++ +1 +(2τ + 1) +m +2 +2 trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +� +trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +� +. +31 + +Based on the previous results we have, +E +� +Y i +τ (θ)Y j +τ (θ) +� += +a2 �τ +2 +�2 +|Σ (θ)|−τ +� +1 +(2τ + 1) +m +2 trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +� +trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +� +− +τ +(τ + 1)m+1 trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +� +trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +� +− +1 +(2τ + 1) +m +2 +1 trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +� +trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +� +− +τ +(τ + 1)m+1 trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +� +trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +� ++ +τ 2 +(τ + 1)m+2 trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +� +trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +� ++ +τ +(τ + 1)m+2 trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +� +trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +� +− +1 +(2τ + 1) +m +2 +1 trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +� +trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +� ++ +τ +(τ + 1)m+2 trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +� +trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +� ++ +4 +(2τ + 1) +m +2 +1 trace +� +Σ (θ)−1 +�∂µ (θ) +∂θi +�T ∂µ (θ) +∂θj +� ++ +2 +(2τ + 1) +m +2 +2 trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +Σ (θ)−1 ∂Σ (θ) +∂θj +� ++ +1 +(2τ + 1) +m +2 +2 trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +� +trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +�� +. +32 + +The previous expression can be written as, +E +� +Y i +τ (θ)Y j +τ (θ) +� += +a2 �τ +2 +�2 +Σ (θ)−τ +� +trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +� +trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +� +� +1 +(2τ + 1) +m +2 − +1 +(2τ + 1) +m +2 +1 − +1 +(2τ + 1) +m +2 +1 + +1 +(2τ + 1) +m +2 +2 +� ++trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +� +trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +� +� +− +τ +(τ + 1)m+1 − +τ +(τ + 1)m+1 + +τ 2 +(τ + 1)m+2 + +τ +(τ + 1)m+2 + +τ +(τ + 1)m+2 +� ++ +4 +(2τ + 1) +m +2 +1 trace +� +Σ (θ)−1 +�∂µ (θ) +∂θi +�T ∂µ (θ) +∂θj +� ++ +2 +(2τ + 1) +m +2 +2 trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +Σ (θ)−1 ∂Σ (θ) +∂θj +�� +. +Therefore, +E +� +Y i +τ (θ)Y j +τ (θ) +� += +� +τ + 1 +τ (2π)mτ/2 +�2 �τ +2 +�2 +|Σ (θ)|−τ +� +4τ 2 +(2τ + 1) +m +2 +2 trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +� +trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +� +− +τ 2 +(1 + τ)m+2 trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +� +trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +� ++ 4 +4 +(2τ + 1) +m +2 +1 trace +� +Σ (θ)−1 +�∂µ (θ) +∂θi +�T ∂µ (θ) +∂θj +� ++ +2 +(2τ + 1) +m +2 +2 trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +Σ (θ)−1 ∂Σ (θ) +∂θj +�� +. +33 + +Finally, +E +� +Y i +τ (θ)Y j +τ (θ) +� += +(τ + 1)2 +� +� +� +� +1 +(2π)mτ/2 |Σ (θ)|1/2 +�2τ +1 +(2τ + 1) +m +2 +2 +� +∆i +2τ∆j +2τ + (2τ + 1) trace +� +Σ (θ)−1 +�∂µ (θ) +∂θi +�T ∂µ (θ) +∂θj +� ++1 +2trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +Σ (θ)−1 ∂Σ (θ) +∂θj +�� +− +� +1 +(2π)mτ/2 |Σ (θ)|1/2 +�2τ +1 +(1 + τ)m+2 ∆i +τ∆j +τ +� +� +� += +(τ + 1)2 Kij +τ (θ) +where Kij +τ (θ) was defined in (8). +Then +√n ∂ +∂θ Hn(θ) +L +−→ +n−→∞ N +� +0, (τ + 1)2 Kτ (θ) +� +and +√n +� +1 +τ + 1 +∂ +∂θ Hn(θ) +� +L +−→ +n−→∞ N (0, Kτ (θ)) . +34 + +6.2 +Appendix B (Proof of Proposition 3) +We have, +∂ +∂θi +Hτ +n(θ) += +−aτ +2 |Σ (θ)|−τ/2 trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +� +� 1 +n +n� +i=1 +exp +� +−τ +2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) +� +− b +� ++a |Σ (θ)|−τ/2 +� 1 +n +n� +i=1 +exp +� +−τ +2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) +� � +−τ +2 +� +� +2 +�∂µ (θ) +∂θi +�T +Σ (θ)−1 (yi − µ (θ)) +− (yi − µ (θ))T +� +Σ (θ)−1 ∂Σ (θ) +∂θj +Σ (θ)−1 +� +(yi − µ (θ)) +�� += +−aτ +2 |Σ (θ)|−τ/2 trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +� +� 1 +n +n� +i=1 +exp +� +−τ +2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) +� +− b +� ++aτ +2 |Σ (θ)|−τ/2 τ +2 +� 1 +n +n� +i=1 +exp +� +−τ +2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) +� +� +2 +�∂µ (θ) +∂θi +�T +Σ (θ)−1 (yi − µ (θ)) +(yi − µ (θ))T +� +Σ (θ)−1 ∂Σ (θ) +∂θj +Σ (θ)−1 +� +(yi − µ (θ)) +�� +. +Therefore, +∂2 +∂θi∂θj +Hτ +n(θ) = +∂ +∂θj +Lτ +1(θ) + +∂ +∂θj +Lτ +2(θ) +being, +Lτ +1(θ) += +−aτ +2 |Σ (θ)|−τ/2 trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +� +� 1 +n +n� +i=1 +exp +� +−τ +2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) +� +− b +� +and +Lτ +2(θ) += +aτ +2 |Σ (θ)|−τ/2 τ +2 +� 1 +n +n� +i=1 +exp +� +−τ +2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) +� +� +2 +�∂µ (θ) +∂θi +�T +Σ (θ)−1 (yi − µ (θ)) +(yi − µ (θ))T +� +Σ (θ)−1 ∂Σ (θ) +∂θi +Σ (θ)−1 +� +(yi − µ (θ)) +�� +. +35 + +We are going to get +∂ +∂θj Lτ +1(θ). +∂ +∂θj +Lτ +1(θ) += +−aτ +2 +� +−τ +2 +� +|Σ (θ)|−τ/2 trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +� +trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +� +� 1 +n +n� +i=1 +exp +� +−τ +2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) +� +− b +� +−aτ +2 |Σ (θ)|−τ/2 trace +� +−Σ (θ)−1 ∂Σ (θ) +∂θj +Σ (θ)−1 ∂Σ (θ) +∂θi ++ Σ (θ)−1 ∂2Σ (θ) +∂θi∂θj +� +� 1 +n +n� +i=1 +exp +� +−τ +2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) +� +− b +� +−aτ +2 |Σ (θ)|−τ/2 trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +� +� 1 +n +n� +i=1 +exp +� +−τ +2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) +� � +−τ +2 +� +� +−2 +�∂µ (θ) +∂θi +�T +Σ (θ)−1 (yi − µ (θ)) +− (yi − µ (θ))T +� +Σ (θ)−1 ∂Σ (θ) +∂θj +Σ (θ)−1 +� +(yi − µ (θ)) +�� += +D1 + D2 + D3, +being +D1 += +aτ +4 +2 +|Σ (θ)|−τ/2 trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +� +trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +� +� 1 +n +n� +i=1 +exp +� +−τ +2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) +� +− b +� +D2 += +−aτ +2 |Σ (θ)|−τ/2 trace +� +−Σ (θ)−1 ∂Σ (θ) +∂θj +Σ (θ)−1 ∂Σ (θ) +∂θi ++ Σ (θ)−1 ∂2Σ (θ) +∂θi∂θj +� +� 1 +n +n� +i=1 +exp +� +−τ +2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) +� +− b +� +36 + +and +D3 += +aτ 2 +4 |Σ (θ)|−τ/2 trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +� +� 1 +n +n� +i=1 +exp +� +−τ +2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) +� +� +−2 +�∂µ (θ) +∂θi +�T +Σ (θ)−1 (yi − µ (θ)) +− (yi − µ (θ))T +� +Σ (θ)−1 ∂Σ (θ) +∂θj +Σ (θ)−1 +� +(yi − µ (θ)) +�� +Now we are going to see some result that will be important in order to get the +convergence in probability of D1, D2 and D3. +Remark 13 We have, +l = 1 +n +n� +i=1 +exp +� +−τ +2 (Y i − µ (θ))T Σ (θ)−1 (Y i − µ (θ)) +� +P +−→ +n−→∞ +1 +(1 + τ)m/2 +Proof. It is clear that +l +P +−→ +n−→∞ EN (µ(θ),Σ(θ)) +� +exp +� +−1 +2 (Y − µ (θ))T +�Σ (θ) +τ +� +(Y − µ (θ)) +�� += +� +exp +� +−1 +2 (Y − µ (θ))T +�Σ (θ) +τ +� +(Y − µ (θ)) +� +fN (µ(θ),Σ(θ)) (y) dy += +1 +(1 + τ)m/2 +� +1 +(2π)m/2 +1 +��� Σ(θ) +τ+1 +��� +1/2 exp +� +−1 +2 (y − µ (θ))T +�Σ (θ) +τ +� +(y − µ (θ)) +� +dy += +1 +(1 + τ)m/2 . +Remark 14 We have, +m = 1 +n +n� +i=1 +exp +� +−τ +2 (Y i − µ (θ))T Σ (θ)−1 (Y i − µ (θ)) +� +−b +P +−→ +n−→∞ +1 +(1 + τ) +m +2 +1 +Proof. By result the previous remark +m +P +−→ +n−→∞ +1 +(1 + τ)m/2 − +τ +(1 + τ) +m +2 +1 = +1 +(1 + τ) +m +2 +1 . +Remark 15 We denote, +n = 1 +n +n� +i=1 +exp +� +−τ +2 (Y i − µ (θ))T Σ (θ)−1 (Y i − µ (θ)) +� +(Y i − µ (θ))T A (Y i − µ (θ)) +37 + +and we have, +n +P +−→ +n−→∞ +trace (AΣ (θ)) +(1 + τ) +m +2 +1 +. +Proof. It is clear that, +n +P +−→ +n−→∞ EN (µ(θ),Σ(θ)) +� +exp +� +−1 +2 (Y − µ (θ))T +�Σ (θ) +τ +�−1 +(Y − µ (θ)) +� +(Y − µ (θ))T A (Y − µ (θ)) +� += +1 +(1 + τ)m/2 +� +1 +(2π)m/2 +1 +��� Σ(θ) +τ+1 +��� +1/2 exp +� +−1 +2 (y − µ (θ))T +�Σ (θ) +τ + 1 +�−1 +(y − µ (θ)) +� +(y − µ (θ))T A (y − µ (θ)) dy += +1 +(1 + τ)m/2 EN(µ(θ), Σ(θ) +1+τ ) +� +(Y − µ (θ))T A (Y − µ (θ)) +� += +1 +(1 + τ) +m +2 +1 trace (AΣ (θ)) . +Based on the previous results we have in relation to D1, +D1 +P +−→ +n−→∞ +τ +4 +|Σ (θ)|− τ +2 +(2π)mτ/2 (1 + τ)m/2 trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +� +trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +� +. +With respect to D2, +D2 +P +−→ +n−→∞ +1 +(2π)m/2 |Σ (θ)|− τ +2 1 +2 +1 +(1 + τ)m/2 trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +Σ (θ)−1 ∂Σ (θ) +∂θi +� +−|Σ (θ)|− τ +2 +(2π)m/2 +1 +2trace +� +Σ (θ)−1 ∂2Σ (θ) +∂θj∂θi +� +. +In a similar way we get for D3 that, +D3 +P +−→ +n−→∞ −τ +4 +|Σ (θ)|− τ +2 +(2π)mτ/2 trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +� +1 +(1 + τ)m/2 trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +� +. +Therefore we have +∂ +∂θj +Lτ +1(θ) +P +−→ +n−→∞ +1 +2 +|Σ (θ)|− τ +2 +(2π)mτ/2 +1 +(1 + τ)m/2 trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +Σ (θ)−1 ∂Σ (θ) +∂θi +� +−1 +2 +|Σ (θ)|− τ +2 +(2π)mτ/2 +1 +(1 + τ)m/2 +� +Σ (θ)−1 ∂2Σ (θ) +∂θj∂θi +� +. +38 + +Now we have, +∂ +∂θj +Lτ +2(θ) += +−aτ 2 +4 |Σ (θ)|− τ +2 trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +� +� +1 +n +n� +i=1 +exp +� +−τ +2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) +� � +2 +�∂µ (θ) +∂θi +�T +Σ (θ)−1 (yi − µ (θ)) + (yi − µ (θ))T +� +Σ (θ)−1 ∂Σ (θ) +∂θj +Σ (θ)−1 +� +(yi − µ (θ)) +�� ++aτ +2 |Σ (θ)|− τ +2 +� 1 +n +n� +i=1 +exp +� +−τ +2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) +� +� +−τ +2 +� � ∂ +∂θj +(yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) +� � +2 +�∂µ (θ) +∂θi +�T +Σ (θ)−1 (yi − µ (θ)) +(yi − µ (θ))T +� +Σ (θ)−1 ∂Σ (θ) +∂θj +Σ (θ)−1 +� +(yi − µ (θ)) +�� ++aτ +2 |Σ (θ)|− τ +2 +� 1 +n +n� +i=1 +exp +� +−τ +2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) +� +∂ +∂θj +� +2 +�∂µ (θ) +∂θi +�T +Σ (θ)−1 (yi − µ (θ)) +� ++ (yi − µ (θ))T +� +Σ (θ)−1 ∂Σ (θ) +∂θj +Σ (θ)−1 +� +(yi − µ (θ)) +�� += +C1 + C2 + C3. +Being, +C1 += +−aτ 2 +4 |Σ (θ)|− τ +2 trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +� � 1 +n +n� +i=1 +exp +� +−τ +2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) +� +� +2 +�∂µ (θ) +∂θi +�T +Σ (θ)−1 (yi − µ (θ)) +(yi − µ (θ))T +� +Σ (θ)−1 ∂Σ (θ) +∂θj +Σ (θ)−1 +� +(yi − µ (θ)) +�� +. +It is clear that, +C1 +P +−→ +n−→∞ −τ +4 +1 +(1 + τ)m/2 +|Σ (θ)|− τ +2 +(2π)mτ/2 trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +� +trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +� +. +It is immediate to see that +C2 += +−aτ 2 +4 |Σ (θ)|− τ +2 1 +n +n� +i=1 +exp +� +−τ +2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) +� +(S1 + S2 + S3 + S4) += +L∗ +1 + L∗ +2 + L∗ +3 + L∗ +4 +39 + +where +L∗ +1 += +−aτ 2 +4 |Σ (θ)|− τ +2 1 +n +n� +i=1 +exp +� +−τ +2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) +� +� +−4 (yi − µ (θ))T Σ (θ)−1 ∂µ (θ) +∂θj +�∂µ (θ) +∂θi +�T +Σ (θ)−1 (yi − µ (θ)) +� +and +L∗ +1 +P +−→ +n−→∞ +|Σ (θ)|− τ +2 +(2π)mτ/2 +τ +(1 + τ)m/2 trace +� +Σ (θ)−1 ∂µ (θ) +∂θj +�∂µ (θ) +∂θi +�T � +. +It is clear that +L∗ +2 +P +−→ +n−→∞ 0 and L∗ +3 +P +−→ +n−→∞ 0. +On the other hand, +L∗ +4 +P +−→ +n−→∞ +τ + 1 +(2π)mτ/2 +τ +4 |Σ (θ)|− τ +2 +1 +(1 + τ)m/2 +� +trace +�� +Σ (θ)−1 ∂Σ (θ) +∂θj +Σ (θ)−1 ∂Σ (θ) +∂θi +� � +Σ (θ)−1 ∂Σ (θ) +∂θi +Σ (θ)−1 + Σ (θ)−1 ∂Σ (θ) +∂θi +Σ (θ)−1 +� +Σ (θ) +1 + τ +� ++ trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +Σ (θ)−1 Σ (θ) +1 + τ +� +trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +Σ (θ)−1 Σ (θ) +1 + τ +�� += 2τ +4 +|Σ (θ)|− τ +2 +(2π)mτ/2 +1 +(1 + τ)m/2+1 trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +Σ (θ)−1 Σ (θ) ∂Σ (θ) +∂θi +� ++τ +4 +|Σ (θ)|− τ +2 +(2π)mτ/2 +1 +(1 + τ)m/2+1 trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +� +trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +� +. +Therefore, +C2 = L∗ +1 + L∗ +2 + L∗ +3 + L∗ +4 +P +−→ +n−→∞ R +being +R += +τ |Σ (θ)|− τ +2 +(2π)mτ/2 (1 + τ)m/2 trace +� +Σ (θ)−1 ∂µ (θ) +∂θj +�∂µ (θ) +∂θj +�T � ++τ +2 +|Σ (θ)|− τ +2 +(2π)mτ/2 (1 + τ)m/2+1 trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +Σ (θ)−1 ∂Σ (θ) +∂θi +� ++τ +4 +|Σ (θ)|− τ +2 +(2π)mτ/2 (1 + τ)m/2+1 trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +� +trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +� +. +40 + +Finally, +−C3 += +aτ +2 |Σ (θ)|− τ +2 1 +n +n� +i=1 +exp +� +−τ +2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) +� +�� +2 ∂ +∂θj +�∂µ (θ) +∂θi +�T +Σ (θ)−1 (yi − µ (θ)) +� ++2 +�∂µ (θ) +∂θi +�T ∂Σ (θ)−1 +∂θi +(yi − µ (θ)) +−2 +�∂µ (θ) +∂θi +�T +Σ (θ)−1 +�∂µ (θ) +∂θj +�T +−2 +�∂µ (θ) +∂θi +�T � +Σ (θ)−1 ∂Σ (θ) +∂θi +Σ (θ)−1 +� +(yi − µ (θ)) ++ (yi − µ (θ))T {− Σ (θ)−1 ∂Σ (θ) +∂θj +Σ (θ)−1 ∂Σ (θ) +∂θi +Σ (θ)−1 ++Σ (θ)−1 ∂2Σ (θ) +∂θi∂θj +Σ (θ)−1 − Σ (θ)−1 ∂Σ (θ) +∂θj +Σ (θ)−1 ∂Σ (θ) +∂θi +Σ (θ)−1 +� +(yi − µ (θ)) += +A∗ +1 + A∗ +2 + A∗ +3 + A∗ +4 + A∗ +5. +It is clear that +A∗ +1 +P +−→ +n−→∞ 0 A∗ +2 +P +−→ +n−→∞ 0 and A∗ +4 +P +−→ +n−→∞ 0. +On the other hand, +A∗ +3 += +−aτ +2 |Σ (θ)|− τ +2 1 +n +n� +i=1 +exp +� +−τ +2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) +� +� +2 +�∂µ (θ) +∂θi +�T +Σ (θ)−1 ∂µ (θ) +∂θi +� +and +A∗ +3 +P +−→ +n−→∞ −(τ + 1) +|Σ (θ)|− τ +2 +(2π)mτ/2 (1 + τ)m/2 +�∂µ (θ) +∂θi +�T +Σ (θ)−1 ∂µ (θ) +∂θj +. +In relation to A∗ +5 we have, +A∗ +5 += +aτ +2 |Σ (θ)|− τ +2 1 +n +n� +i=1 +exp +� +−τ +2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) +� +� +(yi − µ (θ))T +� +Σ (θ)−1 ∂Σ (θ) +∂θj +Σ (θ)−1 ∂Σ (θ) +∂θi +Σ (θ)−1 ++Σ (θ)−1 ∂2Σ (θ) +∂θi∂θj +Σ (θ)−1 − Σ (θ)−1 ∂Σ (θ) +∂θj +Σ (θ)−1 ∂Σ (θ) +∂θj +Σ (θ)−1 +� +(yi − µ (θ)) +� +, +41 + +and +A∗ +5 +P +−→ +n−→∞ − +1 +(2π)mτ/2 +1 +2 |Σ (θ)|− τ +2 +1 +(1 + τ)m/2 trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +Σ (θ)−1 ∂Σ (θ) +∂θi +� ++ +1 +(2π)mτ/2 +1 +2 |Σ (θ)|− τ +2 +1 +(1 + τ)m/2 trace +� +Σ (θ)−1 ∂2Σ (θ) +∂θi∂θ +� +− +1 +(2π)mτ/2 +1 +2 |Σ (θ)|− τ +2 +1 +(1 + τ)m/2 trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +Σ (θ)−1 ∂Σ (θ) +∂θi +� +. +Therefore, +C3 +P +−→ +n−→∞ − (τ + 1) |Σ (θ)|− τ +2 +(2π)mτ/2 +1 +(1 + τ)m/2 +�∂µ (θ) +∂θi +�T +Σ (θ)−1 ∂µ (θ) +∂θj +−|Σ (θ)|− τ +2 +(2π)mτ/2 +1 +(1 + τ)m/2 +1 +2trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +Σ (θ)−1 ∂Σ (θ) +∂θi +� ++|Σ (θ)|− τ +2 +(2π)mτ/2 +1 +(1 + τ)m/2 +1 +2trace +� +Σ (θ)−1 ∂2Σ (θ) +∂θi∂θj +� +−|Σ (θ)|− τ +2 +(2π)mτ/2 +1 +2 +1 +(1 + τ)m/2 trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +Σ (θ)−1 ∂Σ (θ) +∂θi +� +. +We are going to joint all the previous expressions in order to get +∂ +∂θj Lτ +2(θ), +∂ +∂θj +Lτ +2(θ) += +C1 + C2 + C3 += +C1 + L∗ +1 + L∗ +2 + L∗ +3 + L∗ +4 + C3 += +C1 + L∗ +1 + L∗ +2 + L∗ +3 + L∗ +4 + A∗ +1 + A∗ +2 + A∗ +3 + A∗ +4 + A∗ +5. +42 + +Then, +∂ +∂θj +Lτ +2(θ) +P +−→ +n−→∞ −τ +4 +1 +(1 + τ)m/2 +|Σ (θ)|− τ +2 +(2π)mτ/2 trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +� +trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +� ++ +τ |Σ (θ)|− τ +2 +(2π)mτ/2 (1 + τ)m/2 trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +Σ (θ)−1 ∂Σ (θ) +∂θi +� ++τ +2 +|Σ (θ)|− τ +2 +(2π)mτ/2 (1 + τ) +m +2 +1 trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +Σ (θ)−1 ∂Σ (θ) +∂θi +� ++τ +4 +|Σ (θ)|− τ +2 +(2π)mτ/2 (1 + τ)m/2+1 trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +� +trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +� +− (τ + 1) |Σ (θ)|− τ +2 +(2π)mτ/2 +1 +(1 + τ)m/2 +�∂µ (θ) +∂θi +�T +Σ (θ)−1 ∂µ (θ) +∂θj +−|Σ (θ)|− τ +2 +(2π)mτ/2 +1 +(1 + τ)m/2 +1 +2trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +Σ (θ)−1 ∂Σ (θ) +∂θi +� ++|Σ (θ)|− τ +2 +(2π)mτ/2 +1 +(1 + τ)m/2 +1 +2trace +� +Σ (θ)−1 ∂2Σ (θ) +∂θi∂θj +� +−|Σ (θ)|− τ +2 +(2π)mτ/2 +1 +2 +1 +(1 + τ)m/2 trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +Σ (θ)−1 ∂Σ (θ) +∂θi +� +. +Based on the previous results we have +∂2 +∂θi∂θj +Hτ +n(θ) += +∂ +∂θj +Lτ +1(θ) + +∂ +∂θj +Lτ +2(θ) += +D1 + D2 + D3 + C1 + C2 + C3 += +D1 + D2 + D3 + C1 + L∗ +1 + L∗ +2 + L∗ +3 + L∗ +4 ++A∗ +1 + A∗ +2 + A∗ +3 + A∗ +4 + A∗ +5 +43 + +and +∂2 +∂θi∂θj +Hτ +n(θ) +P +−→ +n−→∞ +1 +2 +|Σ (θ)|− τ +2 +(2π)mτ/2 +1 +(1 + τ)m/2 trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +Σ (θ)−1 ∂Σ (θ) +∂θi +� +−1 +2 +|Σ (θ)|− τ +2 +(2π)mτ/2 +1 +(1 + τ)m/2 +� +Σ (θ)−1 ∂2Σ (θ) +∂θj∂θi +� +−τ +4 +1 +(1 + τ)m/2 +|Σ (θ)|− τ +2 +(2π)mτ/2 trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +� +trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +� ++ +τ |Σ (θ)|− τ +2 +(2π)mτ/2 (1 + τ)m/2 trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +Σ (θ)−1 ∂Σ (θ) +∂θi +� ++τ +2 +|Σ (θ)|− τ +2 +(2π)mτ/2 (1 + τ)m/2+1 trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +Σ (θ)−1 ∂Σ (θ) +∂θi +� ++τ +4 +|Σ (θ)|− τ +2 +(2π)mτ/2 (1 + τ)m/2+1 trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +� +trace +� +Σ (θ)−1 ∂Σ (θ) +∂θi +� +− (τ + 1) |Σ (θ)|− τ +2 +(2π)mτ/2 +1 +(1 + τ)m/2 +�∂µ (θ) +∂θi +�T +Σ (θ)−1 ∂µ (θ) +∂θj +−|Σ (θ)|− τ +2 +(2π)mτ/2 +1 +(1 + τ)m/2 +1 +2trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +Σ (θ)−1 ∂Σ (θ) +∂θi +� ++|Σ (θ)|− τ +2 +(2π)mτ/2 +1 +(1 + τ)m/2 +1 +2trace +� +Σ (θ)−1 ∂2Σ (θ) +∂θi∂θj +� +−|Σ (θ)|− τ +2 +(2π)mτ/2 +1 +(1 + τ)m/2 +1 +2trace +� +Σ (θ)−1 ∂Σ (θ) +∂θj +Σ (θ)−1 ∂Σ (θ) +∂θi +� +. +After some algebra we have, +∂2 +∂θi∂θj +Hτ +n(θ) +P +−→ +n−→∞ − (τ + 1) Jij +τ (θ) . +44 + diff --git a/ZtFRT4oBgHgl3EQfPze-/content/tmp_files/load_file.txt b/ZtFRT4oBgHgl3EQfPze-/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..70262b9cc3bbe5cc6af86da43261179c51b7c209 --- /dev/null +++ b/ZtFRT4oBgHgl3EQfPze-/content/tmp_files/load_file.txt @@ -0,0 +1,2706 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf,len=2705 +page_content='Restricted distance-type Gaussian estimators based on density power divergence and their aplications in hypothesis testing A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Felipe, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Jaenada, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Miranda and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Pardo Department of Statistics and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=', Complutense University of Madrid, Spain Abstract Zhang (2019) presented a general estimation approach based on the Gaussian distribution for general parametric models where the likelihood of the data is difficult to obtain or unknown, but the mean and variance- covariance matrix are known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Castilla and Zografos (2021) extended the method to density power divergence-based estimators, which are more ro- bust than the likelihood-based Gaussian estimator against data contami- nation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Here, we present the restricted minimum density power divergence Gaussian estimator (MDPDGE) and study it asymptotic and robustness properties through it asymptotic distribution and influence function, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Restricted estimators are required in many practical situations and provide here constrained estimators to inherent restrictions of the un- derlying distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Further, we derive robust Rao-type test statistics based on the MDPDGE for testing simple a composite null hypothesis and we deduce explicit expression for some main important distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Fi- nally, we empirically evaluate the efficiency and robustness of the method through a simulation study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' AMS 2001 Subject Classification: 62F35, 62J12 Keywords and phrases: Gaussian estimator, Minimum density power diver- gence Gaussian estimator, Robustness, Influence function, Restricted Minimum density power divergence Gaussian estimator, Rao-type tests, Elliptical family of distributions 1 Introduction Let Y 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=', Y n be independent and identically distributed observations from a m-dimensional random vector Y with probability density function fθ(y), θ ∈ Θ ⊂ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' We denote, Eθ [Y ] = µ (θ) and Covθ [Y ] = Σ (θ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' (1) 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='13519v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='ST] 31 Jan 2023 The log-likelihood function, in order to get the maximum likelihood estimator (MLE), is given by, l (θ) = n� i=1 log fθ(yi) being y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=', yn realizations of the m-dimensional random vectors Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=', Yn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' If could be the case that the computation of fθ(y) is difficult or it is unknown but the computation of Eθ [Y ] and Covθ [Y ] is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' In this case, Zhang (2019) proposed to assume that fθ(y) follows a m−dimensional normal distribution with vector mean µ (θ) and variance-covariance matrix Σ (θ) as given in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' From a statistical point of view this procedure can be justified on the basis of the maximum-entropy principle, see Kapur (1989), because the distribution which is consistent with the given information, vector mean and variance- covariance matrix, and at the same time has maximum uncertainty, in terms of Shannon entropy, is the multidimensional normal population with vector mean and variance-covariance given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Hence Zhang (2019) defines the Gaussian likelihood function of θ, under y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=', yn, by lG (θ) = −nm 2 log 2π − n 2 log |Σ (θ)| − 1 2 n� i=1 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) (2) and the Gaussian estimator of θ by, �θG = arg max θ∈Θ lG (θ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' This idea consists of on assuming that the observations y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=', yn are from a m-variate normal population with vector mean µ (θ) and variance-covariance matrix Σ (θ) , the vector mean and variance-covariance matrix associated to the population Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' The Gaussian estimator is related with the Kullback-Leibler divergence and work well in the sense of the asymptotic efficiency but it has, as the classi- cal MLE, serious robustness problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' For this reason Castilla and Zografos (2022) extended the Guassian estimator on the basis of the density power di- vergence (DPD) defining the minimum density power divergence Gaussian esti- mator (MDPDGE),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' by �θ τ G = arg max θ∈Θ⊂Rd Hτ n (θ) (3) being Hτ n (θ) = τ + 1 τ (2π)mτ/2 |Σ (θ)|τ/2 1 n (4) � n� i=1 exp � −τ 2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) � − τ (1 + τ)(m/2)+1 � − 1 τ = a |Σ (θ)|− τ 2 1 n � n� i=1 exp � −τ 2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) � − b � − 1 τ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' 2 where,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' a = τ + 1 τ (2π)mτ/2 and b = τ (1 + τ)(m/2)+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' (5) It is immediate to see that the Gaussian estimator, �θG, of Zhang (2019) can be defined by, �θG = arg max θ∈Θ⊂Rd H0 n (θ) with H0 n (θ) = lim τ→0 Hτ n (θ) = −n 2 log |Σ (θ)|−1 2 n� i=1 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Hence, the MDPDGE is based in the DPD measure, defined by Basu et al (1998), while the Gaussian estimator is based on the Kullback- Leibler divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' All details about the MDPDGE, �θ τ G, can be seen in Castilla and Zografos (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' In particular, in the cited paper, the consistency and the asymptotic distribution of the MDPDGE, �θ τ G, was studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Given Y 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=', Y n independent and identically distributed vectors from the m-dimensional random vector Y , the MDPDGE, �θ τ G, defined in (3) satisfies, √n � �θ τ G − θ � L −→ n−→∞ N(0d, Jτ(θ)−1Kτ(θ)Jτ(θ)−1) (6) being Jτ(θ) = � Jij τ (θ) � i,j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='.,,d and Kτ(θ) = � Kij τ (θ) � i,j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='.,,d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' The elements Jij τ (θ) and Kij τ (θ) of the matrices Jτ (θ) and Kτ (θ) are given by,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Jij τ (θ) = � 1 (2π)m/2 |Σ (θ)|1/2 �τ 1 (1 + τ)(m/2)+2 (7) � (τ + 1) trace � Σ (θ)−1 ∂µ (θ) ∂θ �∂µ (θ) ∂θ �T � +∆i τ∆j τ + 1 2trace � Σ (θ)−1 ∂Σ (θ) ∂θj Σ (θ)−1 ∂Σ (θ) ∂θi �� and Kij τ (θ) = � 1 (2π)m/2 |Σ (θ)|1/2 �2τ 1 (1 + 2τ)(m/2)+2 � ∆i 2τ∆j 2τ (8) + (1 + 2τ) trace � Σ (θ)−1 ∂µ (θ) ∂θ �∂µ (θ) ∂θ �T � +1 2trace � Σ (θ)−1 ∂Σ (θ) ∂θj Σ (θ)−1 ∂Σ (θ) ∂θi �� − � 1 (2π)m/2 |Σ (θ)|1/2 �2τ 1 (1 + τ)m+2 ∆i τ∆j τ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' 3 with ∆i τ = τ 2trace � Σ (θ)−1 ∂Σ (θ) ∂θi � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' On certain occasions we have an additional knowledge about the true param- eter because it satisfies certain restrictions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=', the parameter space is restricted in the way, {θ ∈ Θ/ g(θ) = 0r} , (9) where g : Rd → Rr is a vector-valued function mapping such that the d × r matrix G (θ) = ∂gT (θ) ∂θ (10) exists and is continuous in θ and rank(G (θ)) = r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' also 0r denotes the null vector of dimension r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' In the following the parameter space given in (9) will be denoted by Θ0 because it will represent, in most of the situations, a composite null hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' The superscript T in (10) represents the transpose of the matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' The most popular estimator of θ under the restriction given in (9) is the restricted MLE (RMLE) that is an estimator which maximizes the loglikelihood function subject to the restrictions g(θ) = 0r (see Silvey, 1975).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' The RMLE has the same robustness problems that the MLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Later some restricted estima- tors have been considered in the statistical literature to overcome the problem of robustness of the RMLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' We will only mention the restricted estimators based on divergence measures: In Pardo et al (2002), the restricted minimum Phi-divergence estimator was introduced and its properties studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Basu et al (2018) presented the restricted minimum density power divergence estima- tors (RMDPDE) and studied some applications of them in testing hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' In Ghosh (2015) the theoretical robustness properties of the RMDPDE were studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' More recently Jaenada et al (2022) considered the restricted R´enyi pseudodistance estimator as well as its use in defining Rao-type tests based on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' But if it is not easy to get the fθ(y) or it is unknown the previous estimators can not be obtained and for this reason in this paper we are going to introduce and to study the restricted minimum density power divergence Gaussian esti- mator (RMDPDGE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' In Section 2 we introduce the RMDPDGE and we obtain its asymptotic distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' The rest of the paper goes as follows: Section 3 is devoted to get the influence function of the RMDPDGE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Some statistical appli- cations for testing are presented in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' A simulation study is carried out in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Finally, we present an Appendix in which we have included the proofs of some of the results appearing in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' 2 Restricted minimum density power divergence Gaussian estimators We shall begin giving the definition of the RMDPDGE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Definition 1 Let Y 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=', Y n be independent and identically distributed ob- servations from a m-dimensional random vector Y with Eθ [Y ] = µ (θ) and 4 Covθ [Y ] = Σ (θ), θ ∈ Θ ⊂ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' The RMDPDGE, �θ τ G, is defined by �θ τ G = arg max θ∈Θ/ g(θ)=0r Hτ n (θ) , where Hτ n (θ) was given in (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' The main purpose in this Section is to get the asymptotic distribution of �θ τ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Before presenting the theorem with the asymptotic distribution we shall give some previous results that are necessary in order to proof the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Proposition 2 Let Y 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=', Y n be independent and identically distributed ob- servations from a m-dimensional random vector Y with Eθ [Y ] = µ (θ) and Covθ [Y ] = Σ (θ), θ ∈ Θ ⊂ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Then, √n � 1 τ + 1 ∂Hτ n (θ) ∂θ � L −→ n−→∞ N(0d, Kτ(θ)), where Kτ(θ) was defined in (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' See Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Proposition 3 Let Y 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=', Y n be independent and identically distributed ob- servations from a m-dimensional random vector Y with with Eθ [Y ] = µ (θ) and Covθ [Y ] = Σ (θ), θ ∈ Θ ⊂ Rd, we have, ∂2Hτ n (θ) ∂θ∂θT P −→ n−→∞ − (τ + 1) Jτ(θ), where Jτ(θ) was defined in (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' See Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' In the next theorem we present the asymptotic distribution of the RMD- PDGE, �θ τ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Theorem 4 Let Y 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=', Y n be independent and identically distributed observa- tions from a m-dimensional random vector Y with Eθ [Y ] = µ (θ) and Covθ [Y ] = Σ (θ), θ ∈ Θ ⊂ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Suppose the true distribution of Y belongs to the model and θ ∈ Θ0 is the true parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Then the RMDPDGE �θ τ G of θ obtained under the constraints g(θ) = 0r has the asymptotic distribution, n1/2(�θ τ G − θ) L −→ n−→∞ N(0d, M τ(θ)) (11) where M τ(θ) = P ∗ τ(θ)Kτ (θ) P ∗ τ(θ)T , P ∗ τ(θ) = Jτ(θ)−1 − Qτ(θ)G(θ)T Jτ(θ)−1, (12) Qτ(θ) = J−1 τ (θ)G(θ) � G(θ)T Jτ(θ)−1G(θ) �−1 , (13) and Jτ(θ) and Kτ (θ) were defined in (7) and (8), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' 5 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' The estimating equations for the RMDPDGE are given by � ∂ ∂θHτ n(θ) + G(θ)λn = 0d, g(�θ τ G) = 0r, (14) where λn is a vector of Lagrangian multipliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Now we consider θn = θ0 + mn−1/2, where ||m|| < k, for 0 < k < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' We have, ∂ ∂θ Hτ n(θ)|θ=θn = ∂ ∂θ Hτ n(θ) + ∂2 ∂θT ∂θ Hτ n(θ)|θ=θ∗ (θn − θ) + o(||θn − θ||2) and n1/2 ∂ ∂θ Hτ n(θ) ���� θ=θn = n1/2 ∂ ∂θ Hτ n(θ) + ∂2 ∂θT ∂θ Hτ n(θ)|θ=θ∗ n1/2 (15) (θn − θ) + o(n1/2||θn − θ||2) where θ∗ belongs to the segment joining θ and θ∗ However, o(n1/2||θn − θ||2) = o(n1/2||m||2/n) = o(n−1/2||m||2) = o(Op(1)) = op(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Since lim n→∞ ∂2 ∂θT ∂θ Hτ n(θ) = − (τ + 1) Jτ(θ) we obtain n1/2 ∂ ∂θ Hτ n(θ) ���� θ=θn = n1/2 ∂ ∂θ Hτ n(θ)−(τ + 1) n1/2Jτ(θ)(θn−θ)+op(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' (16) Taking into account that G(θ) is continuous in θ n1/2g(θn) = G(θ)T n1/2(θn − θ) + op(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' (17) The RMDPDGE �θ τ G must satisfy the conditions in (14), and in view of (16) and (17) we have n1/2 ∂ ∂θ Hτ n(θ) − (τ + 1) Jτ(θ)n1/2(�θ τ G − θ) + G(θ)n1/2λn + op(1) = 0p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' (18) From (17) it follows that GT (θ)n1/2(�θ τ G − θ) + op(1) = 0r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' (19) Now we can express equations (18) and (19) in the matrix form as � (τ + 1) Jτ(θ) −G(θ) −GT (θ) 0 � � n1/2(�θ τ G − θ0) n1/2λn � = � n1/2 ∂ ∂θHτ n(θ) 0 � + op(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' 6 Therefore � n1/2(�θ τ G − θ) n1/2λn � = � (τ + 1) Jτ(θ) −G(θ) −GT (θ) 0 �−1 � −n1/2 ∂ ∂θHτ n(θ) 0r � +op(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' But � (τ + 1) Jτ(θ) −G(θ) −GT (θ) 0 �−1 = � L∗ τ(θ) Qτ(θ) Qτ(θ0)T Rτ(θ) � , where L∗ τ(θ) = 1 τ + 1Jτ(θ)−1 − Qτ(θ)G(θ)T Jτ(θ)−1 = 1 τ + 1P ∗ τ(θ) Qτ(θ) = J−1 τ (θ)G(θ) � G(θ)T Jτ(θ)−1G(θ) �−1 Rτ(θ) = G(θ)T Jτ(θ)−1G(θ) P ∗ τ(θ0) and Qτ(θ0) are as given in (12) and (13) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Then, n1/2(�θ τ G − θ) = (τ + 1)−1 P ∗ τ(θ)n1/2 ∂ ∂θ Hτ n(θ) + op(1), (20) and we know by Proposition 2 that n1/2 (τ + 1)−1 ∂ ∂θ Hτ n(θ) L −→ n−→∞ N (0, Kτ (θ)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' (21) Now by (20) and (21) we have the desired result presented in (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Remark 5 Notice that the result in (6) is a special case of the previous theorem when there is no restriction on the parametric space, in the sense that G, defined in (10), is the null matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' In this case the matrix P ∗ τ(θ) given in (12) becomes P ∗ τ(θ) = Jτ(θ)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Therefore, the asymptotic variance-covariance matrix of the unrestricted estimator may be reconstructed from the previous theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' 3 Influence function for the RMDPDGE Based on ∂ |Σ (θ)|−τ/2 ∂θ = − τ 2 |Σ (θ)|−τ/2 trace � Σ (θ)−1 ∂Σ (θ) ∂θ � ∂ ∂θ (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) = −2 � ∂µ(θ) ∂θ �T Σ (θ)−1 (yi − µ (θ))− (yi − µ (θ))T � Σ (θ)−1 ∂Σ (θ) ∂θ Σ (θ)−1 � (yi − µ (θ)) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' 7 we have ∂ ∂θ Hτ n(θ) = 1 n n� i=1 � −a τ 2 |Σ (θ)|−τ/2 trace � Σ (θ)−1 ∂Σ (θ) ∂θ � exp � −τ 2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) � + ba τ 2 |Σ (θ)|−τ/2 trace � Σ (θ)−1 ∂Σ (θ) ∂θ � + a τ 2 |Σ (θ)|−τ/2 exp � −τ 2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) � � 2 �∂µ (θ) ∂θ �T Σ (θ)−1 (yi − µ (θ)) + (yi − µ (θ))T � Σ (θ)−1 ∂Σ (θ) ∂θ Σ (θ)−1 � (yi − µ (θ)) �� = 1 n n� i=1 Ψτ(yi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' θ), where Ψτ(yi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' θ) = a τ 2 |Σ (θ)|−τ/2 � −trace � Σ (θ)−1 ∂Σ (θ) ∂θ � (22) exp � −τ 2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) � + b trace � Σ (θ)−1 ∂Σ (θ) ∂θ � + exp � −τ 2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) � � 2 �∂µ (θ) ∂θ �T Σ (θ)−1 (yi − µ (θ)) + (yi − µ (θ))T � Σ (θ)−1 ∂Σ (θ) ∂θ Σ (θ)−1 � (yi − µ (θ)) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Therefore the estimating equations are, n� i=1 Ψτ(yi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' θ) = 0d, (23) and the MDPDGE is an M-estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Based on the general theory of M- estimators we know that √n � �θ τ G − θ � L −→ n−→∞ N � 0d, S−1MS−1� being S = −E �∂2Hτ n (θ) ∂θ∂θT � and M = Cov �√n ∂ ∂θ Hτ n(θ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Based on Propositions 2 and 3 we have, S = (τ + 1) Jτ (θ) and M = (τ + 1)2 Kτ (θ) 8 and we get the result given in (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' To analyze the robustness of an estimator, Hampel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' (1986) introduced the concept of Influence Function (IF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Since then, the IF have been widely used in the statistical literature to measure robustness in different statistical contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Intuitively, the IF describes the effect of an infinitesimal contamina- tion of the model on the estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Then, IFs associated to locally robust (B- robust) estimators should be bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Let us now obtain the IF of RMDPDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' We consider the contaminated model gε(y) = (1 − ε)fθ(y) + ε∆y, with ∆y the indicator function in y, and we denote �θ τ G,ε = �Tτ(Gε), being Gε the distribution function associated to gε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' By definition �θ τ G,εis the minimizer of Hτ n(θ) subject to g(�θ τ G,ε) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Taking into account that the MDPDGE is an M-estimator we have that the influence function of the MDPDGE is given by IF(y, �Tτ, θ) = Jτ(θ)−1Ψτ(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' θ), (24) where Jτ(θ) was defined in (7) and Ψτ(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' θ) in (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' The influence function of the RMDPDGE will be obtained with the additional condition g(�θ τ G,ε) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Differentiating this last equation gives, at ε = 0, G (θ)T IF(y, �Tτ, θ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' (25) Based on (24) and (25) we have � Jτ(θ) G (θ)T � IF(y, �Tτ, θ) = � Ψτ(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' θ) 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Therefore, � Jτ(θ)T , G (θ) � � Jτ(θ) G (θ)T � IF(y, �Tτ, θ) = Jτ(θ)T Ψτ(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' θ) and the influence function of the RMDPDGE, �θ τ G, is given by � Jτ(θ)T Jτ(θ)) + G (θ) G (θ)T �−1 Jτ(θ)T Ψτ(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' (26) We can observe that the influence function of the �θ τ G, obtained in (26) will be bounded if the influence function of the MDPDGE, �θ τ G, given in (24) is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' In general it is not easy to see if it is bounded or not but in particular situations is not difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' On the other hand if there are not restrictions, G (θ) = 0, and therefore (26) coincides with (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='1 we shall present the expression of Jτ(θ) and ψτ(y, θ) for the exponential and Poisson models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Based on that results we present in Figure 1 the influence function of the MDPDGE, �θ τ G, for θ = 4 and τ = 0, 0, 2 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='8 for the exponential model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' We can see that for τ = 0, the influence function is not bounded and for τ = 0, 2 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='8 is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' This fact points out the robustness of the MDPDGE, �θ τ G, for τ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' 9 Figure 1: Influence function of the MDPDGE for the exponential model with τ = 0 (red), τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 (black) and τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='8 (green) 4 Rao-type tests based on RMDPDGE In the last years many robust test statistics have been introduced in the statis- tical literature based on minimum distance estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' We pay special atten- tion to the procedures based on density power divergence (MDPDE) as well as the procedures based on Renyi’s pseudodistance estimator (MRPE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' The test statistics are essentially of two types: Wald-type tests and Rao-type tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Some references are the following: Basu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' (2013, 2016, 2017, 2018a, 2018b, 2022a, 2022b), Castilla et al (2016, 2021), Jaenada et al (2022a, 2022b), Men´endez et al (1995) and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' In this section we are going to introduce and study the Rao-type tests based on RMDPDGE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' We analyze the case of simple null hypothesis because for com- posite null hypothesis it is necessary a separated paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Note that, 1 √n n� i=1 1 τ + 1Ψτ(yi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' θ) = √n 1 τ + 1 ∂ ∂θ Hτ n(θ), and hence by Proposition 2, 1 √n n� i=1 1 τ + 1Ψτ(yi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' θ) L→ n→∞ N (0p, Kτ (θ)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Definition 6 Let Y 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=', Y n be independent and identically distributed ob- servations from a m-dimensional random vector Y with Eθ [Y ] = µ (θ) and Covθ [Y ] = Σ (θ), θ ∈ Θ ⊂ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' For testing H0 : θ = θ0 versus H1 : θ ̸= θ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' (27) 10 IF(y) 10 8 6 4 2 0 4 5 6 7 8 9 10 11 12 13 14 15 ythe Rao-type test statistic, based on RMDPDGE, is defined by Rτ (θ0) = 1 nU τ n(θ0)T Kτ (θ0)−1 U τ n(θ0) where U τ n(θ) = � 1 τ + 1 n� i=1 Ψ1 τ(yi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' θ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=', 1 τ + 1 �n i=1 Ψd τ(yi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' θ) �T and Ψτ(yi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' θ) = � Ψ1 τ(yi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' θ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='., Ψd τ(yi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' θ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' In the following we shall call Rτ (θ0), ”Rao-type test based on RMDPDGE”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Theorem 7 Let Y 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=', Y n be independent and identically distributed obser- vations from a m- dimensional random vector Y with Eθ [Y ] = µ (θ) and Covθ [Y ] = Σ (θ), θ ∈ Θ ⊂ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Under the null hypothesis given in (27) we have, Rτ (θ0) L→ n→∞ χ2 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' It is clear that 1 √nU τ n(θ) = 1 √n n� i=1 1 τ + 1Ψτ(yi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' θ) = √n 1 τ + 1 ∂ ∂θ Hτ n(θ) L −→ n−→∞ N (0, Kτ (θ)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Now the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Remark 8 Based on previous theorem, if the sample size is large enough, one can use the 100(1 − α) percentile, χ2 d,α, of the chi-square with d degrees of freedom defined by the equation Pr � χ2 d > χ2 d,α � = α, to propose the decision rule: ”Reject H0, with a significant level α, if Rτ (θ0) > χ2 d,α”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Example 9 (Elliptical distributions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' The m-dimensional random vector Y has an elliptical distribution if its characteristic function has the form ϕY (t) = exp � it2µ � ψ �1 2t2Σt � being µ a m−dimensional column vector, Σ a positive definite matrix and ψ(t) the so-called characteristic generator function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' The function ψ may depend on the dimension of random vector Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' In general, it does not follow that Y has a joint density function, fY (y), but this density exists, it is given by fY (y) = cm |Σ|− 1 2 gm �1 2 (y − µ)T Σ−1 (y − µ) � for some density generator function gm which could depend on the dimension m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' The elliptical family of distributions is denoted by Em (µ, Σ,gm) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' In the case the density exists cm is given explicitly by cm = (2π)− m 2 Γ �m 2 � �� x m 2 −1gm (x) dx �−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' 11 For more details about the family Em (µ, Σ,gm) see for instance Fang et al (1987), Gupta and Varga (1993), Cambanis et al (1981), Fang and Zhang (1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' In Fang et al (1987), for instance, can be seen that E [Y ] = µ and Cov [Y ] = cY Σ where cY = −2ψ′(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' In this case, the parameter to be estimated is θ = � µT , Σ � whose dimension is s = m + m(m+1) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' In the following we shall denote µ(θ) instead of µ and Σ (θ) instead of Σ, in order to be consistent with our previous notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' If we are interested in testing H0 : (µ(θ), Σ (θ)) = (µ0, Σ0) versus H1 : (µ(θ), Σ (θ)) ̸= (µ0, Σ0) (28) where µ0 and Σ0 completely known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Hence, We shall reject the null hypothesis given in (28) if Rτ (µ0, Σ0) = 1 nU τ n(µ0, Σ0)T Kτ (µ0, Σ0)−1 U τ n(µ0, Σ0) > χ2 m+ m(m+1) 2 ,α where U τ n(µ0, Σ0 ) = n� i=1 1 τ + 1Ψτ(yi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' µ0, Σ0) with Ψτ(yi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' µ0, Σ0) is obtained from (22) replacing Σ (θ) by cY Σ and µ (θ) by µ and Kτ (µ0, Σ0) is obtained from (8) replacing Σ (θ) by cY Σ and µ (θ) by µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Let us now establish the consistency of the Score-type test based on RMD- PDGE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' In the following we shall denote, Y τ(θ) = a τ 2 |Σ (θ)|−τ/2 � −trace � Σ (θ)−1 ∂Σ (θ) ∂θ � (29) exp � −τ 2 (Y − µ (θ))T Σ (θ)−1 (Y − µ (θ)) � + b trace � Σ (θ)−1 ∂Σ (θ) ∂θ � + exp � −τ 2 (Y − µ (θ))T Σ (θ)−1 (Y − µ (θ)) � � 2 �∂µ (θ) ∂θ �T Σ (θ)−1 (Y − µ (θ)) + (Y − µ (θ))T � Σ (θ)−1 ∂Σ (θ) ∂θ Σ (θ)−1 � (Y − µ (θ)) �� , where a and b were defined in (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' We can observe that ∂ ∂θHn(θ) is the sample mean of a random sample of size n from the m-dimensional population Y τ(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Theorem 10 Let Y 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=', Y n be independent and identically distributed obser- vations from a m- dimensional random vector Y with Eθ [Y ] = µ (θ) and 12 Covθ [Y ] = Σ (θ), θ ∈ Θ ⊂ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Let θ ∈ Θ with θ ̸= θ0, with θ0 defined in (27), and let us assume that Eθ [Y τ(θ0)] ̸= 0d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Then, lim n→∞ P � Rτ (θ0) > χ2 d,α � = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' We have, 1 nU τ n(θ0) = 1 n n� i=1 1 τ + 1Ψτ(Y i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' θ) = 1 τ + 1 ∂ ∂θ Hτ n(θ) P→ n→∞ 1 τ + 1E [Y τ(θ0)] , where Y τ(θ0) was defined in (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Therefore P � Rτ (θ0) > χ2 d,α � = P � 1 nRτ (θ0) > 1 nχ2 d,α � −→ n→∞ I � 1 (τ + 1)2 Eθ [Y τ(θ0)] K−1 τ (θ) ET θ [Y τ(θ0)] > 0 � = 1, where I(·) is the indicator function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Now let us derive the asymptotic distribution of Rτ (θ0) under local Pitman- type alternative hypotheses of the form H1,n : θ = θn, where θn = θ0 +n−1/2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Such results help to determine the asymptotic contiguous power of the Score- type test based on RMDPDGE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Theorem 11 Let Y 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=', Y n be independent and identically distributed ob- servations from a m-dimensional random vector Y with Eθ [Y ] = µ (θ) and Covθ [Y ] = Σ (θ), θ ∈ Θ ⊂ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Under the contiguous alternative hypothesis H1,n : θn = θ0 + n−1/2d, the asymptotic distribution of the Rao-type test based on RMDPDGE, Rτ (θ0) , is a non-central chi-square distribution with d degrees of freedom and non-centrality parameter given by δτ(θ0, d) = dT Jτ (θ0) K−1 τ (θ0) Jτ (θ0) d Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Consider the Taylor series expansion 1 √nU τ n(θn) = 1 √nU τ n(θ0) + 1 n ∂U τ n(θ) ∂θT ���� θ=θ∗ n d, where θ∗ n belongs to the line segment joining θ0 and θ0 + 1 √nd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Now, by propo- sition 3 But 1 n ∂U τ n(θ) ∂θT = 1 τ + 1 ∂2Hτ n (θ) ∂θ ∂θT P −→ n−→∞ −Jτ(θ) Therefore, 1 √nU τ n(θ) ���� θ=θ0+n−1/2d L −→ n→∞ N (−Jτ (θ0) d, Kτ (θ0)) , and Rτ (θ0) L −→ n→∞ χ2 p (δτ(θ0, d)) , 13 with δτ(θ0, d) given by δτ(θ0, d) = dT Jτ (θ0) K−1 τ (θ0) Jτ (θ0) d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Remark 12 The family of Score-type tests, Rτ (θ0) , presented in this Section for simple null hypothesis can be extended to composite null hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' If we are interested in testing H0 : θ ∈ Θ0 = {θ ∈ Θ/ g(θ) = 0r} we can consider the family of Score-type tests given by Rτ � �θ τ G � = 1 nU τ n(�θ τ G)T Qτ(�θ τ G) � Qτ(�θ τ G)Kτ � �θ τ G � Qτ(�θ τ G) �−1 Qτ(�θ τ G)T U τ n(�θ τ G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' (30) However, the analysis of the family of test statistics presented in (30) deserves a new paper, that will be finished soon, in the line of the paper of Basu et al (2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='1 Rao-type test based on MDPDGE for univariate dis- tributions and θ ∈ Θ Let Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='., Yn a random sample from the population Y, with E [Y ] = µ (θ) and V ar [Y ] = σ2 (θ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Based on (23) the estimating equation is given by n� i=1 Ψτ (yi, θ) = 0 with Ψτ (yi, θ) = (τ + 1) � σ2 (θ) �−τ/2 2 (2π)τ/2 �� −∂ log σ2 (θ) ∂θ + ∂ log σ2 (θ) ∂θ �yi − µ (θ) σ (θ) �2 (31) +2∂µ (θ) ∂θ (yi − µ (θ)) 1 σ2 (θ) � exp � − τ 2σ2 (θ) (yi − µ (θ))2 � + τ (1 + τ)3/2 ∂ log σ2 (θ) ∂θ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' The expressions of Jτ (θ) and Kτ (θ) are given by Jτ (θ) = 1 � 2πσ (θ)2� τ 2 1 (1 + τ)5/2 � (τ + 1) σ−2 (θ) �∂µ (θ) ∂θ �2 + τ 2 4 �∂ log σ2 (θ) ∂θ �2 +1 2 �∂ log σ2 (θ) ∂θ �2� 14 and Kτ (θ) = � 1 (2π)1/2 σ (θ) �2τ � 1 (1 + 2τ)5/2 � τ 2 �∂ log σ2 (θ) ∂θ �2 (32) + (1 + 2τ) σ−2 (θ) �∂µ (θ) ∂θ �2 + 1 2 �∂ log σ2 (θ) ∂θ �2� − τ 2 4 (1 + τ)3 �∂ log σ2 (θ) ∂θ �2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Therefore, if we are interesting in testing H0 : θ = θ0 versus H1 : θ ̸= θ0, on the basis of the Rao-type tests based on RMDPDGE, Rτ (θ0) , we shall reject the null hypothesis if Rτ (θ0) = 1 nU τ n (θ0)2Kτ (θ0)−1 > χ2 1,α where U τ n (θ0) = 1 τ + 1 n� i=1 Ψτ (yi, θ) with Ψτ (yi, θ) is defined in (31) and Kτ (θ) is given in (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Let us consider some special cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='1 Poisson Model We shall assume that the random variable Y is Poisson with parameter θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' In this case E [Y ] = V ar [Y ] = θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' The RMDPDGE, for τ > 0, is given by, �θ τ G = arg max θ � τ + 1 τ (2πθ) τ 2 � 1 n n� i=1 exp � − τ 2θ (yi − θ)2� − τ (1 + τ)3/2 � − 1 τ � , and for τ → 0, we have.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' �θG = arg max θ � −1 2 log 2π − 1 2 log θ − 1 n n� i=1 1 2θ (yi − θ)2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' On the other hand, Ψτ (yi, θ) = τ + 1 2 (2πθ) τ 2 θ2 � � −2θ2 + y2 i � exp � − τ 2θ (yi − θ)2� + τθ (1 + τ) 3 2 � and for τ = 0, we obtain the result presented by Zhang (2019) Ψ0 (yi, θ) = 1 2θ2 � −2θ2 + y2 i � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' 15 In relation to Kτ (θ) , we get, Kτ (θ) = � 1 2π �τ 1 2θ2+τ � 1 (1 + 2τ)5/2 � � 2τ 2 + 2θ + 4θτ + 1 � − τ 2 2 (1 + τ)3 �� and hence the Rao-type test based on RMDPDGE, for τ > 0, Rτ (θ0) , for testing, H0 : θ = θ0 versus H1 : θ ̸= θ0, is given by Rτ (θ0) = 1 n 1 � 2 (2πθ) τ 2 θ2�2 � n� i=1 � � −2θ2 0 + y2 i � exp � − τ 2θ0 (yi − θ0)2 � + τθ (1 + τ) 3 2 ��2 Kτ (θ0)−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' For τ = 0 we get R0 (θ0) = 1 4n � n� i=1 �−2θ2 0 + y2 i θ2 0 ��2 K0 (θ0)−1 , with K0 (θ0) = 2θ0 + 1 2θ2 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' We shall reject the null hypothesis if Rτ (θ0) > χ2 1,α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 Exponential model Assume now that the random variable Y is exponential fθ(x) = 1 θ exp � −x θ � , x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' (33) In this case E [Y ] = θ and V ar [Y ] = θ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' The RMDPDGE, for τ > 0, is given by �θ τ G = arg max θ � τ + 1 τ � 1 θ √ 2π �τ � 1 n n� i=1 exp � −τ 2 �yi − θ θ �2� − τ (1 + τ)3/2 � − 1 τ � , and for τ → 0, we have.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' �θG = arg max θ � −1 2 log 2π − log θ − 1 n n� i=1 1 2 �yi − θ θ �2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' On the other hand, Ψτ (yi, θ) = (τ + 1) θτ+3 �√ 2π �τ � � y2 i − yiθ − θ2� exp � −τ 2 �yi − θ θ �2� + τθ2 (1 + τ) 3 2 � , 16 and for τ = 0, we get Ψ0 (yi, θ) = 1 θ3 � y2 i − yiθ + θ2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' The matrix Kτ (θ) has the expression Kτ (θ) = 1 (2π)τ θ2(τ+1) � 1 (1 + 2τ)5/2 � 4τ 2 + 2τ + 3 � − τ 2 (1 + τ)3 � and K0 (θ) = 2 θ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Hence the Rao-type test based on RMDPDGE, for τ > 0, for testing H0 : θ = θ0 versus H1 : θ ̸= θ0, is given by Rτ (θ0) = 1 n 1 θ2τ+6 0 (2π)τ � n� i=1 � � y2 i − yiθ0 − θ2 0 � exp � −τ 2 �yi − θ0 θ0 �2� + τθ2 0 (1 + τ) 3 2 ��2 (34) ×Kτ (θ0)−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' and by R0 (θ0) = 1 2n 1 θ4 0 � n� i=1 �� y2 i − yiθ0 − θ2 0 ���2 (35) for τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' 5 Simulation study We analyze here the performance of the Rao-type tests based on the MDPDGE, Rτ (θ0) , in terms of robustness and efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' We compare the proposed general method assuming Gaussian distribution with Rao-type test statistics based on the true parametric distribution underlying the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' We consider the exponential model with density function fθ0(x) given in (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' For the exponential model, the Rao-type test statistic based on MDPDGE is, for τ > 0, as given in (34) and for τ = 0 as given in (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' To evaluate the robustness of the tests we generate samples from an exponential mixture, f ε θ0(x) = (1 − ε)fθ0(x) + εf2θ0(x), where θ0 denotes the parameter of the exponential distribution and ε is the con- tamination proportion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' The uncontaminated model is thus obtained by setting ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' For comparison purposes we have also considered the robust Rao-type tests based on the restricted MDPDE, introduced and studied in Basu et al (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' 17 The efficiency loss caused by the Gaussian assumption should be advertised by the poorer performance of the Rao-type tests based on the restricted MDPDGE with respect to their analogous based on the restricted MDPDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' For the ex- ponential model, the famility Rao-type test statistics based on the restricted MDPDE is given, for β > 0, as Sβ n(θ0) = � 4β2 + 1 (2β + 1)3 − β2 (β + 1)4 �−1 1 n � 1 θ0 n � i=1 (yi − θ0) exp � −βyi θ0 � + nβ (β + 1)2 �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' For β = 0, the above test reduces to the classical Rao test given by Sn (θ0) = Sβ=0,n (θ0) = �√n ¯Xn − θ0 θ0 �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' We consider the testing problem H0 : θ0 = 2 vs H1 : θ ̸= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' and we empirically examine the level and power of both Rao-type test statistics, the usual test based on the parametric model and the Gaussian-based test by setting the true value of the parameter θ0 = 2 and θ0 = 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Different sample sizes were considered, namely n = 10, 20, 30, 40, 50, 60, 70, 80, 90, 100 and 200, but simulation results were quite similar and so, for brevity, we only report here results for n = 20 and n = 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' The empirical level of the test is computed �αn (ε) = Number of times � Rτ n (θ0) (or Sβ n (θ0)) > χ2 1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='05 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='84146 � Number of simulated samples .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' We set ε = 0%, 5%, 10% and 20% of contamination proportions and perform the Monte-Carlo study over R = 10000 replications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' The tuning parameters τ and β are fixed from a grid of values, namely {0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=', 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='7}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Simulation results are presented in Tables 1 and 2 for n = 20 and n = 40, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' The robustness advantage in terms of level of both Rao-type tests considered, Rτ (θ0) and Sβ n(θ0) with positive values of the tuning with respect to the test statistics with τ = 0 and β = 0 is clearly shown, as their simulated levels are closer to the nominal in the presence of contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Regarding the power of the tests, uncontaminated scenarios there are values at least so good than the corresponding to τ = 0 and β = 0 and for contaminated data the power corresponding to τ > 0 and β > 0 are higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' The loss of efficiency caused by the Guassian assumption can be measured by the discrepancy of the estimated levels and powers between the family of Rao- type tests based on the restricted MDPDGE and the MDPDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' As expected, empirical levels of the test statistics based on the MDPDGE are quite higher than the corresponding levels of the test based in the MDPDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' However, the test statistic based on the parametric model, Sβ n(θ0), is quite conservative and so the corresponding powers are higher than those of the proposed test, Rτ (θ0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Based 18 Table 1: Simulated levels for different contamination proportions and different tuning parametersτ, β = 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='7 for the Rao-type tests Rτ (θ0) and Sβ n(θ0) with for n = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' τ �αn (0) �αn (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='05) �αn (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='10) �αn (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='20) �πn (0) �πn (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='1) �πn (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='15) �πn (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='20) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2601 0.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='7258 19 on the presented results, it seems that the proposed Rao-type test, Rτ (θ0) , performs reasonably well and offers an appealing alternative for situations where the probability density function of the true model is unknown or it is very complicated to work with it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' References [1] Basu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Mandal, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Martin, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' and Pardo, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Testing statistical hypotheses based on the density power divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Annals of the Institute of Statistical Mathematics, 65, 2, 319–348.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' [2] Basu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Mandal, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Martin, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' and Pardo, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Generalized Wald- type tests based on 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Mart´ın, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' and Pardo, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' A Wald-type test statistic for testing linear hypothesis in logistic regression models based on minimum density power divergence estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Electronic Journal of Statistics, 2, 2741–2772.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' [4] Basu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Ghosh, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Martin, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' and Pardo, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' (2018a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Robust Wald- type tests for non-homogeneous observations based on the minimum density power divergence estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Metrika, 81, 5, 493–522.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' [5] Basu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Mandal, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Martin, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' and Pardo, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' (2018b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Testing composite hypothesis based on the density power divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Sankhya B, 80, 2, 222– 262.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' [6] Basu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Ghosh, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Mandal, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Martin, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' and Pardo, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Wald-type tests in GLM with random design based on minimum density power divergence estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Statistical Methods & Applications, 30, 973– 1005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' [7] Basu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' ;' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' On distance-type Gaussian estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Journal of Multivariate Analysis, 188, Paper No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' 104831, 22 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' 20 [11] Castilla, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=';' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' (2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Robust test statistics based on restricted minimum R´enyi’s pseudodistance estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Entropy, 24, 5, Paper No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' 616, 28 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' [23] Kapur, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' (1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Maximum Entropy Models in Science and Engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Wiley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' New Delhi, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' [24] Men´endez, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Morales, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Pardo, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Vajda, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Divergence-based estimation and testing of statistical models of classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Journal of Multivariate Analysis, 54, 2, 329–354.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' 21 [25] Pardo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Pardo, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Zografos, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Minimum φ-divergence estima- tors with constraints in multinomial populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Journal of Statististical Planning and Inference, 104, 1, 221–237.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' [26] Silvey, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' (1975).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Reprinting, Monographs on Statistical Subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Chap- man and Hall, London.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' [27] Zhang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' General Gaussian estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Journal of Multivariate Analysis, 169, 234-247.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' 6 Appendix In the different Sections of the Appendix will be important the following results: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Results in relation to the derivatives (a) ∂Σ (θ) ∂θi = |Σ (θ)| trace � Σ (θ)−1 ∂Σ (θ) ∂θi � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' (b) ∂trace(Σ(θ)) ∂θi = trace �∂Σ (θ) ∂θi � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' (c) ∂Σ (θ) ∂θi −1 = −Σ (θ)−1 ∂Σ (θ) ∂θi Σ (θ)−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Let Y be a normal population with vector mean µ and variance-covariance Σ we have, (a) E � (Y − µ)T A (Y − µ) � = Trace (AΣ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' (b) E � (Y − µ)T A (Y − µ) (Y − µ)T B (Y − µ) � = Trace � AΣ � B + BT � Σ � + Trace (AΣ) Trace (BΣ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' (c) E � (Y − µ)T A (Y − µ) (Y − µ) � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='1 Appendix A (Proof of Proposition 2) The expresion of Hτ n(θ) is given by, Hτ n(θ) = a |Σ (θ)|−τ/2 � 1 n n� i=1 exp � −τ 2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) � − b � −1 τ and we consider the d-dimensional random vector Y τ(θ) defined in (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Ap- plying Central Limit Theorem we have, √n ∂ ∂θ Hτ n(θ) = 1 √n n� i=1 Ψτ(yi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' θ) L −→ n−→∞ N(0m, Sτ(θ0)) with Sτ(θ0) = Cov [Y τ(θ)] = E � Y τ(θ)T Y τ(θ) � 22 because E [Y τ(θ)] = 0d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' We are going to see that E [Y τ(θ)] = 0d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' We have,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='E [Y τ(θ)] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='a τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 |Σ (θ)|−τ/2 E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='−trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='−τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 (Y − µ (θ))T Σ (θ)−1 (Y − µ (θ)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='+ b trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='+ exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='−τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 (Y − µ (θ))T Σ (θ)−1 (Y − µ (θ)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�∂µ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 (Y − µ (θ)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='+ (Y − µ (θ))T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(Y − µ (θ)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='a τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 |Σ (θ)|−τ/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='−trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(τ + 1)m/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(τ + 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 +1 trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(τ + 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 +1 trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='0d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' We can observe that Y τ(θ) is a d-dimensional vector whose j-th component is Y j τ (θ) = a τ 2 |Σ (θ)|−τ/2 � −trace � Σ (θ)−1 ∂Σ (θ) ∂θj � exp � −τ 2 (y − µ (θ))T Σ (θ)−1 (y − µ (θ)) � + b trace � Σ (θ)−1 ∂Σ (θ) ∂θj � + exp � −τ 2 (y − µ (θ))T Σ (θ)−1 (y − µ (θ)) � � −2 �∂µ (θ) ∂θj �T Σ (θ)−1 (y − µ (θ)) + (y − µ (θ))T � Σ (θ)−1 ∂Σ (θ) ∂θj Σ (θ)−1 � (y − µ (θ)) �� , j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=', d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Therefore the element (i, j) of the matrix Sτ(θ0) is given by E � Y i τ (θ)Y j τ (θ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' 23 We are going to get Y i τ (θ)Y j τ (θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' We have, Y i τ (θ)Y j τ (θ) = � a τ 2 |Σ (θ)|−τ/2 � −trace � Σ (θ)−1 ∂Σ (θ) ∂θi � exp � −τ 2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) � + b trace � Σ (θ)−1 ∂Σ (θ) ∂θi � + exp � −τ 2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) � � 2 �∂µ (θ) ∂θi �T Σ (θ)−1 (yi − µ (θ)) + (yi − µ (θ))T � Σ (θ)−1 ∂Σ (θ) ∂θj Σ (θ)−1 � (yi − µ (θ)) ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' � a τ 2 |Σ (θ)|−τ/2 � −trace � Σ (θ)−1 ∂Σ (θ) ∂θj � exp � −τ 2 (y − µ (θ))T Σ (θ)−1 (y − µ (θ)) � + b trace � Σ (θ)−1 ∂Σ (θ) ∂θj � + exp � −τ 2 (y − µ (θ))T Σ (θ)−1 (y − µ (θ)) � � 2 �∂µ (θ) ∂θj �T Σ (θ)−1 (y − µ (θ)) + (y − µ (θ))T � Σ (θ)−1 ∂Σ (θ) ∂θj Σ (θ)−1 � (y − µ (θ)) ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' 24 Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Y i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='τ (θ)Y j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='τ (θ) is given by ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='a2 �τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='|Σ (θ)|−τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='−2τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='−trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='−2τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='−trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='−2τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�∂µ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 (yi − µ (θ)) + (yi − µ (θ))T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(yi − µ (θ)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='−b trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='exp ' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='−2τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 ' 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+page_content='Σ (θ)−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(yi − µ (θ)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�∂µ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 (y − µ (θ)) + (y − µ (θ))T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(y − µ (θ)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' 25 We can write Y i τ (θ)Y j τ (θ) by Y i τ (θ)Y j τ (θ) = a2 �τ 2 �2 |Σ (θ)|−τ {C1 + C2 + C3 + C4 + C5 + C6 + C7 + C8 + C9} and E [Yi(θ)Yj(θ)] = a2 �τ 2 �2 |Σ (θ)|−τ E [C1 + C2 + C3 + C4 + C5 + C6 + C7 + C8 + C9] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' (36) Now we are going to calculate the different expectations appearing in (36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' We have, C1 = trace � Σ (θ)−1 ∂Σ (θ) ∂θi � trace � Σ (θ)−1 ∂Σ (θ) ∂θj � exp � −2τ 2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='E [C1] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(2π)m/2 |Σ (θ)|1/2 exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 (y − µ (θ))T Σ (θ)−1 (y − µ (θ)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 (y − µ (θ))T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(y − µ (θ)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='dy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2τ + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(2π)m/2 ��� Σ(θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2τ+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='1/2 exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 (y − µ (θ))T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2τ + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(y − µ (θ)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='dy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2τ + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='The expression of C2 is given by ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='C2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='−trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='−2τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='26 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='E [C2] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(1 + τ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 +1 trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(2π)m/2 |Σ (θ)|1/2 exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 (y − µ (θ))T Σ (θ)−1 (y − µ (θ)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 (y − µ (θ))T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(y − µ (θ)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='dy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(1 + τ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 +1 trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(1 + τ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(2π)m/2 ��� Σ(θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='τ+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='1/2 exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 (y − µ (θ))T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='τ + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(y − µ (θ)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='dy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(1 + τ)m+1 trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' The expression of C3 is given by C3 = −trace � Σ (θ)−1 ∂Σ (θ) ∂θi � exp � −2τ 2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) � � 2 �∂µ (θ) ∂θi �T Σ (θ)−1 (yi − µ (θ)) + (yi − µ (θ))T � Σ (θ)−1 ∂Σ (θ) ∂θj Σ (θ)−1 � (yi − µ (θ)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Then E [C3] = −trace � Σ (θ)−1 ∂Σ (θ) ∂θi � 1 (1 + 2τ)m/2 � 1 (2π)m/2 ��� Σ(θ) 2τ+1 ��� 1/2 exp � −1 2 (y − µ (θ))T Σ (θ)−1 (y − µ (θ)) � (y − µ (θ))T � Σ (θ)−1 ∂Σ (θ) ∂θj Σ (θ)−1 � (y − µ (θ)) dy = −trace � Σ (θ)−1 ∂Σ (θ) ∂θi � trace � Σ (θ)−1 ∂Σ (θ) ∂θj � 1 (1 + 2τ) m 2 +1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' In relation to C4 we have,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='C4 = −b trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='−τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 (y − µ (θ))T Σ (θ)−1 (y − µ (θ)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='27 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='E [C4] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(1 + τ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 +1 trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(2π)m/2 |Σ (θ)|1/2 exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 (y − µ (θ))T Σ (θ)−1 (y − µ (θ)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='−τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 (y − µ (θ))T (Σ (θ))−1 (y − µ (θ)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='dy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(1 + τ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 +1 trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(1 + τ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(2π)m/2 ��� Σ(θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='τ+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='1/2 exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 (y − µ (θ))T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='τ + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(y − µ (θ)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='dy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(1 + τ)m+1 trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' In relation with C5 we have, C5 = b2 trace � Σ (θ)−1 ∂Σ (θ) ∂θi � trace � Σ (θ)−1 ∂Σ (θ) ∂θj � and E [C5] = τ 2 (1 + τ)m+2 trace � Σ (θ)−1 ∂Σ (θ) ∂θi � trace � Σ (θ)−1 ∂Σ (θ) ∂θj � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' The expression of C6 is given by, C6 = b trace � Σ (θ)−1 ∂Σ (θ) ∂θi � exp � −τ 2 (y − µ (θ))T Σ (θ)−1 (y − µ (θ)) � � 2 �∂µ (θ) ∂θj �T Σ (θ)−1 (y − µ (θ)) + (y − µ (θ))T � Σ (θ)−1 ∂Σ (θ) ∂θj Σ (θ)−1 � (y − µ (θ)) � and E [C6] = τ (1 + τ) m 2 +1 trace � Σ (θ)−1 ∂Σ (θ) ∂θi � trace � Σ (θ)−1 ∂Σ (θ) ∂θj � 1 (1 + τ) m 2 +1 = τ (1 + τ)m+2 trace � Σ (θ)−1 ∂Σ (θ) ∂θi � trace � Σ (θ)−1 ∂Σ (θ) ∂θj � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' The expression of C7 is C7 = −trace � Σ (θ)−1 ∂Σ (θ) ∂θi � exp � −2τ 2 (y − µ (θ))T Σ (θ)−1 (y − µ (θ)) � � 2 �∂µ (θ) ∂θj �T Σ (θ)−1 (y − µ (θ)) + (y − µ (θ))T � Σ (θ)−1 ∂Σ (θ) ∂θj Σ (θ)−1 � (y − µ (θ)) � 28 and E [C7] = − τ (1 + 2τ) m 2 +1 trace � Σ (θ)−1 ∂Σ (θ) ∂θi � trace � Σ (θ)−1 ∂Σ (θ) ∂θj � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' In relation to C7, C8 = b trace � Σ (θ)−1 ∂Σ (θ) ∂θi � exp � −τ 2 (y − µ (θ))T Σ (θ)−1 (y − µ (θ)) � � 2 �∂µ (θ) ∂θj �T Σ (θ)−1 (y − µ (θ)) + (y − µ (θ))T � Σ (θ)−1 ∂Σ (θ) ∂θj Σ (θ)−1 � (y − µ (θ)) � and E [C8] = τ (1 + τ) m 2 +1 trace � Σ (θ)−1 ∂Σ (θ) ∂θi � trace � Σ (θ)−1 ∂Σ (θ) ∂θj � 1 (1 + τ) m 2 +1 = τ (1 + τ)m+2 trace � Σ (θ)−1 ∂Σ (θ) ∂θi � trace � Σ (θ)−1 ∂Σ (θ) ∂θj � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Finally, C9 = exp � −2τ 2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) � � 2 �∂µ (θ) ∂θi �T Σ (θ)−1 (yi − µ (θ)) + (yi − µ (θ))T � Σ (θ)−1 ∂Σ (θ) ∂θi Σ (θ)−1 � (yi − µ (θ)) � � 2 �∂µ (θ) ∂θj �T Σ (θ)−1 (yi − µ (θ)) + (yi − µ (θ))T � Σ (θ)−1 ∂Σ (θ) ∂θj Σ (θ)−1 � (yi − µ (θ)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='C9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='−2τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='4 (yi − µ (θ))T Σ (θ)−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�∂µ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂µ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 (yi − µ (θ)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='+2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�∂µ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 (yi − µ (θ)) (y − µ (θ))T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(yi − µ (θ)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='+2 (yi − µ (θ))T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(yi − µ (θ)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�∂µ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 (yi − µ (θ)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='+ (yi − µ (θ))T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(yi − µ (θ)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='A1 + A2 + A3 + A4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='E [C9] = E [A1] + E [A4] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='29 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='because E [A2] = E [A3] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' We have,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='E [A1] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='exp −2τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 (Y − µ (θ))T Σ (θ)−1 (Y − µ (θ)) 4 (Y − µ (θ))T Σ (θ)−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�∂µ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�T ∂µ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 (Y − µ (θ)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(2π)m/2 |Σ (θ)|1/2 exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 (y − µ (θ))T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2τ + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(y − µ (θ)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(y − µ (θ))T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�∂µ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�T ∂µ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 (y − µ (θ)) dy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(2τ + 1)m/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(2π)m/2 ��� Σ(θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2τ+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='1/2 exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 (y − µ (θ))T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2τ + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(y − µ (θ)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(y − µ (θ))T Σ (θ)−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�∂µ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�T ∂µ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 (y − µ (θ)) dy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(2τ + 1)m/2 trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�∂µ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�T ∂µ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2τ + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(2τ + 1)m/2+1 trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�∂µ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�T ∂µ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' 30 Now we are going to get E [A4] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='E [A4] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='−2τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 (Y − µ (θ))T Σ (θ)−1 (Y − µ (θ)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(Y − µ (θ))T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(Y − µ (θ)) (Y − µ (θ))T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(Y − µ (θ)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(2π)m/2 |Σ (θ)|1/2 exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 (y − µ (θ))T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2τ + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(y − µ (θ)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(y − µ (θ))T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(y − µ (θ)) (y − µ (θ))T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(y − µ (θ)) dy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(2τ + 1)m/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2τ + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 + Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2τ + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(2τ + 1)m/2 trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2τ + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2τ + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(2τ + 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 +2 trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(2τ + 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 +2 trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�T � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(2τ + 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 +2 trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(2τ + 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 +2 trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(2τ + 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 +2 trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' 31 Based on the previous results we have,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Y i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='τ (θ)Y j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='τ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='a2 �τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='|Σ (θ)|−τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(2τ + 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(τ + 1)m+1 trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(2τ + 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 +1 trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(τ + 1)m+1 trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='τ 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(τ + 1)m+2 trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(τ + 1)m+2 trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(2τ + 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 +1 trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(τ + 1)m+2 trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(2τ + 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 +1 trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�∂µ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�T ∂µ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(2τ + 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 +2 trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(2τ + 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 +2 trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' 32 The previous expression can be written as,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Y i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='τ (θ)Y j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='τ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='a2 �τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(2τ + 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(2τ + 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 +1 − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(2τ + 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 +1 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(2τ + 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 +2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='+trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(τ + 1)m+1 − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(τ + 1)m+1 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='τ 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(τ + 1)m+2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(τ + 1)m+2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(τ + 1)m+2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(2τ + 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 +1 trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�∂µ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�T ∂µ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(2τ + 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 +2 trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Therefore, E � Y i τ (θ)Y j τ (θ) � = � τ + 1 τ (2π)mτ/2 �2 �τ 2 �2 |Σ (θ)|−τ � 4τ 2 (2τ + 1) m 2 +2 trace � Σ (θ)−1 ∂Σ (θ) ∂θi � trace � Σ (θ)−1 ∂Σ (θ) ∂θj � − τ 2 (1 + τ)m+2 trace � Σ (θ)−1 ∂Σ (θ) ∂θi � trace � Σ (θ)−1 ∂Σ (θ) ∂θj � + 4 4 (2τ + 1) m 2 +1 trace � Σ (θ)−1 �∂µ (θ) ∂θi �T ∂µ (θ) ∂θj � + 2 (2τ + 1) m 2 +2 trace � Σ (θ)−1 ∂Σ (θ) ∂θi Σ (θ)−1 ∂Σ (θ) ∂θj �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' 33 Finally, E � Y i τ (θ)Y j τ (θ) � = (τ + 1)2 � � � � 1 (2π)mτ/2 |Σ (θ)|1/2 �2τ 1 (2τ + 1) m 2 +2 � ∆i 2τ∆j 2τ + (2τ + 1) trace � Σ (θ)−1 �∂µ (θ) ∂θi �T ∂µ (θ) ∂θj � +1 2trace � Σ (θ)−1 ∂Σ (θ) ∂θi Σ (θ)−1 ∂Σ (θ) ∂θj �� − � 1 (2π)mτ/2 |Σ (θ)|1/2 �2τ 1 (1 + τ)m+2 ∆i τ∆j τ � � � = (τ + 1)2 Kij τ (θ) where Kij τ (θ) was defined in (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Then √n ∂ ∂θ Hn(θ) L −→ n−→∞ N � 0, (τ + 1)2 Kτ (θ) � and √n � 1 τ + 1 ∂ ∂θ Hn(θ) � L −→ n−→∞ N (0, Kτ (θ)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' 34 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 Appendix B (Proof of Proposition 3) We have,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Hτ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='n(θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='−aτ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 |Σ (θ)|−τ/2 trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='n� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='i=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='−τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='− b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='+a |Σ (θ)|−τ/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='n� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='i=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='−τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='−τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�∂µ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 (yi − µ (θ)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='− (yi − µ (θ))T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(yi − µ (θ)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='−aτ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 |Σ (θ)|−τ/2 trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='n� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='i=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='−τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='− b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='+aτ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 |Σ (θ)|−τ/2 τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='n� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='i=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='−τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�∂µ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 (yi − µ (θ)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(yi − µ (θ))T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(yi − µ (θ)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Therefore, ∂2 ∂θi∂θj Hτ n(θ) = ∂ ∂θj Lτ 1(θ) + ∂ ∂θj Lτ 2(θ) being, Lτ 1(θ) = −aτ 2 |Σ (θ)|−τ/2 trace � Σ (θ)−1 ∂Σ (θ) ∂θi � � 1 n n� i=1 exp � −τ 2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) � − b � and Lτ 2(θ) = aτ 2 |Σ (θ)|−τ/2 τ 2 � 1 n n� i=1 exp � −τ 2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) � � 2 �∂µ (θ) ∂θi �T Σ (θ)−1 (yi − µ (θ)) (yi − µ (θ))T � Σ (θ)−1 ∂Σ (θ) ∂θi Σ (θ)−1 � (yi − µ (θ)) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' 35 We are going to get ∂ ∂θj Lτ 1(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Lτ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='1(θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='−aτ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='−τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='|Σ (θ)|−τ/2 trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='n� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='i=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='−τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='− b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='−aτ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 |Σ (θ)|−τ/2 trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='−Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='+ Σ (θ)−1 ∂2Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='n� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='i=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='−τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�∂µ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 (yi − µ (θ)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='− (yi − µ (θ))T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(yi − µ (θ)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Now we are going to see some result that will be important in order to get the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='convergence in probability of D1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' D2 and D3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Remark 13 We have, l = 1 n n� i=1 exp � −τ 2 (Y i − µ (θ))T Σ (θ)−1 (Y i − µ (θ)) � P −→ n−→∞ 1 (1 + τ)m/2 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' It is clear that l P −→ n−→∞ EN (µ(θ),Σ(θ)) � exp � −1 2 (Y − µ (θ))T �Σ (θ) τ � (Y − µ (θ)) �� = � exp � −1 2 (Y − µ (θ))T �Σ (θ) τ � (Y − µ (θ)) � fN (µ(θ),Σ(θ)) (y) dy = 1 (1 + τ)m/2 � 1 (2π)m/2 1 ��� Σ(θ) τ+1 ��� 1/2 exp � −1 2 (y − µ (θ))T �Σ (θ) τ � (y − µ (θ)) � dy = 1 (1 + τ)m/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Remark 14 We have, m = 1 n n� i=1 exp � −τ 2 (Y i − µ (θ))T Σ (θ)−1 (Y i − µ (θ)) � −b P −→ n−→∞ 1 (1 + τ) m 2 +1 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' By result the previous remark m P −→ n−→∞ 1 (1 + τ)m/2 − τ (1 + τ) m 2 +1 = 1 (1 + τ) m 2 +1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Remark 15 We denote, n = 1 n n� i=1 exp � −τ 2 (Y i − µ (θ))T Σ (θ)−1 (Y i − µ (θ)) � (Y i − µ (θ))T A (Y i − µ (θ)) 37 and we have, n P −→ n−→∞ trace (AΣ (θ)) (1 + τ) m 2 +1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' It is clear that, n P −→ n−→∞ EN (µ(θ),Σ(θ)) � exp � −1 2 (Y − µ (θ))T �Σ (θ) τ �−1 (Y − µ (θ)) � (Y − µ (θ))T A (Y − µ (θ)) � = 1 (1 + τ)m/2 � 1 (2π)m/2 1 ��� Σ(θ) τ+1 ��� 1/2 exp � −1 2 (y − µ (θ))T �Σ (θ) τ + 1 �−1 (y − µ (θ)) � (y − µ (θ))T A (y − µ (θ)) dy = 1 (1 + τ)m/2 EN(µ(θ), Σ(θ) 1+τ ) � (Y − µ (θ))T A (Y − µ (θ)) � = 1 (1 + τ) m 2 +1 trace (AΣ (θ)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Based on the previous results we have in relation to D1, D1 P −→ n−→∞ τ 4 |Σ (θ)|− τ 2 (2π)mτ/2 (1 + τ)m/2 trace � Σ (θ)−1 ∂Σ (θ) ∂θi � trace � Σ (θ)−1 ∂Σ (θ) ∂θj � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' With respect to D2, D2 P −→ n−→∞ 1 (2π)m/2 |Σ (θ)|− τ 2 1 2 1 (1 + τ)m/2 trace � Σ (θ)−1 ∂Σ (θ) ∂θj Σ (θ)−1 ∂Σ (θ) ∂θi � −|Σ (θ)|− τ 2 (2π)m/2 1 2trace � Σ (θ)−1 ∂2Σ (θ) ∂θj∂θi � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' In a similar way we get for D3 that, D3 P −→ n−→∞ −τ 4 |Σ (θ)|− τ 2 (2π)mτ/2 trace � Σ (θ)−1 ∂Σ (θ) ∂θi � 1 (1 + τ)m/2 trace � Σ (θ)−1 ∂Σ (θ) ∂θj � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Therefore we have ∂ ∂θj Lτ 1(θ) P −→ n−→∞ 1 2 |Σ (θ)|− τ 2 (2π)mτ/2 1 (1 + τ)m/2 trace � Σ (θ)−1 ∂Σ (θ) ∂θj Σ (θ)−1 ∂Σ (θ) ∂θi � −1 2 |Σ (θ)|− τ 2 (2π)mτ/2 1 (1 + τ)m/2 � Σ (θ)−1 ∂2Σ (θ) ∂θj∂θi � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' 38 Now we have,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Lτ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2(θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='−aτ 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='4 |Σ (θ)|− τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='n� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='i=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='−τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�∂µ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 (yi − µ (θ)) + (yi − µ (θ))T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(yi − µ (θ)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='+aτ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 |Σ (θ)|− τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='n� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='i=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='−τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='−τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� � ∂ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�∂µ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 (yi − µ (θ)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(yi − µ (θ))T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(yi − µ (θ)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='+aτ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 |Σ (θ)|− τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='n� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='i=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='−τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�∂µ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 (yi − µ (θ)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='+ (yi − µ (θ))T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(yi − µ (θ)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='C1 + C2 + C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Being, C1 = −aτ 2 4 |Σ (θ)|− τ 2 trace � Σ (θ)−1 ∂Σ (θ) ∂θj � � 1 n n� i=1 exp � −τ 2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) � � 2 �∂µ (θ) ∂θi �T Σ (θ)−1 (yi − µ (θ)) (yi − µ (θ))T � Σ (θ)−1 ∂Σ (θ) ∂θj Σ (θ)−1 � (yi − µ (θ)) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' It is clear that, C1 P −→ n−→∞ −τ 4 1 (1 + τ)m/2 |Σ (θ)|− τ 2 (2π)mτ/2 trace � Σ (θ)−1 ∂Σ (θ) ∂θi � trace � Σ (θ)−1 ∂Σ (θ) ∂θj � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' It is immediate to see that C2 = −aτ 2 4 |Σ (θ)|− τ 2 1 n n� i=1 exp � −τ 2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) � (S1 + S2 + S3 + S4) = L∗ 1 + L∗ 2 + L∗ 3 + L∗ 4 39 where L∗ 1 = −aτ 2 4 |Σ (θ)|− τ 2 1 n n� i=1 exp � −τ 2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) � � −4 (yi − µ (θ))T Σ (θ)−1 ∂µ (θ) ∂θj �∂µ (θ) ∂θi �T Σ (θ)−1 (yi − µ (θ)) � and L∗ 1 P −→ n−→∞ |Σ (θ)|− τ 2 (2π)mτ/2 τ (1 + τ)m/2 trace � Σ (θ)−1 ∂µ (θ) ∂θj �∂µ (θ) ∂θi �T � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' It is clear that L∗ 2 P −→ n−→∞ 0 and L∗ 3 P −→ n−→∞ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' On the other hand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='L∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='P ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='−→ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='n−→∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='τ + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(2π)mτ/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='4 |Σ (θ)|− τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(1 + τ)m/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 + Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='1 + τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='+ trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='1 + τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='1 + τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='= 2τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='|Σ (θ)|− τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(2π)mτ/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(1 + τ)m/2+1 trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 Σ (θ) ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='+τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='|Σ (θ)|− τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(2π)mτ/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(1 + τ)m/2+1 trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Therefore, C2 = L∗ 1 + L∗ 2 + L∗ 3 + L∗ 4 P −→ n−→∞ R being R = τ |Σ (θ)|− τ 2 (2π)mτ/2 (1 + τ)m/2 trace � Σ (θ)−1 ∂µ (θ) ∂θj �∂µ (θ) ∂θj �T � +τ 2 |Σ (θ)|− τ 2 (2π)mτ/2 (1 + τ)m/2+1 trace � Σ (θ)−1 ∂Σ (θ) ∂θj Σ (θ)−1 ∂Σ (θ) ∂θi � +τ 4 |Σ (θ)|− τ 2 (2π)mτ/2 (1 + τ)m/2+1 trace � Σ (θ)−1 ∂Σ (θ) ∂θj � trace � Σ (θ)−1 ∂Σ (θ) ∂θi � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' 40 Finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='−C3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='aτ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 |Σ (θ)|− τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='n� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='i=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='−τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 ∂ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�∂µ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 (yi − µ (θ)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='+2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�∂µ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�T ∂Σ (θ)−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(yi − µ (θ)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�∂µ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�∂µ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�∂µ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�T � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(yi − µ (θ)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='+ (yi − µ (θ))T {− Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='+Σ (θ)−1 ∂2Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 − Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(yi − µ (θ)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='A∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='1 + A∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 + A∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='3 + A∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='4 + A∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' It is clear that A∗ 1 P −→ n−→∞ 0 A∗ 2 P −→ n−→∞ 0 and A∗ 4 P −→ n−→∞ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' On the other hand, A∗ 3 = −aτ 2 |Σ (θ)|− τ 2 1 n n� i=1 exp � −τ 2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) � � 2 �∂µ (θ) ∂θi �T Σ (θ)−1 ∂µ (θ) ∂θi � and A∗ 3 P −→ n−→∞ −(τ + 1) |Σ (θ)|− τ 2 (2π)mτ/2 (1 + τ)m/2 �∂µ (θ) ∂θi �T Σ (θ)−1 ∂µ (θ) ∂θj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' In relation to A∗ 5 we have,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' A∗ 5 = aτ 2 |Σ (θ)|− τ 2 1 n n� i=1 exp � −τ 2 (yi − µ (θ))T Σ (θ)−1 (yi − µ (θ)) � � (yi − µ (θ))T � Σ (θ)−1 ∂Σ (θ) ∂θj Σ (θ)−1 ∂Σ (θ) ∂θi Σ (θ)−1 +Σ (θ)−1 ∂2Σ (θ) ∂θi∂θj Σ (θ)−1 − Σ (θ)−1 ∂Σ (θ) ∂θj Σ (θ)−1 ∂Σ (θ) ∂θj Σ (θ)−1 � (yi − µ (θ)) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' 41 and A∗ 5 P −→ n−→∞ − 1 (2π)mτ/2 1 2 |Σ (θ)|− τ 2 1 (1 + τ)m/2 trace � Σ (θ)−1 ∂Σ (θ) ∂θj Σ (θ)−1 ∂Σ (θ) ∂θi � + 1 (2π)mτ/2 1 2 |Σ (θ)|− τ 2 1 (1 + τ)m/2 trace � Σ (θ)−1 ∂2Σ (θ) ∂θi∂θ � − 1 (2π)mτ/2 1 2 |Σ (θ)|− τ 2 1 (1 + τ)m/2 trace � Σ (θ)−1 ∂Σ (θ) ∂θj Σ (θ)−1 ∂Σ (θ) ∂θi � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' Therefore, C3 P −→ n−→∞ − (τ + 1) |Σ (θ)|− τ 2 (2π)mτ/2 1 (1 + τ)m/2 �∂µ (θ) ∂θi �T Σ (θ)−1 ∂µ (θ) ∂θj −|Σ (θ)|− τ 2 (2π)mτ/2 1 (1 + τ)m/2 1 2trace � Σ (θ)−1 ∂Σ (θ) ∂θj Σ (θ)−1 ∂Σ (θ) ∂θi � +|Σ (θ)|− τ 2 (2π)mτ/2 1 (1 + τ)m/2 1 2trace � Σ (θ)−1 ∂2Σ (θ) ∂θi∂θj � −|Σ (θ)|− τ 2 (2π)mτ/2 1 2 1 (1 + τ)m/2 trace � Σ (θ)−1 ∂Σ (θ) ∂θj Σ (θ)−1 ∂Σ (θ) ∂θi � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' We are going to joint all the previous expressions in order to get ∂ ∂θj Lτ 2(θ), ∂ ∂θj Lτ 2(θ) = C1 + C2 + C3 = C1 + L∗ 1 + L∗ 2 + L∗ 3 + L∗ 4 + C3 = C1 + L∗ 1 + L∗ 2 + L∗ 3 + L∗ 4 + A∗ 1 + A∗ 2 + A∗ 3 + A∗ 4 + A∗ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' 42 Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Lτ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2(θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='P ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='−→ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='n−→∞ −τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(1 + τ)m/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='|Σ (θ)|− τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(2π)mτ/2 trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='τ |Σ (θ)|− τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(2π)mτ/2 (1 + τ)m/2 trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='+τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='|Σ (θ)|− τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(2π)mτ/2 (1 + τ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 +1 trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='+τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='|Σ (θ)|− τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(2π)mτ/2 (1 + τ)m/2+1 trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='− (τ + 1) |Σ (θ)|− τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(2π)mτ/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(1 + τ)m/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�∂µ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='�T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂µ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='−|Σ (θ)|− τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(2π)mτ/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(1 + τ)m/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='+|Σ (θ)|− τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(2π)mτ/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(1 + τ)m/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂2Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='−|Σ (θ)|− τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(2π)mτ/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(1 + τ)m/2 trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Based on the previous results we have ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Hτ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='n(θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Lτ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='1(θ) + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Lτ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2(θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='D1 + D2 + D3 + C1 + C2 + C3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='D1 + D2 + D3 + C1 + L∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='1 + L∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 + L∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='3 + L∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='+A∗ ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='|Σ (θ)|− τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(2π)mτ/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(1 + τ)m/2 trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='|Σ (θ)|− τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(2π)mτ/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(1 + τ)m/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂2Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='−τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(1 + τ)m/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='|Σ (θ)|− τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(2π)mτ/2 trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='τ |Σ (θ)|− τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(2π)mτ/2 (1 + τ)m/2 trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='+τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='|Σ (θ)|− τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(2π)mτ/2 (1 + τ)m/2+1 trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='+τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='|Σ (θ)|− τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='(2π)mτ/2 (1 + τ)m/2+1 trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='trace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='Σ (θ)−1 ∂Σ (θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='∂θi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='− (τ + 1) |Σ (θ)|− τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFRT4oBgHgl3EQfPze-/content/2301.13519v1.pdf'} +page_content='2 ' 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H. Pang,7, 8 Shreya Anand,9 +Michael Coughlin,10 Ingo Tews,11 Jennifer Barnes,12 Meili Pilloix,2, 13 Weizmann Kiendrebeogo,2 and +Tim Dietrich1, 3 +1Institute of Physics and Astronomy, Theoretical Astrophysics, University Potsdam, Haus 28, Karl-Liebknecht-Str. 24/25, 14476, +Potsdam, Germany +2Artemis, Observatoire de la Cˆote d’Azur, Universit´e Cˆote d’Azur, Boulevard de l’Observatoire, 06304 Nice, France +3Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Am M¨uhlenberg 1, Potsdam 14476, Germany +4Department of Physics and Earth Science, University of Ferrara, via Saragat 1, I-44122 Ferrara, Italy +5INFN, Sezione di Ferrara, via Saragat 1, I-44122 Ferrara, Italy +6INAF, Osservatorio Astronomico d’Abruzzo, via Mentore Maggini snc, 64100 Teramo, Italy +7Nikhef, Science Park 105, 1098 XG Amsterdam, The Netherlands +8Institute for Gravitational and Subatomic Physics (GRASP), Utrecht University, Princetonplein 1, 3584 CC Utrecht, The Netherlands +9Cahill Center for Astrophysics, California Institute of Technology, Pasadena CA 91125, USA +10School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455, USA +11Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA +12Kavli Institute for Theoretical Physics, Kohn Hall, University of California, Santa Barbara, CA 93106, USA +13Laboratoire de Physique et de Chimie de l’Environnement, Universit´e Joseph KI-ZERBO, Ouagadougou, Burkina Faso +(Dated: January 6, 2023) +ABSTRACT +Although being among the closest gamma-ray bursts (GRBs), GRB 211211A poses challenges for +its classification with partially inconclusive electromagnetic signatures. In this paper, we investigate +four different astrophysical scenarios as possible progenitors for GRB 211211A: a binary neutron-star +merger, a black-hole–neutron-star merger, a core-collapse supernova, and an r-process enriched core +collapse of a rapidly rotating massive star (a collapsar). We perform a large set of Bayesian multi- +wavelength analyses based on different models and priors to investigate which astrophysical scenarios +and processes might have been related to GRB 211211A. Our analysis supports previous studies in +which the presence of an additional component, likely related to r-process nucleosynthesis processes, +is required to explain the observed light curves of GRB 211211A, as it can not solely be explained +as a GRB afterglow. Fixing the distance to about 350 Mpc, i.e., the distance of the possible host +galaxy SDSS J140910.47+275320.8, we find a statistical preference for the binary neutron-star merger +scenario and estimate the component masses to be 1.55+0.54 +−0.42M⊙ and 1.34+0.25 +−0.40M⊙. +1. INTRODUCTION +The joint detection of gravitational waves (GWs) +and electromagnetic (EM) signatures originating from +the merger of binary neutron stars (BNSs) on August +17th 2017 (Abbott et al. 2017; Abbott et al. 2017) has +been a breakthrough in multi-messenger astronomy. In +addition to the GW signal GW170817, an associated +kilonova, AT2017gfo, and a gamma-ray burst (GRB), +GRB 170817A, were observed (Abbott et al. 2017). +This multi-messenger detection allowed for an inde- +pendent way of measuring the expansion rate of the +Universe (Abbott et al. 2017), placed new constraints on +the properties of supranuclear-dense matter (Bauswein +et al. 2017; Ruiz et al. 2018; Radice et al. 2018; Most +et al. 2018; Coughlin et al. 2019; Capano et al. 2020; +Dietrich et al. 2020; Huth et al. 2022), and proved that +at least some short GRBs are connected to compact +binary mergers (Abbott et al. 2017). However, it was +also reported that short GRBs could originate from +collapsars (Ahumada et al. 2021), indicating that the +classification of astrophysical scenarios associated with +GRBs is more complex (Zhang et al. (2021); Rossi et al. +(2022)). +Additional signatures associated with GRBs +and their afterglows, such as kilonovae, significantly +help to identify the origin of the progenitors. +The +kilonova AT2017gfo was certainly an exemplary case +for such an EM signal, and spectral features connected +to the creation of new elements (Watson et al. 2019; +Domoto et al. 2022) in the outflowing material have +possibly been observed. +In addition to AT2017gfo, +arXiv:2301.02049v1 [astro-ph.HE] 5 Jan 2023 + +2 +there is a large number of kilonova candidates that +could +be +connected +to +other +GRB +observations, +e.g., +GRB +050709, +GRB +050724A, +GRB +060614, +GRB +061201, +GRB +080905A, +GRB +070724A, +GRB +130603B, +GRB +140903A, +GRB +150101B, +GRB 150424A, GRB 160821B, e.g., +Tanvir et al. +(2013); Berger et al. (2013); Yang et al. (2015); Zhang +et al. (2007); Jin et al. (2015); Yang et al. (2015); Fox +et al. (2005); Hjorth et al. (2005); Covino et al. (2006); +Stratta et al. (2007); Berger et al. (2005); Malesani et al. +(2007); Fong et al. (2016); Troja et al. (2018); Nicuesa +Guelbenzu et al. (2012); Rowlinson et al. (2010); Berger +et al. (2009); Kocevski et al. (2010); Kasliwal et al. +(2017); Jin et al. (2018); Tanvir et al. (2015); Jin et al. +(2018); cf. e.g. Ascenzi et al. (2019) for a review about +some of these kilonova candidates. +The most recent +example that has to be added to the list is the kilonova +candidate connected to GRB 211211A and its optical +and near-infrared counterpart, e.g., Rastinejad et al. +(2022), and Troja et al. (2022), and Mei et al. (2022). +GRB 211211A was discovered on the 11th Decem- +ber 2021 at 13:09:59 (UTC) by the Burst Alert Tele- +scope (BAT) of the Swift Observatory (trigger 1088940, +SNRimg = 11.5, D’Ai et al. 2021). The Fermi Gamma- +ray Burst Monitor detected GRB 211211A indepen- +dently at the exact same trigger time (trigger 211211549, +SNRct = 22.2, Fermi GBM Team 2021). Moreover, the +high-energy space instrument onboard Insight-HXMT +detected GRB 211211A (trigger HEB211211548) during +its routine search (Zhang et al. 2021). The GRB is char- +acterized by a complex emission phase lasting approx- +imately 10 s, and a longer, weaker extended emission +for about 130 s in [15-350] keV (Stamatikos et al. 2021). +Given this duration, GRB 211211A would be classified +as a long GRB typically arising from the core-collapse of +massive stars (e.g., Stanek et al. 2003; Levan et al. 2016) +and not from compact binary mergers. Hence, for a sce- +nario such as GRB 211211A, one would not necessarily +expect to observe an associated kilonova. +About 70 s after the emergence of the prompt emis- +sion, Swift’s X-ray Telescope (XRT) started observing +the source. The X-ray observations showed bright emis- +sion (a flux of 3×10−8ergs−1cm−2 in [0.3–10] keV) with +an exponential decay lasting for hours after the trigger +(Osborne et al. 2021). +The Ultraviolet/Optical Tele- +scope started its observations 92 s later and detected +an optical counterpart within the X-ray localization er- +ror box. Given its close proximity to the galaxy SDSS +J140910.47+275320, an intensive follow-up campaign in- +cluding MITSuME, NEXT, the Nordic Optical Tele- +scope, and the Calar Alto Observatory (Ito et al. 2021; +Jiang et al. 2021; Malesani et al. 2021; de Ugarte Postigo +et al. 2021) was scheduled and the source was observed +across multiple wavelengths. Based on these follow-up +observations and the following analysis, it seems plausi- +ble that SDSS J140910.47+275320.8 was the host galaxy +of GRB 211211A, at 98.6% confidence (Rastinejad et al. +2022). Details about the observation campaign are sum- +marized in Rastinejad et al. (2022). +Rastinejad et al. (2022), Troja et al. (2022), and other +groups explained these observations by invoking a kilo- +nova in association with GRB 211211A. This was sug- +gested for different reasons: (i) the profile of the prompt +emission showed an initially complex structure followed +by an extended softer emission, (ii) a predominant sig- +nature of a supernova was lacking for up to 17 days +post-discovery, (iii) the color evolution of the optical +counterpart had similar properties as AT2017gfo, and +(iv) the offset of the GRB location concerning the cen- +ter of the host galaxy was larger than for typical long +GRBs. +Numerous other groups addressed the origin of +GRB 211211A, e.g., Yang et al. (2022) suggested that +it has similar properties as GRB 060614, another event +associated with a kilonova candidate. +They conclude +that the significant excess in the near-infrared and opti- +cal afterglow at late observations points more towards a +neutron star-white dwarf merger which leaves behind a +rapidly spinning magnetar as a central engine provid- +ing additional heating to the ejecta. +Waxman et al. +(2022) showed that also thermal emission from dust +could explain the observed near-infrared (NIR) data. +Suvorov et al. (2022) mentioned a possible gamma-ray +precursor before the main emission which was caused +by the resonant shattering of one star’s crust prior to +the merger. +In contrast, Gao et al. (2022) concluded +the presence of a strong magnetic field from the pre- +cursor surrounding the central engine of the GRB. This +would result in the prolongation of the accretion pro- +cess and, thus, could explain the duration of the hard +spiky emission detected for GRB 211211A. Similarly, +Xiao et al. (2022) supposes that a magnetar partici- +pated in the merger and caused a quasi-periodic pre- +cursor. +Gompertz et al. (2022) analyzed the spectra +of the prompt emission of GRB 211211A by using syn- +chrotron spectrum models and concluded that the spec- +tral evolution can be explained by a transition from a +fast-cooling to a slow cooling regime, favoring a BNS +merger rather than a neutron-star–black-hole (NSBH) +scenario. Finally, Barnes & Metzger (2023) investigated +the possibility that collapsars could explain the origin of +GRB 211211A and found that the afterglow-subtracted +emission of GRB 211211A is in best agreement for col- +lapsar models with high kinetic energies. + +3 +Following the discussion in the literature, we will use +our nuclear physics and multi-messenger astrophysics +(NMMA) framework (Pang et al. 2022)1 to explore different +astrophysical scenarios for the origin of GRB 211211A. +We will consider the possibility of two merger scenarios, +a BNS merger and an NSBH merger, and in addition +two supernova scenarios, a core-collapse supernova, and +an r-process enriched collapsar. +For our model selec- +tion study, the NMMA framework allows us to simultane- +ously fit the observed data across the full electromag- +netic range with multiple models, e.g., we can simul- +taneously employ GRB afterglow and kilonova models +without the need of splitting the observational data in +chunks and processing them separately, as done in – to +our knowledge – previous studies of GRB 211211A. +2. OBSERVATIONAL DATA +In order to perform our model selection, we collect a +set of multi-wavelength data observed for GRB 211211A +(see Table 2). Concerning the GRB afterglow, we do +not use any data from the prompt emission phase of the +GRB in our analysis. This means that we use available +X-ray data from the Swift X-ray Telescope, in particular, +we use the 0.3 - 10 keV flux light curve observed at late +times (t = 104 s after BAT trigger time) and convert it +to 1 keV flux densities following Gehrels et al. (2008). +For our optical study, we followed Rastinejad et al. +(2022) and included the refined analysis of Swift-UVOT +observations. We contacted the authors of the obser- +vational teams responsible for the GCN reports, espe- +cially for those data which was not analyzed by Rastine- +jad et al. (2022). They provided us with offline results +that we used in this article. +For these data, we cor- +rected the measurements by taking into account the fore- +ground Galactic extinction AV = 0.048 mag (Schlafly & +Finkbeiner 2011). We excluded all photometric results +from observations performed with the Johnson-Cousins +UBVRI system as we do not compute simulated light +curves in these passbands in our Bayesian approach. +Moreover, we also excluded all photometric results from +images taken without filters. +Finally, +we +use +the +6 +GHz +radio +detection +of +GRB 211211A observed 6.27 days after the initial trigger +with a 5σ upper limit flux density of 16 µJy (Rastinejad +et al. 2022). With regard to available GeV data, as re- +ported in Zhang et al. (2022) and Mei et al. (2022), we +do not include this data since our employed GRB model +does not provide mechanisms to explain its origin. +We also re-analyzed data from the 2.3m telescope at +the Centro Astron´omico Hispano en Andaluc´ıa (CAHA), +1 https://github.com/nuclear-multimessenger-astronomy +equipped with the Calar Alto Faint Object Spectro- +graph (CAFOS) and find consistent results with respect +to Rastinejad et al. (2022). Moreover, we exclude the +detection measurement in the i band at 2.68 days post- +discovery from our analysis since we find an upper limit +of 22.6 mag at 5-σ with methods described in Aivazyan +et al. (2022). +3. METHODS +3.1. Bayesian Inference +Our analysis is based on the nuclear physics and multi- +messenger astronomy framework NMMA (Pang et al. 2022) +that allows us to perform joint Bayesian inference runs +of multi-messenger events containing GWs, kilonovae, +supernovae, and GRB afterglow signatures. For this ar- +ticle, we extended the code infrastructure to include the +description of r-process enriched collapsars following the +model of Barnes & Metzger (2022). +We use the EM data of GRB 211211A to investigate +which model or which combination of models describe +the observational data best. According to Bayes’ the- +orem, we compute posterior probability distributions, +p(⃗θ|d, M), for model source parameters ⃗θ under the hy- +pothesis or model M with data d as +p(⃗θ|d, M) = p(d|⃗θ, M)p(⃗θ|M) +p(d|M) +→ P(⃗θ) = L(⃗θ)π(⃗θ) +Z(d) +, +(1) +where P(⃗θ), L(⃗θ), π(⃗θ), and Z(d) are the posterior, +likelihood, prior, and evidence, respectively. +In order +to investigate the plausibility of competing models, we +evaluate the odds ratio O1 +2 for two models M1 and M2 +which is given by +O1 +2 = p(d|M1) +p(d|M2) +p(M1) +p(M2) ≡ B1 +2Π1 +2, +(2) +where B1 +2 and Π1 +2 are the Bayes factor and the prior +odds, respectively. +Under the assumption that the +different astrophysical scenarios considered here are +equally likely to explain GRB 211211A, we impose unity +prior odds, i.e., Π1 +2 = 1, for all comparisons of models +describing these scenarios. Therefore, we simply com- +pute the Bayes factor B1 +2. In our study, we report the +natural logarithm of the Bayes factor, +ln B1 +ref = ln +� p(d|M1) +p(d|Mref) +� +, +(3) +relative to our best fitting model as a reference (ref.), +which we will denote as ln Bref hereafter. +Following +Jeffreys (1961) and Kass & Raftery (1995), we interpret +ln B1 +ref as the evidence favoring our reference model as: + +4 +ln[B1 +ref] < −4.61 +decisive evidence, +−4.61 ≤ ln[B1 +ref] ≤ −2.30 +strong evidence, +−2.30 ≤ ln[B1 +ref] ≤ −1.10 +substantial evidence, +−1.10 ≤ ln[B1 +ref] ≤ 0 +no strong evidence. +However, we point out that these classifications should +only be considered as estimates and that the Bayes +factor is generally a continuous quantity. In addition to +the Bayes factor, we also provide information about the +ratio of the maximum likelihood, or the difference of +the maximum log-likelihood point estimates ln[L1 +2(ˆθ)] +supporting our analysis in Sec. 4.1. +We will denote +this as ln[Lref(ˆθ)] when we compare the maximum +log-likelihood against our reference model. +3.2. Employed models +As described in the introduction, +we investigate +four +different +scenarios +in +our +study +from +which +GRB 211211A could have emerged. In particular, we +consider two merger scenarios: a BNS merger and an +NSBH merger, and two supernova cases: a phenomeno- +logical long GRB supernova template and an r-process +enriched collapsar scenario. +BNS scenario: For this case, we use the kilonova +models of Dietrich et al. (2020) (hereafter ‘BNS-KN- +Bulla’) and of Kasen et al. (2017) (hereafter ‘BNS-KN- +Kasen’). BNS-KN-Bulla is based on the time-dependent +three-dimensional Monte Carlo radiation transfer code +possis (Bulla (2019), Bulla, Mattia (2022)), which com- +putes light curves, spectra, and luminosities for kilono- +vae depending on the viewing-angle θObs. The ejected +material is classified through the dynamical ejecta mass, +M dyn +ej +, and the disk-wind ejecta mass, M wind +ej +. +The +tidal dynamical ejecta component is assumed to be dis- +tributed within a half opening angle Φ. +In the same +way, BNS-KN-Kasen uses the multi-dimensional Monte +Carlo code sedona that solves the multi-wavelength ra- +diation transport equation in a relativistically expanding +medium (Kasen et al. (2006); Roth & Kasen (2015)). In +this paper, we use the one-dimensional model provided +by Kasen et al. (2017), which assumes spherical sym- +metry and uniform composition for our analysis. The +model, ‘BNS-KN-Kasen’, depends on the ejecta mass, +Mej, a characteristic expansion velocity, vej, and the +mass fraction of lanthanides, Xlan, which affects the +opacity. +NSBH scenario: For this case, we also use a possis +model grid of KN spectra tailored to NSBH mergers +which was used in the study of Anand et al. (2021) +(hereafter ‘NSBH-KN-Bulla’). This model depends on +the same model parameters as BNS-KN-Bulla but ex- +cludes the dependence on the half opening angle of the +dynamical ejecta, fixed to Φ = 30◦. +Supernova: In order to assess the possibility of a +typical core-collapse supernova (CCSN) associated with +a long GRB, we use the nugent-hyper model from +sncosmo (Levan et al. 2005) with the absolute magni- +tude, Smax, as the main free parameter. This model is +a template constructed from observations of the super- +nova SN1998bw associated with the long GRB 980425 +and is hereafter abbreviated as ‘SN98bw’. +r-process enriched Collapsar: +Rapidly rotating +massive star core collapses (Burbidge et al. 1957; Qian +& Woosley 1996) are another possible astrophysical site +for r-process nucleosynthesis. As massive stars undergo +a core collapse, material is disrupted and forms an accre- +tion disk which can become neutron-rich through weak +interactions (Beloborodov 2003) and can launch winds +which power emission of r-process-enriched core-collapse +SNe (rCCSNe). +We use the semi-analytic model for +rCCSNe of Barnes & Metzger (2022) (hereafter denoted +as ’SNCol’). The model depends on five free parame- +ters: the total ejecta mass, Mej, a characteristic ejecta +velocity, vej, the 56Ni mass, MNi, the r-process mate- +rial mass, Mrp, and the mixing coordinate, Ψmix. The +ejecta are assumed to be spherically symmetric, with r- +process elements of mass mrp concentrated in an inner +core whose total mass is Ψmixmej, with Ψmix ≤ 1. An +r-process-free envelope surrounds the core, and 56-Ni is +distributed uniformly throughout the core and the en- +velope. The velocity vej is defined such that the total +kinetic energy of the ejecta Ekin is equal to 1 +2Mejv2 +ej.2 +GRB afterglow: For modeling the GRB afterglow +light curves, we employ the semi-analytic model of van +Eerten et al. (2010) and Ryan et al. (2020), available in +the public afterglowpy library (denoted as ‘GRB-M’). +The model computes GRB afterglow emission and takes +the following free parameters as input: the isotropic ki- +netic energy, EK,iso, the viewing angle, θObs, the half- +opening angle of the jet core, θc, the outer truncation +angle of the jet, θw, the interstellar medium density, n, +the electron energy distribution index, p, and the frac- +tions of the shock energy that go into electrons, ϵe, and +magnetic fields, ϵB. The model allows for several an- +gular structures of the GRB jet. For our simulations, +we assume a Gaussian or a top-hat jet structure (here- +after, ‘Gauss’ and ‘top’)3. It is important to note that, +while we try to be agnostic concerning GRB 211211A’s +2 Barnes & Metzger (2023) also compared rCCSNe with obser- +vational data from GRB 211211A. However, not within a Bayesian +approach as employed here and with an updated version of their +model originally described in Barnes & Metzger (2022). +3 In addition, we tested a power law jet structure for which we +found consistent results. + +5 +origin, the GRB-M model that we employ has some lim- +itations. Specifically, it does not include the emission +from the reverse shock that might be important at early +times. Additionally, it does not include the wind-like +interstellar medium, which is expected in the case of a +collapsar. +In Fig. 1, we summarize our approach to analyze +GRB 211211A based on the data set described in Sec. 2. +We employ two different priors for the luminosity dis- +tance, i.e., a narrow Gaussian luminosity distance prior +centered around 350 Mpc as reported by Rastinejad +et al. (2022) and a uniform prior on the luminosity dis- +tance ranging between 0 and 1000 Mpc. +This allows +us to investigate the potential influence of the distance +on the GRB classification. Furthermore, we employ five +models or model combinations to describe the different +astrophysical scenarios. +For the choice of a Gaussian +luminosity distance prior, we report the prior settings +for all parameters of the employed models in Table 3. +Moreover, we use two different GRB jet types, totaling +in 20 Bayesian inference simulations. +4. MULTI-WAVELENGTH ANALYSES +In the following three subsections, 4.1-4.3, we discuss +our results for a narrow Gaussian prior on the luminosity +distance in order to compare with previous studies. In +subsection 4.4, we will investigate the influence of the +distance prior choice and employ a wide uniform prior +on the luminosity distance. +4.1. Model Comparison +As indicated in the introduction, one of the main dif- +ferences between previous studies and our work is that +most previous works fitted first the X-ray and radio +data with a GRB afterglow model, and then used the +afterglow-subtracted optical and near-infrared photom- +etry for fitting a kilonova model. In contrast, we per- +form a joint analysis of the GRB afterglow and a possible +additional contribution such as a kilonova signature or +emission from a rCCSN or CCSN. Moreover, in order to +consider systematic uncertainties arising from different +assumptions made in each model, we employ a 1 mag +uncertainty in our simulations. +In Table 1, we summarize our main findings for the +investigated astrophysical scenarios. We found that the +BNS-GRBKasen +top +model describes the observational data +best, and hence, we pick it as our reference model. Con- +sequently, the Bayes factors and likelihood ratios in Ta- +ble 1 are reported relative to this best-fit inference run. +With reference to Table 1, we show the maximum log- +likelihood light curve fits in Fig. 2 for each assessed sce- +nario, which we will refer to as ”best-fitting light curves” +hereafter. +Comparing only the two different BNS kilonova mod- +els, we find that differences in the Bayes factors are of or- +der unity. We interpret this as a measure of the system- +atic model uncertainty for different employed kilonova +models, given that both BNS-GRBBulla +Gauss/top and BNS- +GRBKasen +Gauss/top should describe the same physical system. +It is worth pointing out that statistical uncertainties, as +stated in the table, are noticeably smaller than model +differences, i.e., our results are dominated by systematic +uncertainties in the underlying light curve models. +Considering +the +differences +between +the +NSBH +and BNS scenarios, +we find strong evidence that +GRB 211211A was connected to a BNS rather than an +NSBH system. This is reflected both in Bayes factors as +well as maximum log-likelihood values as shown in Ta- +ble 1. Comparing the respective best fitting light curves +in Fig. 2, we see that NSBH-GRBBulla +top +fits the NIR-band +data worse compared to GRBKasen +top +. +With regard to the relative Bayes factors for the col- +lapsar scenario, we find that there is decisive evidence +that a BNS scenario is preferred over a collapsar origin +for GRB 211211A. However, it is important to note that +the collapsar model depends on more parameters. Be- +cause of this, Occam’s razor penalizes the model despite +its ability to describe the observational data; cf. Fig. 2. +This ability to describe and fit the observational data +can be estimated from the maximum likelihood ratio re- +sults as given in Tab. 1. +As indicated by Rastinejad et al. (2022), and con- +firmed by our study, we find that a Ni-powered SN event +or an SN-GRB scenario is noticeably less favored com- +pared to a BNS merger. This is depicted in Fig. 2 in +which SN98bw-GRBBulla +top +fails to fit late-time NIR data, +resulting in a larger, negative log-likelihood ratio. +Finally, our study confirms that the BNS-GRBKasen +top +scenario provides decisive evidence when compared with +GRBtop-M simulations, even though the latter sampled +over fewer parameters in respective parameter estima- +tion runs. +Considering the impact of the choice of a +Gaussian vs. top-hat jet structure on our Bayes factor +results, we find a slight preference for the top-hat jet +structure for all assessed scenarios, except for NSBH- +GRBBulla +Gauss. +4.2. Presence of an additional component +Given the overall narrative that GRB 211211A was +a GRB connected to a kilonova, we study the ability +of the GRB-M with top-hat jet structure to describe +the observational data and compare this with two BNS +merger scenarios. For this purpose, we show the best- +fitting light curves for BNS-GRBBulla +top , BNS-GRBKasen +top +, +and GRBtop in Fig. 3. + +6 +Data set +Prior settings +Models +GRB jets +1 +2 +5 +2 +20 +Simulations +1. Compact + Binary +b) NSBH +2. Supernova +b) SN98bw +in four astrophysical scenarios +GRB structures +luminosity distance +1. narrow + Gaussian +2. wide + uniform +1. Gaussian +2. Tophat +a) BNS +a) SNCol +350 += +m += +s +2 +p(D ) +L +DL +1000 +0 +p(D ) +L +DL +Figure 1. Schematic illustration of our comprehensive Bayesian inference campaign performed to analyze GRB 211211A. We +use one observational data set as described in Sec. 2, two prior settings in which we mainly vary the luminosity distance prior +while prior settings for other model parameters remained fixed and are reported in Table 3, five models (including two different +BNS kilonova models) or model combinations for four different astrophysical scenarios, and two GRB jet types (Gaussian and +top-hat), totaling in 20 Bayesian inferences. +Name +Astrophysical +GRB Jet +Model +Bayes factor +Likelihood +Processes +Structure +dimension ln[B1 +ref] +ln[L1 +ref(ˆθ)] +BNS-GRBKasen +top +Kilonova + GRB +Tophat +11 +ref. +ref. +BNS-GRBKasen +Gauss +Kilonova + GRB +Gaussian +12 +-1.01 ± 0.10 +-0.33 +BNS-GRBBulla +top +Kilonova + GRB +Tophat +11 +-0.49 ± 0.10 +-1.15 +BNS-GRBBulla +Gauss +Kilonova + GRB +Gaussian +12 +-1.59 ± 0.10 +-2.13 +NSBH-GRBtop +Kilonova + GRB +Tophat +11 +-3.76 ± 0.10 +-3.82 +NSBH-GRBGauss +Kilonova + GRB +Gaussian +12 +-2.08 ± 0.10 +-4.16 +SNCol-GRBtop +rCCSNe + GRB +Tophat +14 +-10.42 ± 0.11 +-3.04 +SNCol-GRBGauss +rCCSNe + GRB +Gaussian +15 +-10.74 ± 0.11 +-3.58 +SN98bw-GRBtop +CCSNe + GRB +Tophat +8 +-6.93 ± 0.10 +-8.14 +SN98bw-GRBGauss +CCSNe + GRB +Gaussian +9 +-8.05 ± 0.10 +-8.13 +GRBtop +GRB +Tophat +8 +-6.04 ± 0.10 +-7.10 +GRBGauss +GRB +Gaussian +9 +-6.96 ± 0.10 +-7.33 +Table 1. Results for the logarithmic Bayes factors, ln[B1 +ref], and maximum logarithmic likelihood ratios, ln[L1 +ref(ˆθ)], relative to +the best-fit, joint inference using BNS-GRBKasen +top +(ref.). The four investigated scenarios of possible astrophysical origins (BNS, +NSBH, SNCol, and SN98bw) are each being assessed assuming a Gaussian or a Top-hat jet structure. As reference, we list +results for a stand-alone GRB model investigation for both jet structures. +We find that the GRB-M model achieves a good rep- +resentation of the data in almost all bands, except for +the i-band and K-band data at late times (shown in +Fig. 3). In contrast, the joint model inferences of BNS- +GRBBulla +top +and BNS-GRBKasen +top +achieve a better represen- +tation of i-band and K-band data and the observational +data points lie within the estimated 1 magnitude uncer- +tainty (shaded band) of the best-fit light curves. Hence, +our analysis suggests that an additional source of energy +generation is required to generate bright light curves at +late times in the i- and K-band and to fit the observed +data. +We have further investigated the impact of late-time i- +band data on our inference results, in particular, we have +performed analysis runs, not shown in Fig. 3, in which +we have excluded i-band data observed with Gemini- +GMOS two days after trigger time (see Table 2) for BNS- +GRBBulla +top , BNS-GRBKasen +top +, and GRBtop. We found that +BNS-GRBBulla +top , BNS-GRBKasen +top +, and GRBtop perform +almost identically, and predict similar light curves in +the i-band, but also in all other bands. This shows that +late i-band data points are the main source of differ- +ence between the standalone GRB model and BNS-GRB +models. +4.3. Source properties of the potential compact binary +mergers +For the scenario that GRB 211112A was connected to +a compact binary merger, which is favored by our anal- +ysis, we now determine the source properties of the po- +tential progenitor system. For this purpose, we use the + +7 +28 +24 +20 +1 keV +15 +20 +25 +6GHz +26 +22 +18 +u +26 +22 +18 +g +26 +22 +18 +r +26 +22 +18 +i +26 +22 +18 +J +10−2 +10−1 +100 +101 +Time [days] +26 +22 +18 +K +BNS-GRBKasen +top +NSBH-GRBBulla +top +SNCol-GRBtop +SN98bw-GRBtop +Figure 2. +Best fitting light curve from joint Bayesian +inferences listed in Table 1 for possible scenarios: +BNS- +GRBKasen +top +(red), NSBH-GRBtop (green), SNCol-GRBtop (or- +ange), and SN98bw-GRBtop (blue). The observational data +of GRB 211211A in X-ray-1keV, radio-6GHz, UV, optical, +and NIR band as discussed in Sec. 2 are shown as black +dots, whereas black triangles refer to upper detection limits. +inferred GRB afterglow and kilonova properties for both +BNS-KN-Kasen and BNS-KN-Bulla and connect infor- +mation about the ejecta and debris disk to the BNS +properties following Dietrich et al. (2020); cf. Henkel +et al. (2022) for a recent discussion about uncertain- +ties in the employed numerical-relativity informed phe- +nomenological relations. +In Fig. 4, we show our inference results for a possible +BNS source using BNS-GRBKasen +top +, BNS-GRBKasen +Gauss, and +BNS-GRBBulla +top +and contrast these to the prior probabil- +28 +24 +20 +16 +i +28 +24 +20 +16 +J +10−2 +10−1 +100 +101 +Time [days] +28 +24 +20 +16 +K +BNS-GRBKasen +top +GRBtop +BNS-GRBBulla +top +28 +24 +20 +16 +r +Figure 3. Best-fitting light curves from joint Bayesian in- +ferences of BNS-GRBBulla +top +(yellow) and BNS-GRBKasen +top +(red) +compared to a stand-alone GRBtop inference (black) for op- +tical and NIR bands on a logarithmic time scale in days since +trigger time. +ity regions for each parameter, in order to show how con- +straining the observational data is. Comparing inference +results for BNS-GRBKasen +Top +and BNS-GRBKasen +Gauss, we find +that estimated source masses and tidal deformabilies +are very similar. For the top-hat jet structure simula- +tion, we find that a BNS merger with a primary mass of +1.55+0.54 +−0.42M⊙ and a secondary mass of 1.34+0.25 +−0.40M⊙ was +the likely progenitor. The associated dimensionless tidal +deformability of the system lies within ˜Λ = 299+1041 +−274 +. +With regard to a similar analysis for BNS-GRBBulla +Top , +we find a primary mass of 1.56+0.43 +−0.34M⊙ and a sec- +ondary mass of 1.29+0.20 +−0.29M⊙. The corresponding tidal +deformability is 353+598 +−264. Comparing estimated masses +for BNS-GRBKasen +Top +and BNS-GRBBulla +Top , we find overall +good agreement within the stated uncertainties. Con- + +8 +cerning the tidal deformability, we find that the BNS- +KN-Bulla model provides tighter constraints compared +to those extracted with the BNS-KN-Kasen model. We +expect this deviation to originate from the fact that the +BNS-KN-Bulla model provides more detailed informa- +tion on the estimated wind and dynamical ejecta masses, +while the BNS-KN-Kasen model provides a generic es- +timate of the total ejecta mass. +Overall, +our +estimated +masses +are +consistent +with Rastinejad et al. (2022), who concluded that +GRB 211211A originated from a 1.4 M⊙+1.3 M⊙ BNS +merger. We expect that the remaining small differences +are caused by the different analysis of the observed +GRB 211211A data and by the fact that Rastinejad +et al. (2022) assumed the inclination angle, under +which the binary was observed, to be zero. Moreover, +Rastinejad et al. (2022) assumed a fixed equation +of state from the EOS set of Dietrich et al. (2020) +using additional information from Nicholl et al. (2021). +In contrast, we leave the inclination angle as a free +parameter in our analysis and use the updated EOS set +of Huth et al. (2022). This set incorporates information +from +theoretical +nuclear-physics +computations +and +from astrophysical observations of neutron stars such +as Dietrich et al. (2020), but also heavy-ion collision +experimental data. With regard to investigated binary +merger scenarios, we find that the inferred inclination +angle is around θObs ≈ 0.02+0.05 +−0.02, while larger incli- +nation angles of approximately θObs ≈ 0.07+0.11 +−0.06 are +estimated for the two considered supernova scenarios +(see Table 3). +Rastinejad et al. (2022) deduced a total r-process +ejecta mass of Mej = 0.047+0.026 +−0.011M⊙, of which 0.02 M⊙ +correspond to lanthanide-rich ejecta, +0.01 M⊙ +to +intermediate-opacity ejecta, and 0.01 M⊙ to lanthanide- +free material. +With our reference inference result +from BNS-GRBKasen +top +, we find a total ejecta mass of +M BNS +ej,Kasen = 0.021+0.017 +−0.013M⊙ which is broadly in agree- +ment given the uncertainties. +Concerning our analy- +sis based on BNS-GRBBulla +top , we found a total ejecta +mass of M BNS +ej += 0.031+0.033 +−0.018M⊙, of which 0.015M⊙ +can be attributed to lanthanide-rich ejecta, 0.011M⊙ to +intermediate-opacity mass, and 0.002M⊙ to lanthanide- +free material. +For completeness, we have performed a similar inves- +tigation for our NSBH-GRBtop and NSBH-GRBGauss +models to infer the corresponding NSBH properties by +making use of the relations provided in Foucart et al. +(2018) and Kr¨uger & Foucart (2020). Although the ob- +servational data does not provide a strong constraint on +the NSBH source properties, our NSBH-GRBtop anal- +ysis suggests that an NSBH merger with a BH mass +299.40 +597.66 +264.01 +301.17 ++968.83 +_ 276.78 +Figure 4. +Component masses m1,2 and the dimension- +less tidal deformability ˜Λ based on our inference results of +BNS-GRBKasen +Gauss (orange), BNS-GRBKasen +Top +(red) and BNS- +GRBBulla +Top +(blue). Different shadings mark the 68%, 95%, and +99% confidence intervals. For the 1D posterior probability +distributions, we give the 90% confidence interval (dashed +lines) and report median values above each panel. +Grey +shaded areas give the prior probability regions. +of 3.18+8.54 +−2.34M⊙ and an NS mass of 1.39+0.83 +−0.85M⊙ could +have been the progenitor of GRB 211211A, with a total +ejecta mass of M NSBH +ej += +0.008+0.012 +−0.006 M⊙. Likewise, +the BH spin is weakly constrained to χ1 = 0.00+0.57 +−0.74 for +the NSBH-GRBtop inference. Our inferred NS masses +are in agreement with previous GW population analy- +ses (Abbott et al. 2019, 2021a,b) and with the maximum +non-spinning NS mass of 2.7+0.5 +−0.4M⊙ estimated at 90% +credibility by Ye & Fishbach (2022). Within the esti- +mated uncertainties, the inferred BH mass is close to +the NSBH mass gap for which the lightest BH masses +were estimated to be ∼ 5M⊙ (¨Ozel et al. 2010; Farr et al. +2011). +4.4. Influence of the prior choice +Finally, we discuss the influence of a different lu- +minosity distance prior on our results. +The distance +of GRB 211211A was relatively precisely estimated +based on the redshift of the potential host galaxy, +z = 0.0763 ± 0.0002 (Rastinejad et al. 2022). How- +ever, we are generally interested in the influence of a + +750 ++0.54 ++0.54 +1.56 +0.43 +1.55 +57 +-0.34 +0.42 +-0.42 +500 +250 +0 +-0.27 +-0.29 +1000 +2.4 +m2[Mo] += +500 +0 +6 +352.83+ ++1041.31 +273.52 +2000 +4500 +1000 +0 +mul +m? +V +BNS-GRBKasen +BNS-GRBBulla +Gauss +Lop +BNS-GRBKasen +Prior +Top=9 +wide uniform luminosity distance prior on our results. +For this reason, we widen the prior range and allow a +distance between 0 and 1000 Mpc. +Following the procedure in Sec. 4.1, we have com- +puted the logarithmic Bayes factors and found that +BNS-GRBKasen +top +remains to be the best-fitting model. +Moreover, the differences in logarithmic Bayes factors +between BNS-KN-Bulla and BNS-KN-Kasen remain the +same. Overall, the differences with regard to the indi- +vidual Bayes factors as presented in Tab. 1 are small. +However, the SN and collapsar scenarios are now equally +disfavored. Hence, our main conclusions remain valid +also for the wider distance prior. +We have investigated the posterior probability dis- +tributions obtained for a wide uniform distance prior +and compare these with the ones obtained for a narrow +Gaussian distance prior setting. In Fig. 5, we show an +example for the obtained luminosity distance and the +total ejecta mass distributions using GRBKasen +top +. As can +be seen, the wide distance prior leads to a noticeably +weaker constraint on the distance and the total ejecta +mass. +The latter is caused by a degeneracy between +the luminosity distance and the ejecta mass. +Gener- +ally, larger ejecta masses could compensate for larger +distances and vice versa, which explains the shape of the +2D correlation plot of Fig. 5. Similarly (not shown in +the figure), also the SNCol model predicts higher ejecta +masses for larger distances. +With respect to the SN- +GRB and the GRB inferences, the GRB isotropic en- +ergy, log10(EK,iso), tends to increase for larger distances, +which is expected as brighter signals can be detected to +further distances. +5. CONCLUSION +In this paper, we have performed multiple multi- +wavelength analyses for GRB 211211A assuming four +different scenarios, i.e., a BNS merger, an NSBH merger, +an rCCSN, as well as a CCSN. On the basis of joint +multi-wavelength Bayesian inferences combining respec- +tive kilonova or SN models with a gamma-ray burst af- +terglow model, we studied for which scenario we find +the highest statistical evidence to explain the data. We +summarize our main conclusions: +(i) We find strong statistical evidence for a BNS +merger scenario; cf. Table 1. However, we can not +fully rule out other scenarios. +(ii) Our study confirms that GRB 211211A can not +solely be explained as a GRB afterglow and that +an additional emission process (likely related to +r-process nucleosynthesis) is required for a good +Figure 5. Corner plot for BNS-GRBKasen +top +with a narrow +Gaussian luminosity distance prior centered around 350 Mpc +(orange) and a wide uniform luminosity distance prior rang- +ing up to 1000 Mpc (blue). The inferred model parameters +are shown at 68%, 95%, and 99% confidence (shadings from +light to dark). For the 1D posterior probability distributions, +we report the median values and show the 90% confidence +intervals as dashed lines. +description of the observational data, mostly in +late i-band and K-band data. +(iii) Assuming a BNS origin, our study suggests that +this system was a 1.55+0.54 +−0.42M⊙ - 1.34+0.25 +−0.40M⊙ bi- +nary, leading to a total ejecta mass of M BNS +ej += +0.021+0.017 +−0.013M⊙. +Assuming a NSBH origin of +GRB 211211A, our study suggests a 1.39+0.83 +−0.85 - +3.18+8.54 +−2.34M⊙ system with a total ejecta mass of +M NSBH +ej += 0.008+0.012 +−0.006 M⊙. +(iv) The results discussed in Sec. 4.2 showed that near- +infrared data at late times are essential to inves- +tigate the astrophysical origin of interesting tran- +sient objects. +6. ACKNOWLEDGEMENT +This project has received financial support from the +CNRS through the MITI interdisciplinary programs. +S.A. thanks A. de Ugarte Postigo for sharing CAHA +data for this work. +S.A also thanks Rahul Gupta, +Jirong Mao, Robert Strausbaugh, Dong Xu, Jinzhong +Liu, Daniele Malesani, Andrew Levan, and the MIT- + +350.00+3.43 +3.52 +323.66 +1000 +500 +0.36 +0.73 +1500 +log1o(mei)[Mo] +6 +1000 +500 +X +2 +D +log1o(meji) [Mo] +narrow Gaussian prior +wide uniform prior10 +SuME group for their useful comments on their obser- +vations. S.A thanks T. Hussenot for the discussion re- +lated to GRB 211211A. M.B. acknowledges support by +the European Union’s Horizon 2020 Programme under +the AHEAD2020 project (grant agreement n. 871158). +The work of I.T. was supported by the U.S. Department +of Energy, Office of Science, Office of Nuclear Physics, +under Contract No. DE-AC52-06NA25396, by the Lab- +oratory Directed Research and Development program +of Los Alamos National Laboratory under Project No. +20220541ECR, and by the U.S. Department of Energy, +Office of Science, Office of Advanced Scientific Com- +puting Research, Scientific Discovery through Advanced +Computing (SciDAC) NUCLEI program. S. Anand ac- +knowledges support from the National Science Founda- +tion GROWTH PIRE grant No. 1545949. +REFERENCES +Abbott, B. P., Abbott, R., Abbott, T. D., et al. 2017, +ApJL, 848, L13 +Abbott, B. P., Abbott, R., Abbott, T. D., et al. 2017, Phys. +Rev. Lett., 119, 161101. https: +//link.aps.org/doi/10.1103/PhysRevLett.119.161101 +Abbott, B. P., Abbott, R., Abbott, T. 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OBSERVATIONAL DATA SELECTION +MJD [days] +Filter +Telescope/Instrument +Transient [AB mag] +Reference +u-band +0.04 ± 0.005 +u +UVOT +19.7 ± 0.2 +(Rastinejad et al. 2022) +0.06 ± 0.01 +u +UVOT +19.4 ± 0.2 +(Rastinejad et al. 2022) +0.19 ± 0.01 +u +UVOT +19.8 ± 0.1 +(Rastinejad et al. 2022) +0.66 ± 0.01 +u +UVOT +> 20.3 +(Rastinejad et al. 2022) +0.86 ± 0.01 +u +UVOT +> 20.9 +(Rastinejad et al. 2022) +1.19 ± 0.01 +u +UVOT +> 21.9 +(Rastinejad et al. 2022) +g-band +0.260 ± 0.081 +g′ +MITSuME +20.3 ± 0.2 +(Ito et al. 2021) +0.69 ± 0.01 +g +NOT +21.0 ± 0.04 +(Rastinejad et al. 2022) +r-band +0.431 ± 0.020 +r +NEXT +20.25 ± 0.1 +(Jiang et al. 2021) +0.69 ± 0.01 +r +NOT +20.81 ± 0.05 +(Rastinejad et al. 2022) +1.425 ± 0.001 +r′ +GIT +> 21.15 +(Kumar et al. 2021) +i-band +0.68 ± 0.016 +i +CAHA-CAFOS +20.75 ± 0.08 +(Rastinejad et al. 2022) +0.70 ± 0.01 +i +NOT +20.9 ± 0.1 +(Rastinejad et al. 2022) +1.68 ± 0.01 +i +CAHA-CAFOS +22.6 ± 0.15 +(Rastinejad et al. 2022) +5.11 ± < 0.01 +i +Gemini-GMOS +26.03 ± 0.3 +(Rastinejad et al. 2022) +6.08 ± < 0.01 +i +Gemini-GMOS +> 25.49 +(Rastinejad et al. 2022) +J-band +0.445 ± 0.024 +z +NEXT +19.9 ± 0.3 +(Jiang et al. 2021) +4.72 ± 0.02 +H +TNG +> 21.9 +(D’Avanzo et al. 2021) +5.96 ± 0.014 +J +MMT-MMRIS +24.17 ± 0.35 +(Rastinejad et al. 2022) +K-band +4.058 ± 0.005 +K +Gemini-NIRI +22.41 ± 0.14 +(Rastinejad et al. 2022) +5.10 ± 0.005 +K +Gemini-NIRI +22.4 ± 0.2 +(Rastinejad et al. 2022) +6.94 ± 0.02 +K +MMT-MMRIS +23.4 ± 0.3 +(Rastinejad et al. 2022) +7.98 ± 0.01 +K +MMT-MMRIS +23.8 ± 0.3 +(Rastinejad et al. 2022) +Table 2. Multi-wavelength observations of the counterpart and the host galaxy of GRB 211211A. Magnitudes are corrected +for foreground Galactic extinction according to AV = 0.048 mag (Rastinejad et al. 2022). + +13 +B. INFERENCE SETTINGS +All parameter estimation runs were performed using the nuclear physics and multi-messenger astronomy framework +NMMA (Pang et al. 2022). In this framework, joint Bayesian inferences of electromagnetic signals are carried out on the +basis of the nested sampling algorithm implemented in pymultinest (Buchner et al. (2014)). Each simulation used +2048 live points, and the prior settings for each of the employed models, as well as the median values and 90% credible +ranges, are provided in Table 3. +Parameter +Prior +Posterior +BNS-GRBBulla +top +BNS-GRBKasen +top +NSBH-GRBtop +SNCol-GRBtop +SN98bw-GRBtop +GRB-M +log10(EK,iso) [erg] +[47, 55] +52.36+2.14 +−2.15 +51.28+2.86 +−1.38 +51.77+2.79 +−1.67 +51.80+2.21 +−1.56 +50.45+1.08 +−0.73 +θObs [rad] +[0, π +4 ] +0.02+0.05 +−0.02 +0.02+0.05 +−0.02 +0.02+0.05 +−0.02 +0.07+0.07 +−0.06 +0.07+0.14 +−0.05 +θc [rad] +[0.01, +π +10] +0.05+0.08 +−0.04 +0.05+0.09 +−0.04 +0.05+0.08 +−0.04 +0.09+0.10 +−0.06 +0.10+0.14 +−0.07 +log10(n) [cm−3] +[-6, 2] +-0.39+2.38 +−4.78 +-3.46+4.95 +−2.54 +-1.45+3.41 +−4.09 +1.08+0.92 +−3.36 +-4.62+1.41 +−1.37 +p +[2.01, 3] +2.19+0.25 +−0.15 +2.32+0.30 +−0.26 +2.27+0.31 +−0.21 +2.20+0.19 +−0.16 +2.53+0.25 +−0.24 +log10(ϵe) +[-5, 0] +-0.80+0.80 +−1.88 +-0.28+0.28 +−1.81 +-0.48+0.48 +−2.05 +-0.63+0.63 +−1.55 +-0.10+0.10 +−0.23 +log10(ϵB) +[-10, 0] +-4.44+4.42 +−3.60 +-2.01+2.01 +−5.62 +-3.45+3.45 +−4.44 +-4.60+3.66 +−3.52 +-0.68+0.68 +−1.48 +DL[Mpc] +N(350, 2) +350.07+3.42 +−3.46 +350.00+3.43 +−3.52 +350.01+3.46 +−3.49 +350.18+3.60 +−3.59 +349.98+3.68 +−3.36 +BNS-KN-Bulla +log10(M ej +dyn) [M⊙] +[-3, -1] +-1.78+0.70 +−0.62 +log10(M ej +wind) [M⊙] +[-3, -0.5] +-1.98+0.55 +−0.57 +Φ [deg] +[15, 75] +62.42+12.58 +−30.08 +BNS-KN-Kasen +log10(Mej) [M⊙] +[-2.5, -1] +-1.68+0.37 +−0.36 +log10(vej) [c] +[-1.8, -1] +-0.90+0.53 +−0.51 +log10(Xlan) +[-4.5, -1] +-1.88+0.88 +−1.05 +NSBH-KN-Bulla +log10(M ej +dyn) [M⊙] +[-3, -1] +-2.51+0.73 +−0.49 +log10(M ej +wind) [M⊙] +[-3, -0.5] +-2.49+0.67 +−0.51 +SNCol +Mej [M⊙] +[0, 0.5] +0.06+0.05 +−0.04 +MNi [M⊙] +[0, 0.03] +0.00+0.01 +−0.00 +vej [c] +[0, 0.5] +0.21+0.04 +−0.04 +Mrp [M⊙] +[0, 0.05] +0.01+0.01 +−0.01 +Ψmix +[0, 0.9] +0.73+0.17 +−0.35 +SN98bw +Smax +[0, 60] +32.86+24.45 +−24.82 +Table 3. +Model parameters and prior bounds employed in our Bayesian inferences. We report median posterior values at +90 % credibility from simulations that were run with Top-hat jet structure and with a narrow Gaussian luminosity distance +prior N(µ, σ), with mean µ = 350 Mpc and standard deviation σ = 2 Mpc. We employ a conditional prior on the inclination +angle depending on the jet core opening angle, p(θObs|θc), using a truncated Gaussian distribution, NT (µ, σ), where µ = 0 and +σ = θc . + +14 +C. INFERENCE RESULTS +In the following, we present the posterior distribution for our reference model GRBKasen +top +employing a narrow distance +prior centered around 350 Mpc. Figure 6 summarizes our results. As discussed in the main text, we obtain a total +ejecta mass of 10−1.68+0.37 +−0.36 M⊙, which is generally consistent with previous findings in the literature and an average +velocity of 10−0.9+0.53 +−0.51c. Interestingly, our analysis prefers a higher lanthanide fraction compared to the one inferred for +AT2017gfo using the same kilonova models (Coughlin et al. 2018), i.e., we predict a slightly redder kilonova (similar +to Rastinejad et al. (2022) who predict a larger mass of the red component, but opposite to e.g. Mei et al. (2022)). +Considering the obtained GRB posteriors, we find a double peak structure in our posteriors and a clear low n0 - +high ϵB - high p peak, as well as, a high n0 - low ϵB - low p -peak. While this double peak structure might be caused +by the small set of observational data and potential degeneracies, it could also be an indicator of the missing input +physics of the employed GRB models, in particular, the emission from the reverse shock that might be important at +early times and wind interstellar medium density density profile, the absence of which might be responsible for the +high n0 - low ϵB - low p -peak. +0 +500 +1000 +−1.88+0.88 +−1.05 +−2.4 +−2.0 +−1.6 +−1.2 +log10(mej)[M⊙] +0 +1000 +−1.68+0.37 +−0.36 +−1.6 +−1.2 +−0.8 +−0.4 +log10(vej)[c] +0 +500 +1000 +−0.90+0.53 +−0.51 +0.06 +0.12 +0.18 +ι[rad] +0 +2000 +0.02+0.05 +−0.02 +345 +350 +355 +DL [Mpc] +0 +1000 +2000 +350.00+3.43 +−3.52 +51.0 +52.5 +54.0 +log10(E0)[erg] +0 +500 +1000 +51.28+2.86 +−1.38 +−4 +−2 +0 +log10(n0)[cm−3] +0 +500 +1000 +−3.46+4.95 +−2.54 +0.08 +0.16 +0.24 +θc[rad] +0 +2000 +0.05+0.09 +−0.04 +2.25 +2.50 +2.75 +p +0 +500 +1000 +2.32+0.30 +−0.26 +−4.5 +−3.0 +−1.5 +ϵe +0 +2500 +5000 +−0.28+0.28 +−1.81 +−4.0 +−3.2 +−2.4 +−1.6 +log10(χlan) +−7.5 +−5.0 +−2.5 +ϵB +−2.4 +−2.0 +−1.6 +−1.2 +log10(mej)[M⊙] +−1.6 +−1.2 +−0.8 +−0.4 +log10(vej)[c] +0.06 +0.12 +0.18 +ι[rad] +345 +350 +355 +DL [Mpc] +51.0 +52.5 +54.0 +log10(E0)[erg] +−4 +−2 +0 +log10(n0)[cm−3] +0.08 +0.16 +0.24 +θc[rad] +2.25 +2.50 +2.75 +p +−4.5 +−3.0 +−1.5 +ϵe +−7.5 +−5.0 +−2.5 +ϵB +0 +1000 +−2.01+2.01 +−5.62 +Figure 6. Corner plot for BNS-GRBKasen +top +with a narrow Gaussian luminosity distance prior centered around 350 Mpc, in +which we show the inferred parameters at 68%, 95%, and 99% confidence (shadings from light to dark). For the 1D posterior +probability distributions, we report the median values and show the 90% confidence intervals as dashed lines. + diff --git a/b9A0T4oBgHgl3EQfGf_e/content/tmp_files/load_file.txt b/b9A0T4oBgHgl3EQfGf_e/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..244cff2e25d9b070bff5837b1210cbc40545e01b --- /dev/null +++ b/b9A0T4oBgHgl3EQfGf_e/content/tmp_files/load_file.txt @@ -0,0 +1,1612 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf,len=1611 +page_content='Draft version January 6, 2023 Typeset using LATEX twocolumn style in AASTeX62 Model selection for GRB 211211A through multi-wavelength analyses Nina Kunert,1 Sarah Antier,2 Vsevolod Nedora,3 Mattia Bulla,4, 5, 6 Peter T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Pang,7, 8 Shreya Anand,9 Michael Coughlin,10 Ingo Tews,11 Jennifer Barnes,12 Meili Pilloix,2, 13 Weizmann Kiendrebeogo,2 and Tim Dietrich1, 3 1Institute of Physics and Astronomy, Theoretical Astrophysics, University Potsdam, Haus 28, Karl-Liebknecht-Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 24/25,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 14476,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Potsdam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Germany 2Artemis,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Princetonplein 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 3584 CC Utrecht,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' The Netherlands 9Cahill Center for Astrophysics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' California Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Pasadena CA 91125,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' USA 10School of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' University of Minnesota,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Minneapolis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Minnesota 55455,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' USA 11Theoretical Division,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Los Alamos National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Los Alamos,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' NM 87545,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' USA 12Kavli Institute for Theoretical Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Kohn Hall,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' University of California,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Santa Barbara,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' CA 93106,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' USA 13Laboratoire de Physique et de Chimie de l’Environnement,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Universit´e Joseph KI-ZERBO,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Ouagadougou,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Burkina Faso (Dated: January 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2023) ABSTRACT Although being among the closest gamma-ray bursts (GRBs),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' GRB 211211A poses challenges for its classification with partially inconclusive electromagnetic signatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' In this paper, we investigate four different astrophysical scenarios as possible progenitors for GRB 211211A: a binary neutron-star merger, a black-hole–neutron-star merger, a core-collapse supernova, and an r-process enriched core collapse of a rapidly rotating massive star (a collapsar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' We perform a large set of Bayesian multi- wavelength analyses based on different models and priors to investigate which astrophysical scenarios and processes might have been related to GRB 211211A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Our analysis supports previous studies in which the presence of an additional component, likely related to r-process nucleosynthesis processes, is required to explain the observed light curves of GRB 211211A, as it can not solely be explained as a GRB afterglow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Fixing the distance to about 350 Mpc, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=', the distance of the possible host galaxy SDSS J140910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='47+275320.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='8, we find a statistical preference for the binary neutron-star merger scenario and estimate the component masses to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='55+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='54 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='42M⊙ and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='34+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='25 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='40M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' INTRODUCTION The joint detection of gravitational waves (GWs) and electromagnetic (EM) signatures originating from the merger of binary neutron stars (BNSs) on August 17th 2017 (Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2017) has been a breakthrough in multi-messenger astronomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' In addition to the GW signal GW170817, an associated kilonova, AT2017gfo, and a gamma-ray burst (GRB), GRB 170817A, were observed (Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' This multi-messenger detection allowed for an inde- pendent way of measuring the expansion rate of the Universe (Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2017), placed new constraints on the properties of supranuclear-dense matter (Bauswein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Ruiz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Radice et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Most et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Coughlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Capano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Dietrich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Huth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2022), and proved that at least some short GRBs are connected to compact binary mergers (Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' However, it was also reported that short GRBs could originate from collapsars (Ahumada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2021), indicating that the classification of astrophysical scenarios associated with GRBs is more complex (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Rossi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' (2022)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Additional signatures associated with GRBs and their afterglows, such as kilonovae, significantly help to identify the origin of the progenitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' The kilonova AT2017gfo was certainly an exemplary case for such an EM signal, and spectral features connected to the creation of new elements (Watson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Domoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2022) in the outflowing material have possibly been observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' In addition to AT2017gfo, arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='02049v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='HE] 5 Jan 2023 2 there is a large number of kilonova candidates that could be connected to other GRB observations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=', GRB 050709, GRB 050724A, GRB 060614, GRB 061201, GRB 080905A, GRB 070724A, GRB 130603B, GRB 140903A, GRB 150101B, GRB 150424A, GRB 160821B, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=', Tanvir et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' (2013);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Berger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' (2013);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' (2007);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Fox et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' (2005);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Hjorth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' (2005);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Covino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' (2006);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Stratta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' (2007);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Berger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' (2005);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Malesani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' (2007);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Fong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Troja et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Nicuesa Guelbenzu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' (2012);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Rowlinson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' (2010);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Berger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' (2009);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Kocevski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' (2010);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Kasliwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Tanvir et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Ascenzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' (2019) for a review about some of these kilonova candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' The most recent example that has to be added to the list is the kilonova candidate connected to GRB 211211A and its optical and near-infrared counterpart, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=', Rastinejad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' (2022), and Troja et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' (2022), and Mei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' GRB 211211A was discovered on the 11th Decem- ber 2021 at 13:09:59 (UTC) by the Burst Alert Tele- scope (BAT) of the Swift Observatory (trigger 1088940, SNRimg = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='5, D’Ai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' The Fermi Gamma- ray Burst Monitor detected GRB 211211A indepen- dently at the exact same trigger time (trigger 211211549, SNRct = 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='2, Fermi GBM Team 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Moreover, the high-energy space instrument onboard Insight-HXMT detected GRB 211211A (trigger HEB211211548) during its routine search (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' The GRB is char- acterized by a complex emission phase lasting approx- imately 10 s, and a longer, weaker extended emission for about 130 s in [15-350] keV (Stamatikos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Given this duration, GRB 211211A would be classified as a long GRB typically arising from the core-collapse of massive stars (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=', Stanek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Levan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2016) and not from compact binary mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Hence, for a sce- nario such as GRB 211211A, one would not necessarily expect to observe an associated kilonova.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' About 70 s after the emergence of the prompt emis- sion, Swift’s X-ray Telescope (XRT) started observing the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' The X-ray observations showed bright emis- sion (a flux of 3×10−8ergs−1cm−2 in [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='3–10] keV) with an exponential decay lasting for hours after the trigger (Osborne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' The Ultraviolet/Optical Tele- scope started its observations 92 s later and detected an optical counterpart within the X-ray localization er- ror box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Given its close proximity to the galaxy SDSS J140910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='47+275320, an intensive follow-up campaign in- cluding MITSuME, NEXT, the Nordic Optical Tele- scope, and the Calar Alto Observatory (Ito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Malesani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' de Ugarte Postigo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2021) was scheduled and the source was observed across multiple wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Based on these follow-up observations and the following analysis, it seems plausi- ble that SDSS J140910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='47+275320.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='8 was the host galaxy of GRB 211211A, at 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='6% confidence (Rastinejad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Details about the observation campaign are sum- marized in Rastinejad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Rastinejad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' (2022), Troja et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' (2022), and other groups explained these observations by invoking a kilo- nova in association with GRB 211211A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' This was sug- gested for different reasons: (i) the profile of the prompt emission showed an initially complex structure followed by an extended softer emission, (ii) a predominant sig- nature of a supernova was lacking for up to 17 days post-discovery, (iii) the color evolution of the optical counterpart had similar properties as AT2017gfo, and (iv) the offset of the GRB location concerning the cen- ter of the host galaxy was larger than for typical long GRBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Numerous other groups addressed the origin of GRB 211211A, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=', Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' (2022) suggested that it has similar properties as GRB 060614, another event associated with a kilonova candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' They conclude that the significant excess in the near-infrared and opti- cal afterglow at late observations points more towards a neutron star-white dwarf merger which leaves behind a rapidly spinning magnetar as a central engine provid- ing additional heating to the ejecta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Waxman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' (2022) showed that also thermal emission from dust could explain the observed near-infrared (NIR) data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Suvorov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' (2022) mentioned a possible gamma-ray precursor before the main emission which was caused by the resonant shattering of one star’s crust prior to the merger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' In contrast, Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' (2022) concluded the presence of a strong magnetic field from the pre- cursor surrounding the central engine of the GRB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' This would result in the prolongation of the accretion pro- cess and, thus, could explain the duration of the hard spiky emission detected for GRB 211211A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Similarly, Xiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' (2022) supposes that a magnetar partici- pated in the merger and caused a quasi-periodic pre- cursor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Gompertz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' (2022) analyzed the spectra of the prompt emission of GRB 211211A by using syn- chrotron spectrum models and concluded that the spec- tral evolution can be explained by a transition from a fast-cooling to a slow cooling regime, favoring a BNS merger rather than a neutron-star–black-hole (NSBH) scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Finally, Barnes & Metzger (2023) investigated the possibility that collapsars could explain the origin of GRB 211211A and found that the afterglow-subtracted emission of GRB 211211A is in best agreement for col- lapsar models with high kinetic energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 3 Following the discussion in the literature, we will use our nuclear physics and multi-messenger astrophysics (NMMA) framework (Pang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2022)1 to explore different astrophysical scenarios for the origin of GRB 211211A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' We will consider the possibility of two merger scenarios, a BNS merger and an NSBH merger, and in addition two supernova scenarios, a core-collapse supernova, and an r-process enriched collapsar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' For our model selec- tion study, the NMMA framework allows us to simultane- ously fit the observed data across the full electromag- netic range with multiple models, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=', we can simul- taneously employ GRB afterglow and kilonova models without the need of splitting the observational data in chunks and processing them separately, as done in – to our knowledge – previous studies of GRB 211211A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' OBSERVATIONAL DATA In order to perform our model selection, we collect a set of multi-wavelength data observed for GRB 211211A (see Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Concerning the GRB afterglow, we do not use any data from the prompt emission phase of the GRB in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' This means that we use available X-ray data from the Swift X-ray Telescope, in particular, we use the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='3 - 10 keV flux light curve observed at late times (t = 104 s after BAT trigger time) and convert it to 1 keV flux densities following Gehrels et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' For our optical study, we followed Rastinejad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' (2022) and included the refined analysis of Swift-UVOT observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' We contacted the authors of the obser- vational teams responsible for the GCN reports, espe- cially for those data which was not analyzed by Rastine- jad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' They provided us with offline results that we used in this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' For these data, we cor- rected the measurements by taking into account the fore- ground Galactic extinction AV = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='048 mag (Schlafly & Finkbeiner 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' We excluded all photometric results from observations performed with the Johnson-Cousins UBVRI system as we do not compute simulated light curves in these passbands in our Bayesian approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Moreover, we also excluded all photometric results from images taken without filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Finally, we use the 6 GHz radio detection of GRB 211211A observed 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='27 days after the initial trigger with a 5σ upper limit flux density of 16 µJy (Rastinejad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' With regard to available GeV data, as re- ported in Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' (2022) and Mei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' (2022), we do not include this data since our employed GRB model does not provide mechanisms to explain its origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' We also re-analyzed data from the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='3m telescope at the Centro Astron´omico Hispano en Andaluc´ıa (CAHA), 1 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='com/nuclear-multimessenger-astronomy equipped with the Calar Alto Faint Object Spectro- graph (CAFOS) and find consistent results with respect to Rastinejad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Moreover, we exclude the detection measurement in the i band at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='68 days post- discovery from our analysis since we find an upper limit of 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='6 mag at 5-σ with methods described in Aivazyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' METHODS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Bayesian Inference Our analysis is based on the nuclear physics and multi- messenger astronomy framework NMMA (Pang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2022) that allows us to perform joint Bayesian inference runs of multi-messenger events containing GWs, kilonovae, supernovae, and GRB afterglow signatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' For this ar- ticle, we extended the code infrastructure to include the description of r-process enriched collapsars following the model of Barnes & Metzger (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' We use the EM data of GRB 211211A to investigate which model or which combination of models describe the observational data best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' According to Bayes’ the- orem, we compute posterior probability distributions, p(⃗θ|d, M), for model source parameters ⃗θ under the hy- pothesis or model M with data d as p(⃗θ|d, M) = p(d|⃗θ, M)p(⃗θ|M) p(d|M) → P(⃗θ) = L(⃗θ)π(⃗θ) Z(d) , (1) where P(⃗θ), L(⃗θ), π(⃗θ), and Z(d) are the posterior, likelihood, prior, and evidence, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' In order to investigate the plausibility of competing models, we evaluate the odds ratio O1 2 for two models M1 and M2 which is given by O1 2 = p(d|M1) p(d|M2) p(M1) p(M2) ≡ B1 2Π1 2, (2) where B1 2 and Π1 2 are the Bayes factor and the prior odds, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Under the assumption that the different astrophysical scenarios considered here are equally likely to explain GRB 211211A, we impose unity prior odds, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=', Π1 2 = 1, for all comparisons of models describing these scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Therefore, we simply com- pute the Bayes factor B1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' In our study, we report the natural logarithm of the Bayes factor, ln B1 ref = ln � p(d|M1) p(d|Mref) � , (3) relative to our best fitting model as a reference (ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' ), which we will denote as ln Bref hereafter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Following Jeffreys (1961) and Kass & Raftery (1995), we interpret ln B1 ref as the evidence favoring our reference model as: 4 ln[B1 ref] < −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='61 decisive evidence, −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='61 ≤ ln[B1 ref] ≤ −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='30 strong evidence, −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='30 ≤ ln[B1 ref] ≤ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='10 substantial evidence, −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='10 ≤ ln[B1 ref] ≤ 0 no strong evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' However, we point out that these classifications should only be considered as estimates and that the Bayes factor is generally a continuous quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' In addition to the Bayes factor, we also provide information about the ratio of the maximum likelihood, or the difference of the maximum log-likelihood point estimates ln[L1 2(ˆθ)] supporting our analysis in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' We will denote this as ln[Lref(ˆθ)] when we compare the maximum log-likelihood against our reference model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Employed models As described in the introduction, we investigate four different scenarios in our study from which GRB 211211A could have emerged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' In particular, we consider two merger scenarios: a BNS merger and an NSBH merger, and two supernova cases: a phenomeno- logical long GRB supernova template and an r-process enriched collapsar scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' BNS scenario: For this case, we use the kilonova models of Dietrich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' (2020) (hereafter ‘BNS-KN- Bulla’) and of Kasen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' (2017) (hereafter ‘BNS-KN- Kasen’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' BNS-KN-Bulla is based on the time-dependent three-dimensional Monte Carlo radiation transfer code possis (Bulla (2019), Bulla, Mattia (2022)), which com- putes light curves, spectra, and luminosities for kilono- vae depending on the viewing-angle θObs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' The ejected material is classified through the dynamical ejecta mass, M dyn ej , and the disk-wind ejecta mass, M wind ej .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' The tidal dynamical ejecta component is assumed to be dis- tributed within a half opening angle Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' In the same way, BNS-KN-Kasen uses the multi-dimensional Monte Carlo code sedona that solves the multi-wavelength ra- diation transport equation in a relativistically expanding medium (Kasen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' (2006);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Roth & Kasen (2015)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' In this paper, we use the one-dimensional model provided by Kasen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' (2017), which assumes spherical sym- metry and uniform composition for our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' The model, ‘BNS-KN-Kasen’, depends on the ejecta mass, Mej, a characteristic expansion velocity, vej, and the mass fraction of lanthanides, Xlan, which affects the opacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' NSBH scenario: For this case, we also use a possis model grid of KN spectra tailored to NSBH mergers which was used in the study of Anand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' (2021) (hereafter ‘NSBH-KN-Bulla’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' This model depends on the same model parameters as BNS-KN-Bulla but ex- cludes the dependence on the half opening angle of the dynamical ejecta, fixed to Φ = 30◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Supernova: In order to assess the possibility of a typical core-collapse supernova (CCSN) associated with a long GRB, we use the nugent-hyper model from sncosmo (Levan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2005) with the absolute magni- tude, Smax, as the main free parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' This model is a template constructed from observations of the super- nova SN1998bw associated with the long GRB 980425 and is hereafter abbreviated as ‘SN98bw’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' r-process enriched Collapsar: Rapidly rotating massive star core collapses (Burbidge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 1957;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Qian & Woosley 1996) are another possible astrophysical site for r-process nucleosynthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' As massive stars undergo a core collapse, material is disrupted and forms an accre- tion disk which can become neutron-rich through weak interactions (Beloborodov 2003) and can launch winds which power emission of r-process-enriched core-collapse SNe (rCCSNe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' We use the semi-analytic model for rCCSNe of Barnes & Metzger (2022) (hereafter denoted as ’SNCol’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' The model depends on five free parame- ters: the total ejecta mass, Mej, a characteristic ejecta velocity, vej, the 56Ni mass, MNi, the r-process mate- rial mass, Mrp, and the mixing coordinate, Ψmix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' The ejecta are assumed to be spherically symmetric, with r- process elements of mass mrp concentrated in an inner core whose total mass is Ψmixmej, with Ψmix ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' An r-process-free envelope surrounds the core, and 56-Ni is distributed uniformly throughout the core and the en- velope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' The velocity vej is defined such that the total kinetic energy of the ejecta Ekin is equal to 1 2Mejv2 ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='2 GRB afterglow: For modeling the GRB afterglow light curves, we employ the semi-analytic model of van Eerten et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' (2010) and Ryan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' (2020), available in the public afterglowpy library (denoted as ‘GRB-M’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' The model computes GRB afterglow emission and takes the following free parameters as input: the isotropic ki- netic energy, EK,iso, the viewing angle, θObs, the half- opening angle of the jet core, θc, the outer truncation angle of the jet, θw, the interstellar medium density, n, the electron energy distribution index, p, and the frac- tions of the shock energy that go into electrons, ϵe, and magnetic fields, ϵB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' The model allows for several an- gular structures of the GRB jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' For our simulations, we assume a Gaussian or a top-hat jet structure (here- after, ‘Gauss’ and ‘top’)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' It is important to note that, while we try to be agnostic concerning GRB 211211A’s 2 Barnes & Metzger (2023) also compared rCCSNe with obser- vational data from GRB 211211A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' However, not within a Bayesian approach as employed here and with an updated version of their model originally described in Barnes & Metzger (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 3 In addition, we tested a power law jet structure for which we found consistent results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 5 origin, the GRB-M model that we employ has some lim- itations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Specifically, it does not include the emission from the reverse shock that might be important at early times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Additionally, it does not include the wind-like interstellar medium, which is expected in the case of a collapsar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 1, we summarize our approach to analyze GRB 211211A based on the data set described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' We employ two different priors for the luminosity dis- tance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=', a narrow Gaussian luminosity distance prior centered around 350 Mpc as reported by Rastinejad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' (2022) and a uniform prior on the luminosity dis- tance ranging between 0 and 1000 Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' This allows us to investigate the potential influence of the distance on the GRB classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Furthermore, we employ five models or model combinations to describe the different astrophysical scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' For the choice of a Gaussian luminosity distance prior, we report the prior settings for all parameters of the employed models in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Moreover, we use two different GRB jet types, totaling in 20 Bayesian inference simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' MULTI-WAVELENGTH ANALYSES In the following three subsections, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='1-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='3, we discuss our results for a narrow Gaussian prior on the luminosity distance in order to compare with previous studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' In subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='4, we will investigate the influence of the distance prior choice and employ a wide uniform prior on the luminosity distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Model Comparison As indicated in the introduction, one of the main dif- ferences between previous studies and our work is that most previous works fitted first the X-ray and radio data with a GRB afterglow model, and then used the afterglow-subtracted optical and near-infrared photom- etry for fitting a kilonova model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' In contrast, we per- form a joint analysis of the GRB afterglow and a possible additional contribution such as a kilonova signature or emission from a rCCSN or CCSN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Moreover, in order to consider systematic uncertainties arising from different assumptions made in each model, we employ a 1 mag uncertainty in our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' In Table 1, we summarize our main findings for the investigated astrophysical scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' We found that the BNS-GRBKasen top model describes the observational data best, and hence, we pick it as our reference model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Con- sequently, the Bayes factors and likelihood ratios in Ta- ble 1 are reported relative to this best-fit inference run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' With reference to Table 1, we show the maximum log- likelihood light curve fits in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2 for each assessed sce- nario, which we will refer to as ”best-fitting light curves” hereafter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Comparing only the two different BNS kilonova mod- els, we find that differences in the Bayes factors are of or- der unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' We interpret this as a measure of the system- atic model uncertainty for different employed kilonova models, given that both BNS-GRBBulla Gauss/top and BNS- GRBKasen Gauss/top should describe the same physical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' It is worth pointing out that statistical uncertainties, as stated in the table, are noticeably smaller than model differences, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=', our results are dominated by systematic uncertainties in the underlying light curve models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Considering the differences between the NSBH and BNS scenarios, we find strong evidence that GRB 211211A was connected to a BNS rather than an NSBH system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' This is reflected both in Bayes factors as well as maximum log-likelihood values as shown in Ta- ble 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Comparing the respective best fitting light curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2, we see that NSBH-GRBBulla top fits the NIR-band data worse compared to GRBKasen top .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' With regard to the relative Bayes factors for the col- lapsar scenario, we find that there is decisive evidence that a BNS scenario is preferred over a collapsar origin for GRB 211211A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' However, it is important to note that the collapsar model depends on more parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Be- cause of this, Occam’s razor penalizes the model despite its ability to describe the observational data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' This ability to describe and fit the observational data can be estimated from the maximum likelihood ratio re- sults as given in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' As indicated by Rastinejad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' (2022), and con- firmed by our study, we find that a Ni-powered SN event or an SN-GRB scenario is noticeably less favored com- pared to a BNS merger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' This is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2 in which SN98bw-GRBBulla top fails to fit late-time NIR data, resulting in a larger, negative log-likelihood ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Finally, our study confirms that the BNS-GRBKasen top scenario provides decisive evidence when compared with GRBtop-M simulations, even though the latter sampled over fewer parameters in respective parameter estima- tion runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Considering the impact of the choice of a Gaussian vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' top-hat jet structure on our Bayes factor results, we find a slight preference for the top-hat jet structure for all assessed scenarios, except for NSBH- GRBBulla Gauss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Presence of an additional component Given the overall narrative that GRB 211211A was a GRB connected to a kilonova, we study the ability of the GRB-M with top-hat jet structure to describe the observational data and compare this with two BNS merger scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' For this purpose, we show the best- fitting light curves for BNS-GRBBulla top , BNS-GRBKasen top , and GRBtop in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 6 Data set Prior settings Models GRB jets 1 2 5 2 20 Simulations 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Compact Binary b) NSBH 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Supernova b) SN98bw in four astrophysical scenarios GRB structures luminosity distance 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' narrow Gaussian 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' wide uniform 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Gaussian 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Tophat a) BNS a) SNCol 350 = m = s 2 p(D ) L DL 1000 0 p(D ) L DL Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Schematic illustration of our comprehensive Bayesian inference campaign performed to analyze GRB 211211A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' We use one observational data set as described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2, two prior settings in which we mainly vary the luminosity distance prior while prior settings for other model parameters remained fixed and are reported in Table 3, five models (including two different BNS kilonova models) or model combinations for four different astrophysical scenarios, and two GRB jet types (Gaussian and top-hat), totaling in 20 Bayesian inferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Name Astrophysical GRB Jet Model Bayes factor Likelihood Processes Structure dimension ln[B1 ref] ln[L1 ref(ˆθ)] BNS-GRBKasen top Kilonova + GRB Tophat 11 ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' BNS-GRBKasen Gauss Kilonova + GRB Gaussian 12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='33 BNS-GRBBulla top Kilonova + GRB Tophat 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='49 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='15 BNS-GRBBulla Gauss Kilonova + GRB Gaussian 12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='59 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='13 NSBH-GRBtop Kilonova + GRB Tophat 11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='76 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='82 NSBH-GRBGauss Kilonova + GRB Gaussian 12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='16 SNCol-GRBtop rCCSNe + GRB Tophat 14 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='42 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='04 SNCol-GRBGauss rCCSNe + GRB Gaussian 15 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='74 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='58 SN98bw-GRBtop CCSNe + GRB Tophat 8 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='93 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='10 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='14 SN98bw-GRBGauss CCSNe + GRB Gaussian 9 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='05 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='10 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='13 GRBtop GRB Tophat 8 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='10 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='10 GRBGauss GRB Gaussian 9 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='96 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='10 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='33 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Results for the logarithmic Bayes factors, ln[B1 ref], and maximum logarithmic likelihood ratios, ln[L1 ref(ˆθ)], relative to the best-fit, joint inference using BNS-GRBKasen top (ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' The four investigated scenarios of possible astrophysical origins (BNS, NSBH, SNCol, and SN98bw) are each being assessed assuming a Gaussian or a Top-hat jet structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' As reference, we list results for a stand-alone GRB model investigation for both jet structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' We find that the GRB-M model achieves a good rep- resentation of the data in almost all bands, except for the i-band and K-band data at late times (shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' In contrast, the joint model inferences of BNS- GRBBulla top and BNS-GRBKasen top achieve a better represen- tation of i-band and K-band data and the observational data points lie within the estimated 1 magnitude uncer- tainty (shaded band) of the best-fit light curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Hence, our analysis suggests that an additional source of energy generation is required to generate bright light curves at late times in the i- and K-band and to fit the observed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' We have further investigated the impact of late-time i- band data on our inference results, in particular, we have performed analysis runs, not shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 3, in which we have excluded i-band data observed with Gemini- GMOS two days after trigger time (see Table 2) for BNS- GRBBulla top , BNS-GRBKasen top , and GRBtop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' We found that BNS-GRBBulla top , BNS-GRBKasen top , and GRBtop perform almost identically, and predict similar light curves in the i-band, but also in all other bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' This shows that late i-band data points are the main source of differ- ence between the standalone GRB model and BNS-GRB models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Source properties of the potential compact binary mergers For the scenario that GRB 211112A was connected to a compact binary merger, which is favored by our anal- ysis, we now determine the source properties of the po- tential progenitor system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' For this purpose, we use the 7 28 24 20 1 keV 15 20 25 6GHz 26 22 18 u 26 22 18 g 26 22 18 r 26 22 18 i 26 22 18 J 10−2 10−1 100 101 Time [days] 26 22 18 K BNS-GRBKasen top NSBH-GRBBulla top SNCol-GRBtop SN98bw-GRBtop Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Best fitting light curve from joint Bayesian inferences listed in Table 1 for possible scenarios: BNS- GRBKasen top (red), NSBH-GRBtop (green), SNCol-GRBtop (or- ange), and SN98bw-GRBtop (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' The observational data of GRB 211211A in X-ray-1keV, radio-6GHz, UV, optical, and NIR band as discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2 are shown as black dots, whereas black triangles refer to upper detection limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' inferred GRB afterglow and kilonova properties for both BNS-KN-Kasen and BNS-KN-Bulla and connect infor- mation about the ejecta and debris disk to the BNS properties following Dietrich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Henkel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' (2022) for a recent discussion about uncertain- ties in the employed numerical-relativity informed phe- nomenological relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 4, we show our inference results for a possible BNS source using BNS-GRBKasen top , BNS-GRBKasen Gauss, and BNS-GRBBulla top and contrast these to the prior probabil- 28 24 20 16 i 28 24 20 16 J 10−2 10−1 100 101 Time [days] 28 24 20 16 K BNS-GRBKasen top GRBtop BNS-GRBBulla top 28 24 20 16 r Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Best-fitting light curves from joint Bayesian in- ferences of BNS-GRBBulla top (yellow) and BNS-GRBKasen top (red) compared to a stand-alone GRBtop inference (black) for op- tical and NIR bands on a logarithmic time scale in days since trigger time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' ity regions for each parameter, in order to show how con- straining the observational data is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Comparing inference results for BNS-GRBKasen Top and BNS-GRBKasen Gauss, we find that estimated source masses and tidal deformabilies are very similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' For the top-hat jet structure simula- tion, we find that a BNS merger with a primary mass of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='55+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='54 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='42M⊙ and a secondary mass of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='34+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='25 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='40M⊙ was the likely progenitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' The associated dimensionless tidal deformability of the system lies within ˜Λ = 299+1041 −274 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' With regard to a similar analysis for BNS-GRBBulla Top , we find a primary mass of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='56+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='43 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='34M⊙ and a sec- ondary mass of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='29+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='20 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='29M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' The corresponding tidal deformability is 353+598 −264.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Comparing estimated masses for BNS-GRBKasen Top and BNS-GRBBulla Top , we find overall good agreement within the stated uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Con- 8 cerning the tidal deformability, we find that the BNS- KN-Bulla model provides tighter constraints compared to those extracted with the BNS-KN-Kasen model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' We expect this deviation to originate from the fact that the BNS-KN-Bulla model provides more detailed informa- tion on the estimated wind and dynamical ejecta masses, while the BNS-KN-Kasen model provides a generic es- timate of the total ejecta mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Overall, our estimated masses are consistent with Rastinejad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' (2022), who concluded that GRB 211211A originated from a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='4 M⊙+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='3 M⊙ BNS merger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' We expect that the remaining small differences are caused by the different analysis of the observed GRB 211211A data and by the fact that Rastinejad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' (2022) assumed the inclination angle, under which the binary was observed, to be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Moreover, Rastinejad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' (2022) assumed a fixed equation of state from the EOS set of Dietrich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' (2020) using additional information from Nicholl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' In contrast, we leave the inclination angle as a free parameter in our analysis and use the updated EOS set of Huth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' This set incorporates information from theoretical nuclear-physics computations and from astrophysical observations of neutron stars such as Dietrich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' (2020), but also heavy-ion collision experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' With regard to investigated binary merger scenarios, we find that the inferred inclination angle is around θObs ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='02+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='02, while larger incli- nation angles of approximately θObs ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='07+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='11 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='06 are estimated for the two considered supernova scenarios (see Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Rastinejad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' (2022) deduced a total r-process ejecta mass of Mej = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='047+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='026 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='011M⊙, of which 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='02 M⊙ correspond to lanthanide-rich ejecta, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='01 M⊙ to intermediate-opacity ejecta, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='01 M⊙ to lanthanide- free material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' With our reference inference result from BNS-GRBKasen top , we find a total ejecta mass of M BNS ej,Kasen = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='021+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='017 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='013M⊙ which is broadly in agree- ment given the uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Concerning our analy- sis based on BNS-GRBBulla top , we found a total ejecta mass of M BNS ej = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='031+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='033 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='018M⊙, of which 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='015M⊙ can be attributed to lanthanide-rich ejecta, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='011M⊙ to intermediate-opacity mass, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='002M⊙ to lanthanide- free material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' For completeness, we have performed a similar inves- tigation for our NSBH-GRBtop and NSBH-GRBGauss models to infer the corresponding NSBH properties by making use of the relations provided in Foucart et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' (2018) and Kr¨uger & Foucart (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Although the ob- servational data does not provide a strong constraint on the NSBH source properties, our NSBH-GRBtop anal- ysis suggests that an NSBH merger with a BH mass 299.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='40 597.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='66 264.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='01 301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='17 +968.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='83 _ 276.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='78 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Component masses m1,2 and the dimension- less tidal deformability ˜Λ based on our inference results of BNS-GRBKasen Gauss (orange), BNS-GRBKasen Top (red) and BNS- GRBBulla Top (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Different shadings mark the 68%, 95%, and 99% confidence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' For the 1D posterior probability distributions, we give the 90% confidence interval (dashed lines) and report median values above each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Grey shaded areas give the prior probability regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='18+8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='54 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='34M⊙ and an NS mass of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='39+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='83 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='85M⊙ could have been the progenitor of GRB 211211A, with a total ejecta mass of M NSBH ej = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='008+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='012 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='006 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Likewise, the BH spin is weakly constrained to χ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='57 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='74 for the NSBH-GRBtop inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Our inferred NS masses are in agreement with previous GW population analy- ses (Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2019, 2021a,b) and with the maximum non-spinning NS mass of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='7+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='4M⊙ estimated at 90% credibility by Ye & Fishbach (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Within the esti- mated uncertainties, the inferred BH mass is close to the NSBH mass gap for which the lightest BH masses were estimated to be ∼ 5M⊙ (¨Ozel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Farr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Influence of the prior choice Finally, we discuss the influence of a different lu- minosity distance prior on our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' The distance of GRB 211211A was relatively precisely estimated based on the redshift of the potential host galaxy, z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='0763 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='0002 (Rastinejad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' How- ever, we are generally interested in the influence of a 750 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='54 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='54 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='56 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='43 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='55 57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='42 500 250 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='29 1000 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='4 m2[Mo] = 500 0 6 352.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='83+ +1041.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='31 273.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='52 2000 4500 1000 0 mul m?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' V BNS-GRBKasen BNS-GRBBulla Gauss Lop BNS-GRBKasen Prior Top=9 wide uniform luminosity distance prior on our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' For this reason, we widen the prior range and allow a distance between 0 and 1000 Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Following the procedure in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='1, we have com- puted the logarithmic Bayes factors and found that BNS-GRBKasen top remains to be the best-fitting model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Moreover, the differences in logarithmic Bayes factors between BNS-KN-Bulla and BNS-KN-Kasen remain the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Overall, the differences with regard to the indi- vidual Bayes factors as presented in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 1 are small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' However, the SN and collapsar scenarios are now equally disfavored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Hence, our main conclusions remain valid also for the wider distance prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' We have investigated the posterior probability dis- tributions obtained for a wide uniform distance prior and compare these with the ones obtained for a narrow Gaussian distance prior setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 5, we show an example for the obtained luminosity distance and the total ejecta mass distributions using GRBKasen top .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' As can be seen, the wide distance prior leads to a noticeably weaker constraint on the distance and the total ejecta mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' The latter is caused by a degeneracy between the luminosity distance and the ejecta mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Gener- ally, larger ejecta masses could compensate for larger distances and vice versa, which explains the shape of the 2D correlation plot of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Similarly (not shown in the figure), also the SNCol model predicts higher ejecta masses for larger distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' With respect to the SN- GRB and the GRB inferences, the GRB isotropic en- ergy, log10(EK,iso), tends to increase for larger distances, which is expected as brighter signals can be detected to further distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' CONCLUSION In this paper, we have performed multiple multi- wavelength analyses for GRB 211211A assuming four different scenarios, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=', a BNS merger, an NSBH merger, an rCCSN, as well as a CCSN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' On the basis of joint multi-wavelength Bayesian inferences combining respec- tive kilonova or SN models with a gamma-ray burst af- terglow model, we studied for which scenario we find the highest statistical evidence to explain the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' We summarize our main conclusions: (i) We find strong statistical evidence for a BNS merger scenario;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' However, we can not fully rule out other scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' (ii) Our study confirms that GRB 211211A can not solely be explained as a GRB afterglow and that an additional emission process (likely related to r-process nucleosynthesis) is required for a good Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Corner plot for BNS-GRBKasen top with a narrow Gaussian luminosity distance prior centered around 350 Mpc (orange) and a wide uniform luminosity distance prior rang- ing up to 1000 Mpc (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' The inferred model parameters are shown at 68%, 95%, and 99% confidence (shadings from light to dark).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' For the 1D posterior probability distributions, we report the median values and show the 90% confidence intervals as dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' description of the observational data, mostly in late i-band and K-band data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' (iii) Assuming a BNS origin, our study suggests that this system was a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='55+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='54 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='42M⊙ - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='34+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='25 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='40M⊙ bi- nary, leading to a total ejecta mass of M BNS ej = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='021+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='017 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='013M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Assuming a NSBH origin of GRB 211211A, our study suggests a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='39+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='83 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='85 - 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='18+8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='54 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='34M⊙ system with a total ejecta mass of M NSBH ej = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='008+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='012 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='006 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' (iv) The results discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='2 showed that near- infrared data at late times are essential to inves- tigate the astrophysical origin of interesting tran- sient objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' ACKNOWLEDGEMENT This project has received financial support from the CNRS through the MITI interdisciplinary programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' thanks A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' de Ugarte Postigo for sharing CAHA data for this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='A also thanks Rahul Gupta, Jirong Mao, Robert Strausbaugh, Dong Xu, Jinzhong Liu, Daniele Malesani, Andrew Levan, and the MIT- 350.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='00+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='43 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='52 323.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='66 1000 500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='73 1500 log1o(mei)[Mo] 6 1000 500 X 2 D log1o(meji) [Mo] narrow Gaussian prior wide uniform prior10 SuME group for their useful comments on their obser- vations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='A thanks T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Hussenot for the discussion re- lated to GRB 211211A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' acknowledges support by the European Union’s Horizon 2020 Programme under the AHEAD2020 project (grant agreement n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 871158).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' The work of I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' was supported by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Department of Energy, Office of Science, Office of Nuclear Physics, under Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' DE-AC52-06NA25396, by the Lab- oratory Directed Research and Development program of Los Alamos National Laboratory under Project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 20220541ECR, and by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Department of Energy, Office of Science, Office of Advanced Scientific Com- puting Research, Scientific Discovery through Advanced Computing (SciDAC) NUCLEI program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Anand ac- knowledges support from the National Science Founda- tion GROWTH PIRE grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 1545949.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' REFERENCES Abbott, B.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=', Liu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=', & Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2022, arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='09675 Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=', Xiong, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=', Li, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2021, GRB Coordinates Network, 31236, 1 ¨Ozel, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=', Psaltis, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=', Narayan, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=', & McClintock, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2010, The Astrophysical Journal, 725, 1918 12 APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' OBSERVATIONAL DATA SELECTION MJD [days] Filter Telescope/Instrument Transient [AB mag] Reference u-band 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='005 u UVOT 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='2 (Rastinejad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2022) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='06 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='01 u UVOT 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='2 (Rastinejad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2022) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='19 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='01 u UVOT 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='1 (Rastinejad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2022) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='66 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='01 u UVOT > 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='3 (Rastinejad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2022) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='86 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='01 u UVOT > 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='9 (Rastinejad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2022) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='19 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='01 u UVOT > 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='9 (Rastinejad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2022) g-band 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='260 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='081 g′ MITSuME 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='2 (Ito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2021) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='69 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='01 g NOT 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='04 (Rastinejad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2022) r-band 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='431 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='020 r NEXT 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='1 (Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2021) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='69 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='01 r NOT 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='81 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='05 (Rastinejad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2022) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='425 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='001 r′ GIT > 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='15 (Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2021) i-band 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='68 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='016 i CAHA-CAFOS 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='75 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='08 (Rastinejad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2022) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='70 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='01 i NOT 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='1 (Rastinejad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2022) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='68 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='01 i CAHA-CAFOS 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='15 (Rastinejad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2022) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='11 ± < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='01 i Gemini-GMOS 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='03 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='3 (Rastinejad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2022) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='08 ± < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='01 i Gemini-GMOS > 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='49 (Rastinejad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2022) J-band 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='445 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='024 z NEXT 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='3 (Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2021) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='72 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='02 H TNG > 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='9 (D’Avanzo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2021) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='96 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='014 J MMT-MMRIS 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='17 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='35 (Rastinejad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2022) K-band 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='058 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='005 K Gemini-NIRI 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='41 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='14 (Rastinejad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2022) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='10 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='005 K Gemini-NIRI 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='2 (Rastinejad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2022) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='94 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='02 K MMT-MMRIS 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='3 (Rastinejad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2022) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='98 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='01 K MMT-MMRIS 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='3 (Rastinejad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2022) Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Multi-wavelength observations of the counterpart and the host galaxy of GRB 211211A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Magnitudes are corrected for foreground Galactic extinction according to AV = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='048 mag (Rastinejad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 13 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' INFERENCE SETTINGS All parameter estimation runs were performed using the nuclear physics and multi-messenger astronomy framework NMMA (Pang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' In this framework, joint Bayesian inferences of electromagnetic signals are carried out on the basis of the nested sampling algorithm implemented in pymultinest (Buchner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' (2014)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Each simulation used 2048 live points, and the prior settings for each of the employed models, as well as the median values and 90% credible ranges, are provided in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Parameter Prior Posterior BNS-GRBBulla top BNS-GRBKasen top NSBH-GRBtop SNCol-GRBtop SN98bw-GRBtop GRB-M log10(EK,iso) [erg] [47, 55] 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='36+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='14 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='15 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='28+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='86 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='38 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='77+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='79 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='67 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='80+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='21 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='56 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='45+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='08 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='73 θObs [rad] [0, π 4 ] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='02+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='02+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='02+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='07+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='07 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='07+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='14 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='05 θc [rad] [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='01, π 10] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='05+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='08 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='05+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='09 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='05+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='08 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='09+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='10+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='14 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='07 log10(n) [cm−3] [-6, 2] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='39+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='38 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='78 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='46+4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='95 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='54 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='45+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='41 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='09 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='08+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='92 −3.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='20+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='19 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='16 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='53+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='25 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='24 log10(ϵe) [-5, 0] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='80+0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='63+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='63 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='10+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='23 log10(ϵB) [-10, 0] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='44+4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='42 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='60 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='01+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='01 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='62 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='45+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='45 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='44 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='60+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='66 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='68+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='68 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='48 DL[Mpc] N(350, 2) 350.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='07+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='42 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='46 350.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='00+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='43 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='52 350.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='01+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='46 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='49 350.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='18+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='60 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='59 349.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='98+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='68 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='36 BNS-KN-Bulla log10(M ej dyn) [M⊙] [-3, -1] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='78+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='70 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='62 log10(M ej wind) [M⊙] [-3, -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='5] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='98+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='55 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='57 Φ [deg] [15, 75] 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='42+12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='58 −30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='08 BNS-KN-Kasen log10(Mej) [M⊙] [-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='5, -1] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='68+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='37 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='36 log10(vej) [c] [-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='8, -1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='90+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='53 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='51 log10(Xlan) [-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='5, -1] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='88+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='88 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='05 NSBH-KN-Bulla log10(M ej dyn) [M⊙] [-3, -1] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='51+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='73 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='49 log10(M ej wind) [M⊙] [-3, -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='5] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='49+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='67 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='51 SNCol Mej [M⊙] [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='5] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='06+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='04 MNi [M⊙] [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='03] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='00 vej [c] [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='5] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='21+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='04 Mrp [M⊙] [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='05] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='01+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='01 Ψmix [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='9] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='73+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='17 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='35 SN98bw Smax [0, 60] 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='86+24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='45 −24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='82 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Model parameters and prior bounds employed in our Bayesian inferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' We report median posterior values at 90 % credibility from simulations that were run with Top-hat jet structure and with a narrow Gaussian luminosity distance prior N(µ, σ), with mean µ = 350 Mpc and standard deviation σ = 2 Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' We employ a conditional prior on the inclination angle depending on the jet core opening angle, p(θObs|θc), using a truncated Gaussian distribution, NT (µ, σ), where µ = 0 and σ = θc .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 14 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' INFERENCE RESULTS In the following, we present the posterior distribution for our reference model GRBKasen top employing a narrow distance prior centered around 350 Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Figure 6 summarizes our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' As discussed in the main text, we obtain a total ejecta mass of 10−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='68+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='37 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='36 M⊙, which is generally consistent with previous findings in the literature and an average velocity of 10−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='9+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='53 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='51c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Interestingly, our analysis prefers a higher lanthanide fraction compared to the one inferred for AT2017gfo using the same kilonova models (Coughlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 2018), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=', we predict a slightly redder kilonova (similar to Rastinejad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' (2022) who predict a larger mass of the red component, but opposite to e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Mei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' (2022)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Considering the obtained GRB posteriors, we find a double peak structure in our posteriors and a clear low n0 - high ϵB - high p peak, as well as, a high n0 - low ϵB - low p -peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' While this double peak structure might be caused by the small set of observational data and potential degeneracies, it could also be an indicator of the missing input physics of the employed GRB models, in particular, the emission from the reverse shock that might be important at early times and wind interstellar medium density density profile, the absence of which might be responsible for the high n0 - low ϵB - low p -peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' 0 500 1000 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='88+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='88 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='05 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='4 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='0 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='6 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='2 log10(mej)[M⊙] 0 1000 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='68+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='37 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='36 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='6 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='8 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='4 log10(vej)[c] 0 500 1000 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='90+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='53 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='18 ι[rad] 0 2000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='02+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='02 345 350 355 DL [Mpc] 0 1000 2000 350.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='00+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='43 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='52 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='0 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='5 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='0 log10(E0)[erg] 0 500 1000 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='28+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='86 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='38 −4 −2 0 log10(n0)[cm−3] 0 500 1000 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='46+4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='95 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='24 θc[rad] 0 2000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='05+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='09 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='04 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='50 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='75 p 0 500 1000 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='32+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='30 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='26 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='5 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='0 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='5 ϵe 0 2500 5000 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='28+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='28 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='81 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='0 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='2 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='4 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='6 log10(χlan) −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='5 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='0 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='5 ϵB −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='4 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='0 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='6 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='2 log10(mej)[M⊙] −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='6 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='8 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='4 log10(vej)[c] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='18 ι[rad] 345 350 355 DL [Mpc] 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='0 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='5 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='0 log10(E0)[erg] −4 −2 0 log10(n0)[cm−3] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='24 θc[rad] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='50 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='75 p −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='5 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='0 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='5 ϵe −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='5 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='0 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='5 ϵB 0 1000 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='01+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='01 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content='62 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' Corner plot for BNS-GRBKasen top with a narrow Gaussian luminosity distance prior centered around 350 Mpc, in which we show the inferred parameters at 68%, 95%, and 99% confidence (shadings from light to dark).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} +page_content=' For the 1D posterior probability distributions, we report the median values and show the 90% confidence intervals as dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9A0T4oBgHgl3EQfGf_e/content/2301.02049v1.pdf'} diff --git a/b9AzT4oBgHgl3EQfZvxg/content/tmp_files/2301.01356v1.pdf.txt b/b9AzT4oBgHgl3EQfZvxg/content/tmp_files/2301.01356v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..58fec7d1d4ea81fd45baec215ca34b8c42a33433 --- /dev/null +++ b/b9AzT4oBgHgl3EQfZvxg/content/tmp_files/2301.01356v1.pdf.txt @@ -0,0 +1,2660 @@ +Provably Fast and Space-Efficient Parallel Biconnectivity +Xiaojun Dong +UC Riverside +xdong038@ucr.edu +Letong Wang +UC Riverside +lwang323@ucr.edu +Yan Gu +UC Riverside +ygu@cs.ucr.edu +Yihan Sun +UC Riverside +yihans@cs.ucr.edu +Abstract +Biconnectivity is one of the most fundamental graph prob- +lems. The canonical parallel biconnectivity algorithm is the +Tarjan-Vishkin algorithm, which has 𝑂(𝑛 +𝑚) optimal work +(number of operations) and polylogarithmic span (longest de- +pendent operations) on a graph with 𝑛 vertices and 𝑚 edges. +However, Tarjan-Vishkin is not widely used in practice. We +believe the reason is the space-inefficiency (it generates an +auxiliary graph with 𝑂(𝑚) edges). In practice, existing par- +allel implementations are based on breath-first search (BFS). +Since BFS has span proportional to the diameter of the graph, +existing parallel BCC implementations suffer from poor per- +formance on large-diameter graphs and can be even slower +than the sequential algorithm on many real-world graphs. +We propose the first parallel biconnectivity algorithm +(FAST-BCC) that has optimal work, polylogarithmic span, +and is space-efficient. Our algorithm first generates a skele- +ton graph based on any spanning tree of the input graph. +Then we use the connectivity information of the skeleton to +compute the biconnectivity of the original input. All the steps +in our algorithm are highly-parallel. We carefully analyze +the correctness of our algorithm, which is highly non-trivial. +We implemented FAST-BCC and compared it with exist- +ing implementations, including GBBS, Slota and Madduri’s +algorithm, and the sequential Hopcroft-Tarjan algorithm. +We ran them on a 96-core machine on 27 graphs, including +social, web, road, 𝑘-NN, and synthetic graphs, with signif- +icantly varying sizes and edge distributions. FAST-BCC is +the fastest on all 27 graphs. On average (geometric means), +FAST-BCC is 5.1× faster than GBBS, and 3.1× faster than +the best existing baseline on each graph. +CCS Concepts: • Theory of computation → Shared mem- +ory algorithms; Graph algorithms analysis; Parallel al- +gorithms. +Keywords: Parallel Algorithms, Graph Algorithms, Bicon- +nectivity +1 +Introduction +Computing the biconnected components is one of the most +fundamental graph problems. Given an undirected graph +𝐺 = (𝑉, 𝐸) with 𝑛 = |𝑉 | vertices and 𝑚 = |𝐸| edges, a +connected component (CC) is a maximal subset in 𝑉 such +that every two vertices in it are connected by a path. A +biconnected component (BCC) (or blocks) is a maximal +subset 𝐶 ⊆ 𝑉 such that 𝐶 is connected and remains con- +nected after removing any vertex 𝑣 ∈ 𝐶. In this paper, we use +BCC (or CC) for both the biconnected (or connected) com- +ponent in the graph and the problem to compute all BCCs +(or CCs). BCC has extensive applications such as planarity +testing [7, 23, 44], centrality computation [46, 57, 58], and +network analysis [6, 54]. +Sequentially, the Hopcroft-Tarjan algorithm [43] can com- +pute the BCCs of a graph in 𝑂(𝑛 + 𝑚) time. However, this +algorithm requires generating a spanning tree of 𝐺 based +on the depth-first search (DFS), which is considered hard to +be parallelized [55]. Later, Tarjan and Vishkin proposed the +canonical parallel BCC algorithm along with the Euler-tour +technique [63]. It uses an arbitrary spanning tree (AST) (a +spanning tree with any possible shape) of the graph instead +of the depth-first tree. Tarjan-Vishkin algorithm has𝑂(𝑛+𝑚) +optimal work (number of operations) and polylogarithmic +span (longest dependent operations), assuming an efficient +parallel CC algorithm. +Although the Tarjan-Vishkin algorithm is theoretically +considered “optimal” in work and span, significant challenges +still remain in achieving a high-performance implementa- +tion in practice. The main issue in Tarjan-Vishkin is space- +inefficiency. Tarjan-Vishkin generates an auxiliary graph +𝐺 ′ = (𝑉 ′, 𝐸′) (which we refer to as the skeleton), where +every edge 𝑒 ∈ 𝐸 maps to a vertex in 𝑉 ′. Tarjan and Vishkin +showed that computing CC on𝐺 ′ gives the BCC on𝐺, and we +refer to this step as the connectivity phase. This skeleton- +connectivity framework is adopted in many later papers. +It first generates a skeleton, which is an auxiliary graph +𝐺 ′ from 𝐺, and then finds the CCs on 𝐺 ′ that reflects BCC +information on the input graph 𝐺. Unfortunately, in Tarjan- +Vishkin, generating the skeleton 𝐺 ′ (which takes 𝑂(𝑚) extra +space) and computing CC on 𝐺 ′ greatly increase the memory +usage and slow down the performance. +In practice, most existing parallel BCC implementations +also follow the skeleton-connectivity framework but over- +come the space issue by using other skeletons based on +breadth-first search (BFS) trees [24, 25, 28, 30, 38, 62, 66]. +They are based on sparse certificates [26], and more discus- +sions are given in Sec. 3.3. These algorithms either use skele- +tons with 𝑂(𝑛) size [24, 25, 28, 38, 66] or maintain implicit +skeletons with 𝑂(𝑛) auxiliary space [30, 62]. We say a BCC +algorithm is space-efficient if it uses 𝑂(𝑛) auxiliary space +(other than the input graph). However, since computing BFS +has span proportional to the graph, these BFS-based algo- +rithms can be fast on small-diameter graphs (e.g., social and +web graphs), but have poor performance on large-diameter +graphs (e.g., 𝑘-nn and road graphs). In our experiments, we +observe that existing parallel implementations can even be +1 +arXiv:2301.01356v1 [cs.DS] 3 Jan 2023 + +Ours GBBS SM'14 SEQ +Ours GBBS SM'14 SEQ +Social +YT +5.88 +4.36 +3.15 +1.00 +K-NN +HH5 +7.01 1.14 +n +1.00 +OK +30.51 19.91 +5.66 +1.00 +CH5 +4.11 0.37 +n +1.00 +LJ +17.92 11.77 +n +1.00 +GL2 +6.24 1.64 +n +1.00 +TW +34.21 17.42 +2.40 +1.00 +GL5 +8.53 1.44 +n +1.00 +FT +39.26 18.93 10.22 +1.00 +GL10 10.59 4.31 +n +1.00 +MEAN 21.23 12.75 +4.57 +1.00 +GL15 11.88 5.91 +n +1.00 +Web +GG +8.92 +5.65 +n +1.00 +GL20 11.84 6.88 +n +1.00 +SD +29.74 16.46 +n +1.00 +COS5 14.16 6.86 +n +1.00 +CW +30.37 17.52 +n +1.00 +MEAN +8.68 2.42 +- +1.00 +HL14 32.46 19.96 +n +1.00 +Synthetic +SQR 18.50 1.59 10.56 1.00 +HL12 33.99 29.15 +n +1.00 +REC +12.48 0.36 3.02 1.00 +MEAN 24.53 15.68 +- +1.00 +SQR' +8.06 0.85 +n +1.00 +Road +CA +5.15 +0.55 +n +1.00 +REC' +7.81 0.48 +n +1.00 +USA +6.69 +0.49 +0.60 +1.00 +Chn7 11.97 0.04 0.08 1.00 +GE +10.77 +1.43 +2.44 +1.00 +Chn8 11.97 0.04 0.06 1.00 +MEAN +7.18 +0.73 +1.21 +1.00 +MEAN 11.30 0.27 0.18 1.00 +TOTAL MEAN 12.89 2.50 0.96 1.00 +1 +0 +.5 +2 +4 +8 +16 32 >32 +MEAN = geometric mean +n = no support +Figure 1. The heatmap of relative speedup for parallel +BCC algorithms over the sequential Hopcroft-Tarjan algo- +rithm [43] using 96 cores (192 hyper-threads). Larger/green +means better. The numbers indicate how many times a parallel +algorithm is faster than sequential Hopcroft-Tarjan (< 1 means +slower). The two baseline algorithms are from [30, 62]. Complete +results are in Tab. 2. +slower than sequential Hopcroft-Tarjan on many real-world +graphs (see GBBS [30] and SM’14 [62] in Fig. 1). +In this paper, we give the first space-efficient (𝑂(𝑛) +auxiliary space) parallel biconnectivity algorithm that +has efficient 𝑂(𝑚 + 𝑛) work and polylogarithmic span. +Our skeleton 𝐺 ′ is based on an arbitrary spanning tree (AST). +Unlike Tarjan-Vishkin, our 𝐺 ′ is a subgraph of 𝐺 and can +be maintained implicitly in 𝑂(𝑛) auxiliary space. The key +idea is to carefully identify some fence edges, which indicate +the “boundaries” of the BCCs. At a high level, we categorize +all graph edges into fence tree edges, plain (non-fence) tree +edges, back edges, and cross edges. Our skeleton 𝐺 ′ contains +the plain tree edges and cross edges. Using 𝑂(𝑛) space, we +can efficiently determine the category of each edge in 𝐺. +When processing the skeleton, we use the input graph 𝐺 +but skip the fence and back edges. We show that the BCC +information of𝐺 can be constructed from the CC information +of 𝐺 ′ plus some simple postprocessing. Since our algorithm +is based on Fencing an Arbitrary Spanning Tree, we call +our algorithm FAST-BCC. More details of FAST-BCC are +in Fig. 2. We note that conceptually our algorithm is simple, +but the correctness analysis is highly non-trivial. +We implement our theoretically-efficient FAST-BCC al- +gorithm and compare it to the state-of-the-art parallel BFS- +based BCC implementations GBBS [30] and SM’14 [62]. We +also compare FAST-BCC to the sequential Hopcroft-Tarjan +algorithm. We test 27 graphs, including social, web, road, +𝑘-NN, and synthetic graphs, with significantly varying sizes +and edge distributions. The details of the graphs and results +are given in Tab. 2. We also show the relative running time +in Fig. 1, normalized to the sequential Hopcroft-Tarjan. +On a machine with 96 cores, FAST-BCC is the fastest on +all tested graphs. We use the geometric means to compare +the “average” performance across multiple graphs. Due to +work- and space-efficiency, our algorithm running on one +core is competitive with Hopcroft-Tarjan (2.8× slower on +average). Polylogarithmic span leads to good parallelism +for all types of graphs (15–66× self-relative speedup on +average). On small-diameter graphs (social and web graphs), +although GBBS and SM’14 also achieve good parallelism, +FAST-BCC is still 1.2–2.1× faster than the best of the two, +and is 5.9–39× faster than sequential Hopcroft-Tarjan. For +large-diameter graphs (road, 𝑘-nn, grid, and chain graphs), +existing BFS-based implementations can perform worse than +Hopcroft-Tarjan. Due to the low span, FAST-BCC is 1.7–295× +faster than GBBS (10× on average), and 4.1–18.5× faster than +sequential Hopcroft-Tarjan (9.2× on average). On all graphs, +FAST-BCC is 3.1× faster on average than the best of the three +existing implementations. Our code is publicly available [35]. +2 +Preliminaries +Computational Model. We use the work-span (or work- +depth) model for fork-join parallelism with binary forking to +analyze parallel algorithms [14, 29], which is recently used in +many papers on parallel algorithms [3, 9, 10, 12, 13, 15–21, 32– +34, 40, 41, 60, 69]. We assume a set of threads that share a +common memory. A process can fork two child software +threads to work in parallel. When both children complete, +the parent process continues. The work of an algorithm is +the total number of instructions and the span (depth) is the +length of the longest sequence of dependent instructions in +the computation. We say an algorithm is work-efficient if its +work is asymptotically the same as the best sequential algo- +rithm. We can execute the computation using a randomized +work-stealing scheduler [5, 22] in practice. We assume unit- +cost atomic operation compare_and_swap(𝑝, 𝑣old, 𝑣new) (or +CAS), which atomically reads the memory location pointed +to by 𝑝, and write value 𝑣new to it if the current value is 𝑣old. +It returns true if successful and false otherwise. +Notation. Given an undirected graph 𝐺 = (𝑉, 𝐸), we use +𝑛 = |𝑉 |, 𝑚 = |𝐸|. Let diam(𝐺) be the diameter of 𝐺, and 𝑥–𝑦 +be an edge between 𝑥 and 𝑦. CC and BCC are as defined +in Sec. 1. An articulation point (or cut vertex) is a vertex +s.t. removing it increases the number of CCs. A bridge (or +cut edge) is an edge s.t. removing it increases the number of +CCs. A spanning tree𝑇 of a connected graph 𝐺 is a spanning +subgraph of 𝐺 that contains no cycles. The spanning forest is +defined similarly if 𝐺 is disconnected. For simplicity, we as- +sume 𝐺 is connected, but our algorithm and implementation +work on any graph. Given a graph 𝐺 and a rooted spanning +tree 𝑇, an edge is a tree edge if it is in 𝑇. A non-tree edge +is a back edge if one endpoint is the ancestor of the other +endpoint, and a cross edge otherwise. Fig. 2 Step 3 shows an +2 + +𝐺 = (𝑉, 𝐸) : Input Graph +𝑇 = (𝑉, 𝐸𝑇 ): A spanning tree in 𝐺 +𝑎,𝑏,𝑐,𝑢, 𝑣,ℎ,𝑤,𝑥,𝑦,𝑧,𝑢′, 𝑣′,𝑐′ · · · ∈ 𝑉 +: Vertices in 𝐺 +𝑥–𝑦 ∈ 𝐸 +: An edge in 𝐺 +𝐶,𝐶𝑖 +: A BCC in 𝐺 +𝑇𝑢 +: 𝑢’s subtree in 𝑇 +ℎ𝐶 +: The BCC head of 𝐶 +𝑝(𝑢) +: 𝑢’s parent in 𝑇 +𝑥 ~ 𝑦 +: A tree path in 𝑇 +𝑃 = 𝑥–𝑦–· · · : A path +𝐺′ +: The skeleton +Fence edge: (𝑝(𝑣), 𝑣) ∈ 𝐸𝑇 , � (𝑥,𝑦) ∈ 𝐸, s.t. 𝑥 ∈ 𝑇𝑣 and 𝑦 ∉ 𝑇𝑝 (𝑣) +(no edge from 𝑣’s subtree escapes from 𝑝(𝑣)’s subtree) +Plain edge : (𝑝(𝑣), 𝑣) ∈ 𝐸𝑇 , (𝑝(𝑣), 𝑣) is not a fence edge +Back edge, Cross edge : Edges in 𝐸 \ 𝐸𝑇 , defined as usual +Skeleton 𝐺′ = (𝑉, 𝐸′) in FAST-BCC : 𝐸′ = {plain & cross edges} +Table 1. Notations and terminologies in this paper. +illustration. If 𝑇 is a BFS tree, there are no back edges; if 𝑇 is +a DFS tree, there are no cross edges. We use 𝑥 ~𝑦 to denote +the tree path between 𝑥 and 𝑦 on 𝑇. We denote the parent +of vertex 𝑢 as 𝑝(𝑢), and the subtree of 𝑢 as 𝑇𝑢. The notation +used in this paper is given in Tab. 1. +We use 𝑂(𝑓 (𝑛)) with high probability (whp) in 𝑛 to mean +𝑂(𝑐𝑓 (𝑛)) with probability at least 1 − 𝑛−𝑐 for 𝑐 ≥ 1. +Euler tour technique (ETT). ETT is proposed by Tarjan +and Vishkin [63] in their BCC algorithm to root a spanning +tree. Later, ETT becomes a widely-used primitive in both +sequential and parallel settings, including computational ge- +ometry [2], graph algorithms [4, 27, 65], maintaining subtree +or tree path sums [29], and many others. ETT is needed in +Tarjan-Vishkin because when an arbitrary spanning tree is +generated for a graph (e.g., from a CC algorithm), it is not +rooted, and thus we do not have the parent-child information +for the vertices. Given an unrooted tree 𝑇 with 𝑛 − 1 edges, +ETT finds an Euler tour of 𝑇, which is a cycle traversing +each edge in 𝑇 exactly twice (once in each direction). ETT +first constructs a linked list on the 2𝑛 − 2 directed tree edges, +and runs list ranking on it. We refer the audience to the text- +books on parallel algorithms [45, 56] for more details on ETT. +Using the semisort algorithm from [14, 42] and list ranking +from [14], ETT costs 𝑂(𝑛) expected work and 𝑂(log𝑛) span +whp. Given𝑇, we can set any vertex as the root of𝑇, and use +ETT to determine the directions of the edges. We can then +determine the parent of any vertex, and whether an edge is +a tree edge, back edge, or cross edge in 𝑂(1) work. +3 +Existing BCC Algorithms +This section reviews the existing BCC algorithms and imple- +mentations. We will use the skeleton-connectivity framework +to describe the existing BCC algorithms. The skeleton phase +generates a skeleton 𝐺 ′ from 𝐺, which is an auxiliary graph. +Then the connectivity phase computes the connectivity on +𝐺 ′ to construct the BCCs of 𝐺. Existing BCC algorithms can +be categorized by how the skeleton 𝐺 ′ is generated. The +Hopcroft-Tarjan algorithm uses DFS-based skeletons; the +Tarjan-Vishkin Algorithm generates a skeleton based on an +arbitrary spanning tree (AST); almost all other BCC algo- +rithms (see Sec. 3.3) use BFS-based skeletons. +3.1 +The Hopcroft-Tarjan Algorithm +Sequentially, Hopcroft-Tarjan BCC algorithm [43] has 𝑂(𝑛 + +𝑚) work using a depth-first search (DFS) tree 𝑇. Based on +𝑇, two tags first[·] and low[·] are assigned to each vertex. +first[𝑣] is the preorder number of each vertex in 𝑇. low[𝑣] +gives the earliest (smallest preorder) vertex incident on any +vertex 𝑢 ∈ 𝑇𝑣 via a non-tree edge and 𝑢 itself. More formally, +low[𝑣] = min{𝑤1[𝑢] | 𝑢 ∈ 𝑉 is in the subtree rooted at 𝑣} +𝑤1[𝑢] = min{{first[𝑢]} ∪ {first[𝑢′] | (𝑢,𝑢′) ∉ 𝑇 }} +To compute the BCCs, an additional stack is maintained. +Each time we visit a new edge, it is pushed into the stack. +When an articulation point 𝑝(𝑢) is found by 𝑢 (low[𝑢] ≥ +first[𝑝(𝑢)]), edges are popped from the stack until 𝑢–𝑝(𝑢) is +popped. These edges and the relevant vertices form a BCC. +Conceptually, the skeleton in Hopcroft-Tarjan is the DFS +tree without the “fence edges” of 𝑢–𝑝(𝑢) when low[𝑢] ≥ +first[𝑝(𝑢)]. This insight also inspires our BCC algorithm. +3.2 +The Tarjan-Vishkin Algorithm +Hopcroft-Tarjan uses a DFS tree as the skeleton, but DFS is in- +herently serial and hard to be parallelized [55]. To parallelize +BCC, the Tarjan-Vishkin algorithm [63] uses an arbitrary +spanning tree (AST) instead of a DFS tree. This spanning +tree 𝑇 can be obtained by any parallel CC algorithm. The +algorithm then uses ETT (which is also proposed in that pa- +per) to root the tree 𝑇 (see Sec. 2). Then the algorithm builds +a skeleton 𝐺 ′ = (𝐸, 𝐸′) and runs a connectivity algorithm +on it. We describe 𝐺 ′ in more details in Appendix A, and +only briefly review it here. The vertices in 𝐺 ′ correspond +to the edges in 𝐺1. To determine the edges in 𝐺 ′, the algo- +rithm uses four tags (first[·], last[·], low[·], and high[·]) for +each vertex. Here first[𝑢] and last[𝑢] are the first and last +appearance of vertex 𝑢 in the Euler tour (note that this is +not the same first[·] in Hopcroft-Tarjan, but conceptually +equivalent). low[·] is the same as defined in Hopcroft-Tarjan, +and high[·] is defined symmetrically: +high[𝑣] = max{𝑤2[𝑢] | 𝑢 ∈ 𝑉 is in the subtree rooted at 𝑣} +𝑤2[𝑢] = max{{first[𝑢]} ∪ {first[𝑢′] | (𝑢,𝑢′) ∉ 𝑇 }} +All tags can be computed in 𝑂(𝑛 + 𝑚) expected work and +𝑂(log𝑛) span whp using ETT. Tarjan-Vishkin then finds the +CCs on 𝐺 ′ to compute the BCCs of 𝐺. However, 𝐺 ′ in Tarjan- +Vishkin can be large, making the algorithm less practical. +Assuming an efficient ETT and a parallel CC algorithm, +Tarjan-Vishkin uses 𝑂(𝑛 + 𝑚) optimal expected work and +polylogarithmic span. However, the space-inefficiency ham- +pers the practicability of Tarjan-Vishkin since 𝐺 ′ contains 𝑚 +vertices and 𝑂(𝑚) edges. In our experiments, Tarjan-Vishkin +takes up to 11× extra space than our FAST-BCC or GBBS. On +our machine with 1.5TB memory, Tarjan-Vishkin ran out of +memory when processing the Clueweb graph [52], although +1In a later paper [37], it was shown that the number of vertices in 𝐺′ can +be reduced to 𝑂 (𝑛), but |𝐸′| is still 𝑂 (𝑚). +3 + +it only takes about 300GB to store the graph (see discus- +sions in Sec. 6.3). The large space usage forbids running +Tarjan-Vishkin on large-scale graphs on most multicore ma- +chines. Even for small graphs, high space usage can increase +memory footprint and slow down the performance. +Some existing BCC implementations (e.g., GBBS [30] and +TV-filter [28]) were also described as Tarjan-Vishkin algo- +rithms, probably since they also use the skeleton-connectivity +framework. We note that their correctness relies on BFS- +based skeletons (i.e., sparse certificates [26]), and we catego- +rized them below together with a few other algorithms. +3.3 +Other Existing Algorithms / Implementations +Before Tarjan-Vishkin, Savage and JáJá [59] showed a par- +allel BCC algorithm based on matrix-multiplication with +𝑂(𝑛3 log𝑛) work. Tsin and Chin [64] gave an algorithm that +uses an AST-based skeleton. It is quite similar to Tarjan- +Vishkin, but uses 𝑂(𝑛2) work. +To achieve space-efficiency, many later parallel BCC algo- +rithms use BFS-based skeletons [24, 25, 28, 30, 38, 48, 62, 66]. +Many of them use the similar idea of sparse certificates [26]. +BCC is much simpler with a BFS tree—all non-tree edges +are cross edges with both endpoints in the same or adja- +cent levels. Cong and Bader’s TV-filter algorithm [28] uses +the skeleton as the BFS tree 𝑇 and an arbitrary spanning +tree/forest for 𝐺 \ 𝑇 (𝑂(𝑛) total size). Slota and Madduri’s +algorithms [62] and Dhulipala et al.’s algorithm [30] use the +skeletons as the input graph𝐺 excluding𝑂(𝑛) vertices/edges. +The other algorithms [24, 25, 38, 66] use a BFS tree as the +skeleton, and compute connectivity dynamically. All these +algorithms are space-efficient. Their skeleton graphs either +have 𝑂(𝑛) size [24, 25, 28, 38, 66] or can be implicitly repre- +sented using 𝑂(𝑛) information [30, 62]. However, the span +to generate a BFS tree is proportional to the diameter of the +graph, which is inefficient for large-diameter graphs. +3.4 +Space-Efficient BCC Representation +Since some vertices (articulation points) appear in multiple +BCCs (see Fig. 2 as an example), we need a representation of +all BCCs in a space-efficient manner (𝑂(𝑛) space). We use a +commonly used representation [10, 30, 38] in our algorithm. +Given a spanning tree 𝑇, we assign a label for each vertex +except for the root of 𝑇, indicating which BCC this vertex is +in. For all vertices with the same label, we find another vertex +called the component head (see details in Sec. 4.1) attached +to this label, and all these vertices and the component head +form a BCC. An example of this representation is given in +Fig. 2. It is easy to see that this representation uses 𝑂(𝑛) +space since we have 𝑛 − 1 labels for all vertices and at most +𝑛 − 1 component heads. +4 +The FAST-BCC Algorithm +In this section, we present our FAST-BCC algorithm with +analysis. Our algorithm is the first parallel BCC algorithm +that is work-efficient, space-efficient, and has polylogarith- +mic span. Recall that BFS-based algorithms are space-efficient, +Algorithm 1: The FAST-BCC algorithm +Input: An undirected graph 𝐺 = (𝑉, 𝐸) +Output: The labels 𝑙[·] for vertices, and the component head +for each BCC +1 Compute the spanning forest 𝐹 of 𝐺 +⊲ First CC +2 Root all trees in 𝐹 using the Euler tour technique ⊲ Rooting +3 Compute tags (e.g., low, high) of each vertex based on the +Euler tour +⊲ Tagging +4 Compute the vertex label 𝑙[·] using connectivity on 𝐺 with +edges satisfying InSkeleton(𝑢, 𝑣) = true +⊲ Last CC +5 ParallelForEach 𝑢 ∈ 𝑉 with 𝑙[𝑢] ≠ 𝑙[𝑝(𝑢)] +6 +Set the component head of 𝑙[𝑢] as 𝑝(𝑢) +7 Function InSkeleton(𝑢, 𝑣)⊲ Decide if 𝑢–𝑣 is in skeleton 𝐺′ +8 +if (𝑢, 𝑣) is a tree edge then +9 +return ¬ Fence(𝑢, 𝑣) and ¬ Fence(𝑣,𝑢) +10 +else return ¬ Back(𝑢, 𝑣) and ¬ Back(𝑣,𝑢) +11 Function Fence(𝑢, 𝑣) +⊲ Decide if tree edge is fence edge +12 +return first[𝑢] ≤ low[𝑣] and last[𝑢] ≥ high[𝑣] +13 Function Back(𝑢, 𝑣) ⊲ Decide if non-tree edge is back edge +14 +return first[𝑢] ≤ first[𝑣] and last[𝑢] ≥ first[𝑣] +but BFS itself does not parallelize well. Tarjan-Vishkin is +based on AST and is highly parallel, but generating the skele- +ton is space-inefficient. To achieve both high parallelism and +space efficiency, we need novel algorithmic insights. +Interestingly, our key idea is to revisit the sequential DFS- +based Hopcroft-Tarjan algorithm (Sec. 3.1). Although DFS +is inherently sequential, the insights in Hopcroft-Tarjan in- +spire our parallel BCC algorithm. The (implicit) skeleton +in Hopcroft-Tarjan is simple and the skeleton size is small +(𝑂(𝑛)). Unlike many later parallel BCC algorithms with the +high-level ideas to combine cycles (based on Fact 4.2), the +idea in Hopcroft-Tarjan is the “fencing” condition as follows. +When computing the CC on the skeleton 𝐺 ′ (the DFS tree) +and traversing the edge from 𝑣 to 𝑝(𝑣), the CC on𝐺 ′ (BCC on +𝐺) is fenced if low[𝑣] ≥ first[𝑝(𝑣)]. This condition partitions +the DFS tree 𝑇 into multiple CCs that correspond to BCCs +in 𝐺. Note that 𝐺 ′ in Hopcroft-Tarjan only contains edges +from the DFS tree, because there are no cross edges in DFS +trees and all back edge information is captured by low[·]. +Now we try to generalize this idea to an arbitrary span- +ning tree (AST). Directly using the “fencing” condition in +Hopcroft-Tarjan does not work since we need to deal with +cross edges. Note that a fence edge𝑣–𝑝(𝑣) in Hopcroft-Tarjan +means that vertices in 𝑢’s subtree do not have an edge that +escapes (i.e., the other endpoint is outside) 𝑝(𝑢)’s subtree. We +define our fence edges also based on this condition. More for- +mally, we say a tree edge (𝑢, 𝑣) where𝑢 = 𝑝(𝑣) is a fence edge +if there is no edge (𝑥,𝑦) ∈ 𝐸 such that 𝑥 ∈ 𝑇𝑣 and 𝑦 ∉ 𝑇𝑢. In- +tuitively, it means 𝑣’s subtree𝑇𝑣 is “isolated” from other parts +outside 𝑝(𝑣)’s subtree, and only interacts with the outside +world through 𝑝(𝑣). To get an equivalent condition for an +AST, we borrow the idea from Tarjan-Vishkin and also com- +pute four axillary arrays first[·], last[·], low[·], and high[·]. +4 + +Input Graph 𝑮: contains 3 BCCs +{s, u}, {r, s, t, v, w, x}, {t, y, z}. +Step 1: First CC. Find the CCs of 𝐺 +and a spanning tree (forest). +r +s +v +w +u +x +y +z +t +r +s +u +v +w +x +y +z +t +Step 2.2: Set any vertex as the +root and run list ranking. The +result implies the tree edge +directions. +Step 3: Tagging. +Step 4.1: Find the CCs of the +skeleton 𝐺′ only with cross and +plain edges in 𝐺 (solid edges in +Step 3). Ignore the root. +Step 4.2: Assign the +component head to +each CC in 𝐺′. Each +CC in 𝐺′ with its +component head is +a BCC. +r +s +u +v +w +x +y +z +t +r +s +u +v +w +x +y +z +t +Step 2.1: Based on the +spanning tree 𝑇 of 𝐺. Create +the linked list of the Euler +tour of 𝑇. +Step 2: Rooting. Generate rooted spanning trees. +r +s +v +w +u +x +y +z +t +r +s +u +v +w +x +y +z +t +Fence edge +Plain edge +Back edge +Cross edge +Compute tags (first/last/low/high/…) for each +vertex. Use these tags to identify fence, plain, +cross, and back edges. +BCC1 {s, u}: +Head s + {u} +BCC2 {s, t, v, w, x}: +Head r + {s, t, v, w, x} +BCC3 {t, y, z}: +Head t + {y, z} +Tree edge: +Non-tree edge: +Step 4: Last CC. Run CC on the skeleton. +Figure 2. The outline of the FAST-BCC algorithm and a running example. The four steps are explained in detail in Sec. 4.1. +The “fencing” condition then becomes low[𝑣] ≥ first[𝑝(𝑣)] +and high[𝑣] ≤ last[𝑝(𝑣)]. A non-fence tree edge is referred +to as a plain edge. Note that the information for back edges +is already captured by the low[·] and high[·] arrays, which +will also be used to decide fence edges. Our algorithm will +ignore back edges as in Hopcroft-Tarjan, and our skeleton 𝐺 ′ +contains plain tree edges and cross edges. Since the main +approach in our algorithm is Fencing an Arbitrary Spanning +Tree, we call our algorithm FAST-BCC. We note that the +high-level idea of fencing (find some special edges on the +spanning tree) is also used in some existing work [10, 30, 62]. +Our design of the skeleton and the fencing condition is the +first to achieve work-efficiency, polylogarithmic span, and +space-efficiency for the BCC problem. +The outline of the algorithm is given in Fig. 2, and the +pseudocode is in Alg. 1. Although our fencing algorithm +is simple, we note that formally proving the correctness +(Sec. 4.2) is highly non-trivial. +4.1 +Algorithmic Details +Our FAST-BCC algorithm has four steps: First-CC (gener- +ate spanning trees), Rooting (root the spanning trees using +ETT), Tagging (compute first[·], last[·], 𝑤1[·], 𝑤2[·], low[·], +high[·], 𝑝[·]), and Last-CC (run CC on the skeleton and post- +processing). In the skeleton-connectivity framework, the +first three steps are the skeleton phase (compute the skele- +ton 𝐺 ′), and the last step is the connectivity phase (run CC +on 𝐺 ′ to find all BCCs in 𝐺). +First-CC (Step 1 in Fig. 2, Line 1 in Alg. 1). This step finds +all CCs in 𝐺 and generates a spanning forest 𝐹 of 𝐺. For +simplicity, in the following, we focus on one CC and its +spanning tree 𝑇, which is unrooted at this moment. If 𝐺 +contains multiple CCs, they are simply processed in parallel. +Running CC only requires 𝑂(𝑛) auxiluary space. +Rooting (Step 2 in Fig. 2, Line 2 in Alg. 1). We use the Euler +tour technique (ETT) in Sec. 2 to root 𝑇, which implies the +tree edge directions (Fig. 2, Step 2). ETT requires 𝑂(𝑛) space. +Tagging (Step 3 in Fig. 2, Line 3 in Alg. 1). This step generates +the tags used in the algorithm, including 𝑤1[·], 𝑤2[·], low[·], +high[·], first[·], last[·] (same as in Tarjan-Vishkin, see Sec. 3) +and the parent array 𝑝[·]. low[·] and high[·] values are com- +puted by looping over all edges and getting arrays 𝑤1 and 𝑤2, +and applying 𝑛 1D range-minimum queries (RMQ). This step +takes in 𝑂(𝑛 + 𝑚) work and 𝑂(log𝑛) span [14]. These tags +will help to decide the four edge types (see details below). +All the tag arrays have size 𝑂(𝑛). +Last-CC (Step 4 in Fig. 2, Line 4–6 in Alg. 1). As men- +tioned, our skeleton graph 𝐺 ′ contains plain tree edges and +cross edges. To achieve space efficiency, we do not explic- +itly store 𝐺 ′. Since 𝐺 ′ is a subgraph of 𝐺, we can directly +use 𝐺 but skip the fence edges and back edges, which can +be determined using the tags generated in Step 3 (Line 7– +14). Then we compute the CCs on the skeleton 𝐺 ′ (Line 4), +which assigns a label 𝑙[𝑣] to each vertex (Fig. 2, Step 4.1). In +Lem. 4.11, we show that if two vertices are connected in 𝐺 ′, +they must be biconnected on the input graph 𝐺. We then +assign the head to each label (Lines 5 and 6) by looping over +all fence edges (Fig. 2, Step 4.2). For a fence edge 𝑢–𝑝(𝑢), if +𝑢 and 𝑝(𝑢) have different labels (Line 5), 𝑝(𝑢) (intuitively) +isolates vertices below 𝑢 with the other parts in the graph. +Thus, we assign 𝑝(𝑢) as the component head of 𝑢’s CC in 𝐺 ′. +We prove the correctness of this step in Lem. 4.9 and 4.12. +This step also only requires 𝑂(𝑛) auxiluary space, which is +needed by running CC on 𝐺 but skip certain edges. +4.2 +Correctness for the FAST-BCC Algorithm +We now prove the correctness of our algorithm. Note that our +algorithm will identify the spanning forest in the first step +and deal with each CC respectively. For simplicity, through- +out the section, we focus on one CC in 𝐺. +In the following, when we use the concepts about a span- +ning tree of the graph (e.g., root, parent, child, and subtree), +we refer to the specific spanning tree identified in Step 1 +of our algorithm, and use 𝑇 to represent it. Recall that 𝑇𝑢 +denotes the subtree rooted at vertex 𝑢, and 𝑢 ~𝑣 denotes the +tree path on 𝑇 from 𝑢 to 𝑣. Some other notation is given +in Tab. 1. In a spanning tree, we say a node 𝑢 is shallower +(deeper) than 𝑣 if 𝑢 is closer (farther) to the root than 𝑣. We +use node and vertex interchangeably. +5 + +Algorithm 1 is correct +Fact 4.1 +Two BCCs +𝐶1 ∩ 𝐶2 ≤ 1 +Fact 4.2 +Cycle ⇒ BCC +Lemma 4.3 +A BCC is +connected on 𝑇 +Lemma 4.4 +BCC head ⟺ +articulation point +Lemma 4.6 +Property of +plain tree edge +Lemma 4.5 +The skeleton 𝐺′ is generated +correctly +Lemma 4.8 +(inductive) +Non-head vertices +in a BCC get the +same label +Lemma 4.9 +BCC head is +identified as +component head +Theorem 4.7 +Every BCC in 𝐺 is +identified by Alg. 1 +Lemma 4.11 +(constructive) +vertices with the +same label are +biconnected +Lemma 4.12 +Component head for +label 𝑙 is biconnected +with all vertices with +label 𝑙 +Theorem 4.10 +Every BCC identified by +Alg. 1 is biconnected +Figure 3. The structure of the correctness proof for Alg. 1. +We note that although Alg. 1 is simple, the correctness +proof is sophisticated. We show the relationship of facts, +lemmas, and theorems in Fig. 2. The proofs for Fact 4.1 and 4.2 +and Lem. 4.3 to 4.5 are given in Appendix B, and here we +mainly focus on the proofs that reflect some key ideas in our +new algorithm. +We first show some facts for BCCs based on the definition. +Fact 4.1. Two BCCs share at most one common vertex. +Fact 4.2. For a cycle in a graph, all vertices on the cycle are +in the same BCC. +Lemma 4.3. Given a graph 𝐺, vertices in each BCC 𝐶 ⊆ 𝑉 +must also be connected in an arbitrary spanning tree 𝑇 for 𝐺. +Since each BCC 𝐶 must be connected in the spanning tree +in 𝑇, there must exist a unique shallowest node in this BCC +on 𝑇. We call this shallowest node the BCC head of the +BCC 𝐶, and denote it as ℎ𝐶. +Lemma 4.4. Each non-root BCC head is an articulation point. +An articulation point must be a BCC head. +Lemma 4.5. The function InSkeleton (Line 7) in Alg. 1 can +correctly skip the fence and back edges. +Next, we show a useful property of the plain tree edges. +Lemma 4.6. For a plain tree edge 𝑥–𝑦 where 𝑥 is the parent +of 𝑦, let 𝑧 be 𝑥’s parent, then 𝑥,𝑦,𝑧 are biconnected. +Proof. Since 𝑥–𝑦 is not a fence edge, there must be an edge +𝑎–𝑏, s.t. 𝑎 ∈ 𝑇𝑦 and 𝑏 ∉ 𝑇𝑥. The cycle 𝑦 ~𝑎–𝑏 ~𝑧–𝑥–𝑦 then +contains 𝑥, 𝑦, and 𝑧. Due to Fact 4.2, 𝑥, 𝑦, and 𝑧 are in the +same BCC. +□ +Next, we show that Alg. 1 can correctly identify all BCCs. +We will show two directions. First, if two vertices 𝑢 and 𝑣 are +biconnected, Alg. 1 must put them in a BCC. Second, for any +two vertices 𝑢 and 𝑣 in a BCC found by Alg. 1, they must be +biconnected. +Theorem 4.7. For 𝑢, 𝑣 ∈ 𝑉 , if they are biconnected, Alg. 1 +assigns them to the same BCC. +To prove Thm. 4.7, we discuss two cases: 1) one of 𝑢 and 𝑣 +is a BCC head, and 2) neither of them is a BCC head. +Lemma 4.8. For a BCC 𝐶 and two vertices 𝑢, 𝑣 ∈ 𝐶 \ {ℎ𝐶}, +they are connected in the skeleton 𝐺 ′ and will get the same +label in Alg. 1. +Proof. If all tree edges connecting 𝐶 \ {ℎ𝐶} are plain tree +edges, 𝑢 and 𝑣 are already connected in 𝐺 ′. Next, we show +that the two endpoints of every fence edge are also connected +in 𝐺 ′. To do so, we first sort (only conceptually) all vertices +in 𝐶 \ {ℎ𝐶} by their depth in 𝑇. Then we inductively show +from bottom up (deep to shallow) that, given a vertex 𝑣 ∈ 𝐶, +𝑇𝑣 ∩ 𝐶 (𝑣’s subtree in 𝐶) is connected in 𝐺 ′. +The base case is the deepest vertices in𝐶\{ℎ𝐶}. In this case, +their subtree contains only one vertex so they are connected. +We now consider the inductive step—if for all vertices +with depth ≥ 𝑑, their subtrees in 𝐶 are connected in 𝐺 ′, +then for all vertices with depth 𝑑 − 1, their subtrees in 𝐶 +are also connected in 𝐺 ′. Consider a vertex 𝑢 ∈ 𝐶 \ {ℎ𝐶} +with depth 𝑑 − 1. If 𝑢 has only one child 𝑣 in 𝐶, then 𝑢–𝑣 is +a plain tree edge since otherwise 𝑣’s subtree cannot escape +𝑢’s subtree and 𝑢 is an articulation point (disconnecting 𝑣 +and 𝑝(𝑢)), contradicting Lem. 4.4. Assume 𝑢 has multiple +children 𝑐1, . . . ,𝑐𝑘 in 𝐶. Let 𝑢–𝑣 be a fence edge that is not +in 𝐺 ′, where 𝑣 = 𝑐𝑖 is a child of 𝑢. We will show that 𝑢 and 𝑣 +are still connected in 𝐺 ′. +Since 𝑢 is not a BCC head, 𝑝(𝑢) must also be in 𝐶. Based +on the definition of BCC, if we remove 𝑢, 𝑣 and 𝑝(𝑢) are +still connected 𝐶. Let the path be 𝑃 = 𝑣–𝑥1–𝑥2–...–𝑥𝑘–𝑝(𝑢) +where 𝑥𝑖 ∈ 𝐶 and 𝑥𝑖 ≠ 𝑢. We will construct a path in 𝐺 ′ from +𝑃 that connects 𝑣 and 𝑢. Let 𝑥𝑗+1 be the first vertex on path +𝑃 that is not in 𝑇𝑢. We will use the path 𝑣 = 𝑥0–𝑥1–𝑥2–...–𝑥𝑗. +All nodes in this path have depths ≥ 𝑑. Due to the induction +hypothesis, if some of the edges are back or fence edges, we +can replace them with the paths in 𝐺 ′, and denote this path +as 𝑃 ′. Then, since 𝑥𝑗+1 ∉ 𝑇𝑢 is connected to 𝑥𝑗 ∈ 𝑇𝑢, all edges +on tree path 𝑥𝑗 ~𝑢 are plain tree edges. As a result, 𝑢 and 𝑣 +are connected in 𝐺 ′ using the path 𝑃 ′ from 𝑣 to 𝑥𝑗, and the +tree path from 𝑥𝑗 to 𝑢 (all edges are in 𝐺 ′). By the induction, +all vertices in 𝐶 \ {ℎ𝐶} are connected in 𝐺 ′, and hence get +the same label after Line 4. +□ +Lemma 4.9. Any BCC head will be correctly identified as a +component head in Alg. 1. +Proof. Consider a BCC 𝐶 and its BCC head ℎ𝐶. Among all +the children of ℎ𝐶, a subset 𝑆 of them are in the same BCC 𝐶. +Consider any 𝑐 ∈ 𝑆. We will show that the edge 𝑐–ℎ𝐶 must +be identified correctly in Line 5. +We first show that 𝑐–ℎ𝐶 must be a fence. If ℎ𝐶 is the root +of 𝑇, and in this case, all tree edges connecting to ℎ𝐶 are +fence edges. Otherwise, this can be inferred from the contra- +positive of Lem. 4.6. If 𝑐–ℎ𝐶 is a plain tree edge, 𝑐, ℎ𝐶, and +𝑝(ℎ𝐶) must be biconnected, which means 𝑝(ℎ𝐶) is also in +the BCC 𝐶. This contradicts the assumption that ℎ𝐶 is the +shallowest node (BCC head) in the BCC. +We then show that after we run the CC on the skeleton 𝐺 ′ +(Line 4), ℎ𝐶 and 𝑐 have different labels (i.e., ℎ𝐶 and 𝑐 are not +6 + +connected in 𝐺 ′). Assume to the contrary that there exists a +path 𝑃 from 𝑐 to ℎ𝐶 on 𝐺 ′. Consider the last node 𝑡 on the +path before ℎ𝐶. Because ℎ𝐶–𝑐 is a fence edge and is ignored +in 𝐺 ′, 𝑐 ≠ 𝑡. We discuss three cases. (1) 𝑡 is not in the ℎ𝐶’s +subtree 𝑇ℎ𝐶. Consider the first edge 𝑥–𝑦 on the path 𝑃 such +that 𝑥 ∈ 𝑇ℎ𝐶 and 𝑦 ≠ 𝑇ℎ𝐶. Since 𝑥–𝑦 escapes ℎ𝐶’s subtree, +the tree path 𝑃 ′ = 𝑥 ~ℎ𝐶 only contains plain tree edges. +Let 𝑐′ be ℎ𝐶’s child on the path 𝑃 ′. From Lem. 4.6, 𝑐′, ℎ𝐶, +and 𝑝(ℎ𝐶) are biconnected. In this case, ℎ𝐶–𝑐 ~𝑥 ~𝑐′–ℎ𝐶 is +a cycle, and Fact 4.2 shows that 𝑐′, ℎ𝐶 and 𝑐 are biconnected. +The contrapositive of Fact 4.1 indicates that 𝑐′, ℎ𝐶, 𝑐, and +𝑝(ℎ𝐶) are all biconnected, contradicting the assumption that +ℎ𝐶 is the BCC head (the shallowest node in the BCC). (2) +𝑡 ∈ 𝑇ℎ𝐶, but 𝑡 is not ℎ𝐶’s child. This is impossible because +𝑡–ℎ𝐶 is a back edge, which is not in 𝐺 ′. (3) 𝑡 is a child of ℎ𝐶. +This case is similar to (1). By replacing 𝑐′ in the previous +proof by 𝑡, we can get the same contradiction. Combining +all cases proves that there is no path in 𝐺 ′ between ℎ𝐶 and +its children in 𝐶, so 𝑙[ℎ𝐶] is different from the labels of its +children in 𝐶. +□ +Combining Lem. 4.8 and 4.9, we can prove Thm. 4.7. +We then show the other direction—all the BCCs computed +by Alg. 1 are indeed biconnected. +Theorem 4.10. If two vertices 𝑢 and 𝑣 are identified as in the +same BCC by Alg. 1, they must be biconnected. +Similar to the previous proof, we consider two cases: (1) +none of the two vertices is a component head (they are con- +nected in 𝐺 ′), proved in Lem. 4.11, and (2) one of them is +identified as a component head in Line 6, proved in Lem. 4.12. +Lemma 4.11. If two vertices 𝑢 and 𝑣 are connected in the +skeleton 𝐺 ′, they are biconnected. +Proof. Since 𝑢 and 𝑣 are connected in 𝐺 ′, there exists a path +𝑃 from 𝑢 to 𝑣 only using edges in 𝐺 ′. Let 𝑃 be 𝑢 = 𝑝0–𝑝1–...– +𝑝𝑘−1–𝑝𝑘 = 𝑣. We will show that after removing any vertex +𝑝𝑖 where 1 ≤ 𝑖 < 𝑘 on 𝑃, 𝑝𝑖−1 and 𝑝𝑖+1 are still connected, +meaning that 𝑢 and 𝑣 are biconnected. We summarize all +possible local structures in three cases, based on whether +𝑝𝑖−1 (and 𝑝𝑖+1) is a child of 𝑝𝑖 in 𝑇. +Case 1: both 𝑝𝑖−1 and 𝑝𝑖+1 are 𝑝𝑖’s children. Since 𝑝𝑖−1–𝑝𝑖 is +not a fence edge, there must be an edge 𝑥–𝑦 s.t. 𝑥 ∈ 𝑇𝑝𝑖−1 and +𝑦 ∉ 𝑇𝑝𝑖. Similarly, for 𝑝𝑖–𝑝𝑖+1, there exists an edge (𝑥 ′,𝑦′) +s.t. 𝑥 ′ ∈ 𝑇𝑃𝑖+1 and 𝑦′ ∉ 𝑇𝑃𝑖. Hence, without using 𝑝𝑖, 𝑝𝑖−1 and +𝑝𝑖+1 are still connected by the path 𝑝𝑖−1 ~𝑥–𝑦 ~𝑦′–𝑥 ′ ~𝑝𝑖+1. +Here since 𝑦,𝑦′ ∉ 𝑇𝑝𝑖, 𝑦 ~𝑦′ does not contain 𝑝𝑖. +Case 2: one of 𝑝𝑖−1 and 𝑝𝑖+1 is 𝑝𝑖’s child. WLOG, assume +𝑝𝑖−1 is the child. Since 𝑝𝑖−1–𝑝𝑖 is not a fence edge, there must +be an edge 𝑥–𝑦 such that 𝑥 ∈ 𝑇𝑝𝑖−1 and 𝑦 ∉ 𝑇𝑝𝑖. Also, since +𝑝𝑖+1 is either the parent of 𝑝𝑖 or connected to 𝑝𝑖 using a cross +edge, 𝑝𝑖+1 ∉ 𝑇𝑝𝑖. Hence, without using 𝑝𝑖, 𝑝𝑖−1 and 𝑝𝑖+1 are +still connected using the path 𝑝𝑖−1 ~𝑥–𝑦 ~𝑝𝑖+1. +Case 3: neither 𝑝𝑖−1 nor 𝑝𝑖+1 is a child of 𝑝𝑖, and neither of +them is in 𝑇𝑝𝑖 (otherwise they are connected by a back edge). +Without using 𝑝𝑖, 𝑝𝑖−1 and 𝑝𝑖+1 are still connected using the +tree path 𝑝𝑖−1 ~𝑝𝑖+1. +Since removing any vertex on the path 𝑃 does not discon- +nect the path, all vertices in the same CC of the skeleton are +biconnected. +□ +Lemma 4.12. If Line 6 in Alg. 1 assigns ℎ as the component +head of a connected component (CC) 𝐶 in the skeleton 𝐺 ′, then +ℎ is biconnected with 𝐶. +Proof. First of all, assume ℎ is assigned as the component +head because of its child 𝑐, where ℎ–𝑐 is a fence edge. We +will show that the connected component 𝐶 in 𝐺 ′ containing +𝑐 is biconnected with ℎ. There are two cases. +Case 1: 𝐶 only contains vertices in 𝑇𝑐. This means that no +vertices in 𝑇𝑐 have a cross edge to another vertex outside 𝑇𝑐. +Therefore, either all edges incident on 𝑐′ ∈ 𝑇𝑐 do not escape +from 𝑇𝑐, or some node 𝑐′ ∈ 𝑇𝑐 is connected to nodes outside +𝑇𝑐 via back edges. In the former case, all the edges connecting +𝑐 and its children are fence edges, and thus 𝐶 only contains +𝑐. In this case, ℎ is trivially biconnected with 𝐶. In the latter +case, assume 𝑥 ∈ 𝑇𝑐 ∩𝐶 has a back edge connected to 𝑦 ∉ 𝑇𝑐. +Note that 𝑦 can only be ℎ—if 𝑦 is ℎ’s ancestor, then edge +𝑥–𝑦 escapes 𝑇ℎ, so ℎ–𝑐 is a plain tree edge (contradiction). +Therefore, we can find a cycle ℎ–𝑐 ~𝑥–ℎ. From Fact 4.2, ℎ,𝑐,𝑥 +are biconnected, and ℎ is in the same BCC as 𝑐 and 𝑥, and +thus all vertices in 𝐶 (Lem. 4.11 and Fact 4.1). +Case 2: 𝐶 contains both vertices in 𝑇𝑐 and some vertices +in 𝑇ℎ \𝑇𝑐. Hence, there exists a cross edge 𝑥–𝑦, where 𝑥 ∈ 𝑇𝑐 +and 𝑦 ∉ 𝑇𝑐. We can find a cycle ℎ,~𝑥–𝑦 ~ℎ. From Fact 4.2, +ℎ,𝑐,𝑢 are biconnected. ℎ is in the same BCC as 𝑐 and 𝑢. +□ +Combining Lem. 4.11 and 4.12 proves Thm. 4.10. +Thm. 4.7 shows that if two vertices are put in the same +BCC by Alg. 1, they are biconnected in𝐺. Thm. 4.10 indicates +that two vertices biconnected in 𝐺 will be put in the same +BCC by Alg. 1. Lem. 4.5 indications back edges and fence +edges are identified correctly by Alg. 1. Combining them +together indicates that Alg. 1 is correct. +4.3 +Cost Bounds for the FAST-BCC Algorithm +We now analyze the cost bounds of the algorithm. +Theorem 4.13. Alg. 1 computes the BCCs of a graph𝐺 with𝑛 +vertices and 𝑚 edges using 𝑂(𝑛 +𝑚) expected work, 𝑂(log3 𝑛) +span whp, and 𝑂(𝑛) auxiliary space (other than the input). +Proof. The first and last steps compute the graph connectiv- +ity twice. Graph connectivity can be computed in 𝑂(𝑛 + 𝑚) +expected work and 𝑂(log3 𝑛) span whp [61]. In Step 2, ETT +can be performed 𝑂(𝑛) expected work and 𝑂(log𝑛) span +whp (see Sec. 2). In Step 3, computing low[·] and high[·] ar- +rays based on RMQ takes 𝑂(𝑚) work and 𝑂(log𝑛) span [14]. +Adding all pieces together gives the work and span bounds. +For the space, all arrays for the tags have size 𝑂(𝑛). As +mentioned, we do not generate the skeleton explicitly. In +the last step, we try all the edges in 𝐺 but skip the back and +fence edges. In all, the auxiliary space needed is 𝑂(𝑛). +□ +7 + +5 +Implementation Details +We discuss some implementation details of FAST-BCC in +this section. +Connectivity. Connectivity is used twice in FAST-BCC. +The only existing parallel CC implementation with good +theoretical guarantee we know of is the SDB algorithm [61] +(an initial version of GBBS is based on this algorithm). A +recent paper by Dhulipala et al. [31] gave 232 parallel CC +implementations, many of which outperformed the SDB +algorithm, but no analysis of work-efficiency was given. A +more recent version of GBBS uses the UF-Async algorithm +in [31] to compute CC. To achieve efficiency both in theory +and in practice, FAST-BCC uses the LDD-UF-JTB algorithm +from [31] and we provide a new analysis that this algorithm +is indeed theoretically-efficient. +LDD-UF-JTB consists of two steps. It first runs a low- +diameter decomposition (LDD) algorithm [53] to find a de- +composition (partition of vertices) of the graph such that +each component has a low diameter and the number of edges +crossing different components is bounded. The second step +is to use a union-find structure by Jayanti et al. [47] to union +components connected by cross-component edges. We now +show the bounds of this algorithm. +Theorem 5.1. The LDD-UF-JTB algorithm computes the CCs +of a graph 𝐺 with 𝑛 vertices and 𝑚 edges using 𝑂(𝑛 + 𝑚) +expected work and 𝑂(log3 𝑛) span whp. +Therefore, using LDD-UF-JTB for CC preserves the cost +bounds in Thm. 4.13. We prove Thm. 5.1 in Appendix B.6. +We optimized LDD-UF-JTB using the hash bag and local +search techniques proposed from [67]. These optimizations +are only used in computing CCs in our algorithm, and we do +not claim them as contributions of this paper. In our tests, +using these optimizations improves the performance of FAST- +BCC by 1.5× on average (up to 5×). Some results are shown +in Fig. 6. We note that among all 232 CC algorithms in [31], +no one is constantly faster, and the relative performance +is decided by the input graph properties. In FAST-BCC, we +currently use the same CC algorithm for all graphs, and we +acknowledge that using the fastest CC algorithm on each +graph can further improve the performance of FAST-BCC. +We choose LDD-UF-JTB mainly because it is theoretically- +efficient, and also can generate CC as a by-product efficiently. +Spanning Forest. The spanning forest of 𝐺 is obtained as a +by-product of Step 1, which saves all edges to form the CCs. +We then re-order the vertices in the compressed sparse row +(CSR) format to let each CC be contiguous. +Euler Tour Technique (ETT). We use the standard ETT +to root the spanning trees (see Sec. 2). We replicate each +undirected edge in 𝑇 into two directed edges and semisort +them [42], so edges with the same first endpoint are con- +tiguous. Then we construct a circular linked list as the Euler +circuit. Assume a vertex 𝑣 has 𝑘 in-coming neighbors 𝑢1, 𝑢2, +· · · , 𝑢𝑘. For every incoming edge of 𝑣 except for the last one, +we link it to its next outgoing edge (i.e., 𝑢𝑖–𝑣 is linked to +𝑣–𝑢𝑖+1 for 1 ≤ 𝑖 < 𝑘). For the last incoming edge, we link it +to the first outgoing edge of 𝑣 (i.e., 𝑢𝑘–𝑣 is linked to 𝑣–𝑢1). +After we obtain the Euler circuit of the tree, we flatten +the linked list to an array by list ranking, and acquire the +Euler tour order of each vertex. For list ranking, we coarsen +the base cases by sampling √𝑛 nodes. We start from these +nodes in parallel, with each node sequentially following the +pointers until it visits the next sample. Then we compute +the offsets of each sample by prefix sum, pass the offsets to +other nodes by chasing the pointers from the samples, and +scatter all nodes into a contiguous array. +Computing Tags. We use several tags 𝑤1, 𝑤2, first, last, +low, and high for each vertex, defined the same as Tarjan- +Vishkin [63] (see Sec. 3). We use CAS operations to compute +first and last as they represent the first and last appearances +of a vertex in the Euler tour order. For each tree edge (𝑢, 𝑣), if +first[𝑢] < first[𝑣], we set 𝑝(𝑣) = 𝑢, or vice versa. Computing +low and high are similar, so we only discuss low here. We first +initialize 𝑤1[𝑣] with first[𝑣] for each 𝑣 ∈ 𝑉 . Then it traverses +all non-tree edges𝑢–𝑣 and updates 𝑤1[𝑢] and 𝑤1[𝑣] with the +minimum of first[𝑢] and first[𝑣]. We build a parallel sparse +table [14] on 𝑤1 to support range minimum queries. Note +that first[𝑣] and last[𝑣] reflect the range of 𝑣’s subtree in the +Euler tour order. Thus, low[𝑣] can be computed by finding +the minimum element in 𝑤1[·] in the range between first[𝑣] +and last[𝑣]. high[·] can be computed similarly. +6 +Experiments +Setup. We run our experiments on a 96-core (192 hyper- +threads) machine with four Intel Xeon Gold 6252 CPUs, and +1.5 TB of main memory. We implemented all algorithms in +C++ using ParlayLib [11] for fork-join parallelism and some +parallel primitives (e.g., sorting). We use numactl -i all +in experiments with more than one thread to spread the +memory pages across CPUs in a round-robin fashion. We +run each test for 10 times and report the median. +We tested on 27 graphs, including social networks, web +graphs, road graphs, 𝑘-NN graphs, and synthetic graphs. +The information of the graphs is given in Tab. 2. In addi- +tion to commonly-used benchmarks of social, web and, road +graphs, we also use 𝑘-NN graphs and synthetic graphs. 𝑘- +NN graphs are widely used in machine learning algorithms +(see discussions in [68]). In 𝑘-NN graphs, each vertex is a +multi-dimensional data point and has 𝑘 edges pointing to +its 𝑘-nearest neighbors (excluding itself). We also create six +synthetic graphs, including two grids (SQR and REC), two +sampled grids (SQR’ and REC’, each edge is created with +probability 0.6), and two chains (Chn7 and Chn8). SQR and +SQR’ have sizes 104 ×104. REC and REC’ have sizes 103 ×105. +Each row and column in grid graphs are circular. Chn7 and +Chn8 have sizes 107 and 108. The tested graphs cover a wide +range of sizes and edge distributions. +8 + +𝒏 +𝒎 +𝑫 +#BCC +|BCC1|% +Ours +GBBS +SM’ +SEQ +𝑻best +Notes +par. +seq. spd. par. +seq. +spd. +14 +/ours +Social +YT 1.13M 5.98M +23 +673,661 +39.83% 0.030 0.465 15.6 0.040 0.435 10.8 +0.059 0.175 +1.35 +com-youtube [70] +OK 3.07M 234M +9 +68,117 +97.76% 0.103 3.08 +30.0 0.158 4.86 +30.8 +0.297 3.14 +1.53 +com-orkut [70] +LJ 4.85M 85.7M +19 +1,133,883 +75.61% 0.104 3.02 +28.9 0.159 3.34 +21.0 +n +1.87 +1.52 +soc-LiveJournal1 [8] +TW 41.7M +2.41B +23 +1,936,001 +95.33% +1.44 +52.9 +36.7 +2.83 +95.2 +33.7 +20.5∗ +49.2 +1.96 +Twitter [49] +FT 65.6M +3.61B +37 14,039,045 +78.50% +3.10 +129 +41.6 +6.44 +260 +40.5 +10.9 +122 +2.07 +Friendster [70] +Web +GG +876K 8.64M +24 +175,274 +73.31% 0.029 0.534 18.7 0.045 0.530 11.8 +n +0.255 +1.58 +web-Google [51] +SD 89.2M +3.88B +35 16,189,065 +80.36% +3.11 +134 +43.2 +5.61 +213 +38.0 +n +92.3 +1.81 +sd_arc [52] +CW +978M +74.7B +254 81,809,602 +86.48% +22.9 +1464 64.0 +39.7 +1526 +38.4 +n +695 +1.73 +ClueWeb [52] +HL14 +1.72B +124B +366 124,406,075 +83.25% +31.1 +2057 66.0 +50.7 +2113 +41.7 +n +1011 +1.63 +Hyperlink14 [52] +HL12 +3.56B +226B +650 410,853,262 +80.63% +89.1 +5435 61.0 +104 +5985 +57.6 +n +3027 +1.17 +Hyperlink12 [52] +Road +CA 1.97M 5.53M +857 +381,366 +79.55% 0.040 0.824 20.6 0.372 1.05 +2.82 +n +0.206 +5.15 +roadnet-CA [51] +USA 23.9M 57.7M +8,263 +7,390,330 +66.90% 0.336 12.1 +36.0 +4.64 +15.1 +3.25 +3.73∗ +2.25 +6.69 +RoadUSA [1] +GE 12.3M 32.3M +2,240 +2,482,488 +78.67% 0.267 11.1 +41.6 +2.02 +11.4 +5.66 +1.14∗ +2.88 +7.54 +Germany [1] +𝒌-NN +HH5 2.05M 13.0M +1,859 +17,408 +62.55% 0.073 1.60 +22.0 0.447 1.52 +3.41 +n +0.509 +6.16 +Household [36, 68], 𝑘=5 +CH5 4.21M 29.7M 14,479 +299 +15.41% 0.128 2.85 +22.2 +1.44 +2.38 +1.66 +n +0.528 +4.11 +CHEM [39, 68], 𝑘=5 +GL2 24.9M 65.4M 13,333 10,940,922 +0.03% 0.402 13.8 +34.5 +1.53 +16.9 +11.0 +n +2.51 +3.80 +GeoLife [68, 71], 𝑘=2 +GL5 24.9M 157M 21,600 +1,009,434 +30.07% 0.472 19.1 +40.5 +2.80 +19.4 +6.92 +n +4.03 +5.93 +GeoLife [68, 71], 𝑘=5 +GL10 24.9M 305M +3,824 +51,465 +86.38% 0.668 29.2 +43.8 +1.64 +23.5 +14.3 +n +7.07 +2.46 +GeoLife [68, 71], 𝑘=10 +GL15 24.9M 453M +3,664 +23,149 +91.11% 0.751 34.4 +45.8 +1.51 +25.9 +17.1 +n +8.92 +2.01 +GeoLife [68, 71], 𝑘=15 +GL20 24.9M 602M +2,805 +13,619 +93.96% 0.861 39.2 +45.6 +1.48 +28.6 +19.3 +n +10.2 +1.72 +GeoLife [68, 71], 𝑘=20 +COS5 +321M +1.96B +1,180 +85,283 +99.74% +8.46 +382 +45.2 +17.5 +392 +22.4 +n +120 +2.07 +Cosmo50 [50, 68], 𝑘=5 +Synthetic +SQR +100M 400M 10,000 +1 100.00% +1.32 +43.4 +32.9 +15.4 +44.2 +2.87 +20.3∗ +24.4 +11.7 +2D grid 104 × 104 +REC +100M 240M 50,500 +1 100.00% +1.35 +43.6 +32.4 +47.0 +34.6 0.735 13.1∗ +16.8 +12.5 +2D grid 103 × 105 +SQR’ +100M 400M 10,256 23,836,580 +70.65% +1.31 +50.1 +38.1 +12.5 +60.9 +4.88 +n +10.6 +8.06 +sampled SQR +REC’ +100M 240M 69,014 23,826,514 +70.66% +1.37 +46.8 +34.3 +22.4 +58.9 +2.63 +n +10.7 +7.81 +sampled REC +Chn7 +10M +20M 107 − 1 +107 − 1 +0.00% 0.278 13.1 +46.9 +81.6 +19.7 0.241 40.5∗ +3.33 +12.0 +Chain of size 107 +Chn8 +100M 200M 108 − 1 +108 − 1 +0.00% +3.25 +152 +46.9 +957 +307 +0.320 703∗ +38.9 +12.0 +Chain of size 108 +Table 2. Graph information, running times (in seconds), and speedups. 𝑇best/ours (highlighted in yellow) is the fastest time of the +other implementations / our time, both using all cores. “𝑛” = number of vertices. “𝑚” = number of edges. “𝐷” = approximate diameter. +“#BCC” = number of BCCs. “|BCC1|%” = percentage of the largest BCCs. “GBBS” = GBBS’s implementation [30]. “SM’14” = Slota and +Madduri’s algorithm [62] (the faster of the two proposed algorithms). Since SM’14 has scalability issues (see Fig. 4), we report the 16-core +time if it is faster, and denote as (∗). “SEQ” = Hopcroft-Tarjan BCC algorithm [43]. Details about the baselines are introduced in Sec. 6. The +fastest runtime for each graph is underlined. Red numbers are parallel runtime slower than the sequential algorithm. “par.” = parallel running +time (on 192 hyper-threads). “seq.” = sequential running time (on 1 thread). “spd.” = self-relative speedup. “n” = no support, because SM’14 +only works on connected graphs. +For real-world directed graphs, we symmetrize them to +test BCC. We call the social and web graphs low-diameter +graphs as they have smaller diameters (mostly within a few +hundreds). We call the road, 𝑘-NN, and synthetic graphs +large-diameter graphs as their diameters are mostly more +than a thousand. When comparing the average running times +across multiple graphs, we always take the geometric mean +of the numbers. +Baseline Algorithms. We call all existing algorithms that +we compare to the baselines. We implement the highly- +optimized sequential Hopcroft-Tarjan [43] algorithm for +comparison, referred to as SEQ or the sequential baseline. We +compare the number of BCCs reported by each algorithm +with SEQ to verify correctness. +We also compare to two most recent available BCC im- +plementations GBBS [30], and Slota and Madduri [62]. We +use SM’14 to denote the better of the two BCC algorithms in +Slota and Madduri [62]. On many graphs, we observe that +SM’14 is faster on 16 threads than using all 192 threads, in +which case we report the lower time of 16 and 192 threads. +Through correspondence with the authors, we understand +that SM’14 requires the input graph to be connected, so we +only report the running time when it gives the correct an- +swers. As few graphs we tested are entirely connected, we +focus on comparisons with GBBS and SEQ. We also compare +our breakdown and sequential running times with GBBS +since GBBS can process most of the tested graphs2. +Unfortunately, we cannot find any existing implementa- +tions for Tarjan-Vishkin to compare with. We are aware of +2GBBS updated a new version after this paper was accepted, so we also +updated the numbers using their latest version (Nov. 2022). Some new +features in the latest version greatly improved their BCC performance. +9 + +1248 24 96 +0.2 +1 +5 +20 +TW +1248 24 96 +SD +1248 24 96 +USA +1248 24 96 +GL5 +1248 24 96 +REC +FAST-BCC +GBBS +SM'14 +Figure 4. Scalability curves for different BCC algorithms. In +each plot, 𝑥-axis is core counts (last data point is 96 core with +hyperthreading) and 𝑦-axis is speedups normalized to SEQ (the +sequential Hopcroft-Tarjan algorithm). Higher is better. SEQ is 1. +two papers that implemented Tarjan-Vishkin [28, 37]. Ed- +wards and Vishkin’s implementation [37] is on the XMT +architecture and they did not release their code. Cong and +Bader’s code [28] is released, but it was written in 2005 and +uses some system functions that are no longer supported +on our machine. For a full comparison, we implemented a +faithful Tarjan-Vishkin from the original paper [63]. As en- +gineering Tarjan-Vishkin is not the main focus of this paper, +we mainly use it to evaluate the memory usage. +We note that the implementations for both GBBS and +SM’14 exclude the postprocessing to compute the actual +BCCs, but only report the number of BCCs at the end of the +algorithm. We include this step in FAST-BCC, although this +postprocessing only takes at most 2% of the total running +time in all our tests. +6.1 +Overall Performance +We present the running time of all algorithms in Tab. 2. +Our FAST-BCC is faster than all baselines on all graphs, +mainly due to the theoretical efficiency—work- and space- +efficiency enables competitive sequential times over the +Hopcroft-Tarjan sequential algorithm, and polylogarithmic +span ensures good speedup for all graphs. +Sequential Running Time. We first compare the sequen- +tial running time of SEQ, GBBS, and FAST-BCC. SEQ and +FAST-BCC use 𝑂(𝑛 + 𝑚) work. To enable parallelism, both +FAST-BCC and GBBS traverse all edges multiple times (run- +ning CC twice in Steps 1 and 4, and computing low/high for +the skeleton in Step 3). We describe more details about GBBS +implementation in Sec. 6.2. On average, our sequential time +is 2.8× slower than SEQ, but is 10% faster than GBBS. +Scalability and Parallelism. To measure parallelism, we +report the scalability curves for FAST-BCC, GBBS and SM’14 +on some representative graphs (Fig. 4). For fair comparison, +the speedup numbers in Fig. 4 are normalized to the running +time of SEQ. On these five graphs, FAST-BCC is the only +algorithm that scales to all processors. It also outperforms +GBBS and SM’14 on all graphs with all numbers of threads +(expect REC on 2 cores). We noticed that SM’14 suffers from +scalability issues, and the best performance can be achieved +at around 16 threads. Hence, we report SM’14’s better run- +ning time of 16 and 192 threads in Tab. 2. GBBS has similar +issues on a few graphs. However, as GBBS’s performance +does not drop significantly as core count increases, we con- +sistently report GBBS’s time on 192 threads in Tab. 2. +Comparing the self-relative speedup with GBBS, our aver- +age self-relative speedups on both low-diameter graphs and +large-diameter graphs are 36×. On large-scale low-diameter +graphs with sufficient parallelism, the self-relative speedup +can be up to 66×. Even on large-diameter graphs, FAST-BCC +achieves up to 47× self-relative speedup. In comparison, +the self-relative speedup of GBBS’s BFS-based algorithm +is 29× on low-diameter graphs and 3.7× on large-diameter +graphs. This makes GBBS only 11% faster than SEQ on large- +diameter graphs (and can be slower on some graphs), while +ours is 5.1–18.5× better. Overall, our parallel running time +is 10× faster on large-diameter graphs and 1.6× faster on +low-diameter graphs than GBBS. On some graphs, SM’14 +achieves better performance than GBBS, but FAST-BCC is +1.7–11.1× faster than SM’14 on all the graphs. +To verify that GBBS’s performance is bottlenecked by +BFS, we created 𝑘-NN graphs GL2–20 from the set of points +but with different values of 𝑘. When increasing 𝑘 over 5, +the graphs have more edges but smaller diameters. For both +FAST-BCC and SEQ, the running times increase when 𝑘 +grows due to more edges (and thus more work), but the +trend of GBBS’s running time is decreasing. This indicates +that the BFS is the dominating part of running time for +GBBS, and the performance on GBBS is bottlenecked by +the 𝑂(Diam(𝐺) log𝑛) span. +6.2 +Performance Breakdown +To understand the performance gain of FAST-BCC over prior +parallel BFS-based BCC algorithms, we compare our perfor- +mance breakdown with GBBS in Fig. 5. We choose GBBS +because it can process all graphs. Since GBBS is also in the +skeleton-connectivity framework, we use the same four step +names for GBBS as in FAST-BCC, but there are a few dif- +ferences. (1) For First-CC, FAST-BCC generates a spanning +forest while GBBS only finds all CCs. (2) For Rooting, FAST- +BCC uses ETT to root the tree while GBBS applies BFS on +all CCs to find the spanning trees. (3) The task for Tagging is +almost the same, but GBBS computes fewer tags than FAST- +BCC since it is based on BFS trees. FAST-BCC uses 1D RMQ +queries that are theoretically-efficient, while GBBS uses a +bottom-up traversal on the BFS tree. (4) For Last-CC, both +algorithms run CC algorithms on the skeletons to find BCCs. +We start with discussing the two steps First-CC and Last- +CC that use connectivity. GBBS can be faster than our al- +gorithm in First-CC on some graphs. The reason is that our +algorithm also constructs the spanning forest in First-CC, +while GBBS has to run BFS in Rooting to generate the BFS +spanning forest. In Last-CC, the two algorithms achieves +similar performance, and in many cases, FAST-BCC is faster. +We note that the CC algorithm is independent with the BCC +10 + +0.00 +0.02 +0.04 +YT +0.0 +0.1 +OK +0.0 +0.1 +LJ +0 +1 +2 +TW +0.0 +2.5 +5.0 +FT +0.00 +0.02 +0.04 +GG +First CC +Rooting +Tagging +Last CC +0 +2 +4 +SD +0 +20 +40 +CW +0 +20 +40 +HL14 +0 +50 +100 +HL12 +0.0 +0.2 +CA +0 +2 +4 +USA +0 +1 +2 +GE +0.0 +0.2 +0.4 +HH5 +0.0 +0.5 +1.0 +CH5 +0 +1 +GL2 +0 +2 +GL5 +0 +1 +GL10 +GBBS +Ours +0 +1 +GL15 +GBBS +Ours +0 +1 +GL20 +GBBS +Ours +0 +10 +COS5 +GBBS +Ours +0 +10 +SQR +GBBS +Ours +0 +20 +40 +REC +GBBS +Ours +0 +5 +10 +SQR' +GBBS +Ours +0 +10 +20 +REC' +GBBS +Ours +0 +50 +Chn7 +GBBS +Ours +0 +500 +1000 +Chn8 +Figure 5. BCC breakdown. 𝑦-axis is the running time in seconds. +algorithm itself. Both the CC algorithm used in our imple- +mentation and GBBS are based on algorithms in an existing +paper [31]. As mentioned, based on the results in [31], the +“best” CC algorithm can be very different for different types +of graphs. One can also plug in any CC algorithms to FAST- +BCC or GBBS BCC algorithm to achieve better performance +for specific input graphs. +In the Rooting step (generate rooted spanning trees, the +red bar), FAST-BCC is significantly faster than GBBS. GBBS +is based on a BFS tree, and after computing the CCs of in- +put graph 𝐺, it has to run BFS on 𝐺 again, which results in +𝑂(𝑚 + 𝑛) work and 𝑂(Diam(𝐺) log𝑛) span. In comparison, +FAST-BCC obtains the spanning trees from the First-CC step, +and only uses ETT in the Rooting step with 𝑂(𝑛) expected +work and 𝑂(log𝑛) span whp. As shown in Fig. 5, this step +for GBBS is the dominating cost for large-diameter graphs, +and this is likely the case for other parallel BCC algorithms +using BFS-based skeletons. FAST-BCC almost entirely saves +the cost in this step (13× faster on average on large-diameter +graphs). For low-diameter graphs, the two algorithms per- +form similarly—FAST-BCC is about 1.1× faster in this step. +In the Tagging step (the green bars), both FAST-BCC and +GBBS compute the tags such as low and high. Since FAST- +BCC uses an AST, the values of the arrays are computed +using 1D range-minimum query (see Sec. 4.1) with 𝑂(log𝑛) +span. GBBS computes them by a bottom-up traversal on +the BFS tree, with 𝑂(Diam(𝐺) log𝑛) span. Hence, on large- +diameter, GBBS also consumes much time on this step, and +FAST-BCC is 1.2–830× faster than GBBS. On low-diameter +graphs, GBBS also gets sufficient parallelism, and the per- +formance for both algorithms are similar. +In summary, on all graphs, FAST-BCC is faster than GBBS +mainly due to the efficiency in the Rooting and Tagging step, +and the reason is that our algorithm has polylogarithmic +span, while GBBS relies on the BFS spanning tree and re- +quires 𝑂(Diam(𝐺) log𝑛) span. +6.3 +The Tarjan-Vishkin Algorithm +Although engineering the Tarjan-Vishkin (TV) Algorithm +is not the focus of this paper, for completeness, we also im- +plemented the faithful Tarjan-Vishkin algorithm [63]. We +mainly use it to measure the space usage and get a sense on +how Tarjan-Vishkin compares to other existing BCC algo- +rithms. We report the relative space usage of FAST-BCC, TV, +and GBBS in Fig. 7, normalized to the most space-efficient +implementation. Because of the space inefficiency, our TV +implementation cannot run on the three largest graphs (CW, +HL14, and HL12) on our machines with 1.5TB memory. We +note that the smallest among them (CW) only takes about +300GB to store the graph, and our algorithm uses 572GB +memory to process it. GBBS is slightly more space-efficient +than FAST-BCC, and takes about 20% less space than us. The +reason is that they need to compute fewer number of tags +than FAST-BCC. Regarding running time, we take the aver- +age of the running times for each algorithm on each category +of the graph instances, and normalize to FAST-BCC. +Ours +GBBS +Our-TV +SEQ +Social +1 +1.67 +10.1 +21.2 +Web +1 +1.69 +7.61 +16.3 +Road +1 +9.89 +1.94 +7.18 +𝑘-NN +1 +3.58 +4.13 +8.68 +Synthetic +1 +14.9 +3.96 +17.0 +Due to the cost to explicitly construct the skeleton, TV +performs slowly on small-diameter graphs, and is slower +than GBBS even on 𝑘-NN graphs. On all these graphs, the +speedup for TV on 96 cores over SEQ is only 1.4–3×. This is +consistent with the findings in prior BCC papers [28, 62]. TV +works well on road and synthetic graphs due to small edge- +to-vertex ratio, so the 𝑂(𝑚) work and space for generating +the skeleton does not dominate the running time. In this +case, polylogarithmic span allows TV to perform consistently +better than GBBS. On all graphs, TV is faster than SEQ on +96 cores, but slower than FAST-BCC. +11 + +7 +Conclusion +In this paper, we propose the FAST-BCC (Fencing on Arbi- +trary Spanning Tree) algorithm for parallel biconnectivity. +FAST-BCC has 𝑂(𝑚 + 𝑛) expected optimal work, polylog- +arithmic span (high parallelism), and uses 𝑂(𝑛) auxiliary +space (space-efficient). The theoretical efficiency also en- +ables high performance. 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In International World Wide Web Conference (WWW). +247–256. +A +More details and discussion for the +Tarjan-Vishkin Algorithm +To parallelize BCC, the Tarjan-Vishkin algorithm [63] uses +an arbitrary spanning tree (AST)𝑇 instead of a DFS tree. The +spanning tree can be obtained by any parallel CC algorithm. +13 + +The first is the famous Euler tour technique (ETT) that effi- +ciently roots a tree (see Sec. 2). Our algorithm also uses ETT. +The second technique is to build the skeleton 𝐺 ′ based on +an AST. Unfortunately, 𝐺 ′ in Tarjan-Vishkin is very large, +making the algorithm less practical. +Given the input graph 𝐺 = (𝑉, 𝐸) and an AST 𝑇, 𝐺 ′ = +(𝐸, 𝐸′) where 𝐸′ consists of (𝑒1,𝑒2) (𝑒1,𝑒2 ∈ 𝐸 are edges in +𝐺) iff. one of the following conditions hold: +• 𝑒1 = (𝑢, 𝑝(𝑢)), 𝑒2 = (𝑢, 𝑣) in 𝐺 \𝑇, and 𝑢, 𝑣 ∈ 𝑉, first[𝑣] < +first[𝑢]. +• 𝑒1 = (𝑢, 𝑝(𝑢)), 𝑒2 = (𝑣, 𝑝(𝑣)), and (𝑢, 𝑣) is a cross edge in +𝐺 \𝑇. +• 𝑒1 = (𝑢, 𝑣), where 𝑣 = 𝑝(𝑢) is not the root in 𝑇, and 𝑒2 = +(𝑣, 𝑝(𝑣)), and there exists a non-tree edge (𝑥,𝑦) such that +𝑥 ∈ 𝑇𝑢 and 𝑦 ∉ 𝑇𝑣. +The above relationships can be determined by using the +four axillary arrays first[·], last[·], low[·], and high[·] as +mentioned in Sec. 3.2. +In fact, we can prove that FAST-BCC is equivalent to +Tarjan-Vishkin. However, the analysis for Tarjan-Vishkin +is also quite involved (we refer to JáJá’s textbook for a good +reference [45]). Hence, we give a standalone analysis for +FAST-BCC in Sec. 4.2, since we feel that understanding the +analysis of Tarjan-Vishkin (correctness and cost bounds) and +the analysis in Sec. 4.2 is in a similar level of difficulty. +In addition, we believe that the five algorithms we de- +scribed and tested experimentally (Hopcroft-Tarjan, Tarjan- +Vishkin, FAST-BCC, GBBS, SM’14) are similar, once we put +them in the skeleton-connectivity framework and explicitly +specify what the skeleton graph 𝐺 ′ is in each algorithm. Thus, +the skeleton-connectivity framework brings in a different +angle to understand parallel BCC algorithms, and eventually +helps us come up with FAST-BCC that is simple and efficient. +B +Additional Proofs +The proofs here are not very complicated, and should have +been shown previously. We provide them here mainly for +completeness since the proofs of other lemmas in Sec. 4.2 +use them. +B.1 +Proof of Fact 4.1 +Proof. Assume to the contrary that two BCCs𝐶1 and𝐶2 share +at least two common vertices. After removing an arbitrary +vertex from 𝐶1 ∪ 𝐶2, there is at least one common vertex +remaining. WLOG, we assume 𝑢 is a remaining common +vertex. Because 𝐶1 and 𝐶2 are BCCs and 𝑢 is in both of them, +all the remaining vertices in 𝐶1 ∪ 𝐶2 are connected to 𝑢, so +they remain in the same CC as 𝑢. Therefore, 𝐶1 ∪𝐶2 is a BCC, +which contradicts with that 𝐶1 and 𝐶2 are two BCCs. +□ +B.2 +Proof of Fact 4.2 +Proof. We first rewrite the cycle starts and ends with 𝑣𝑖 as +𝑣0–𝑣1–𝑣2–...–𝑣 𝑗–...–𝑣𝑘 (𝑣0 = 𝑣𝑘 = 𝑣𝑖). All the other vertices +on the cycle appear exactly once. Then, there exists at least +two disjoint paths that connect 𝑣𝑖 and 𝑣 𝑗: one is 𝑣0–...–𝑣 𝑗, the +other one is𝑣 𝑗–...–𝑣𝑘. Removing any vertex other than𝑣𝑖 and +𝑣 𝑗 disconnects at most one of the two paths, while the other +path still connects 𝑣𝑖 and 𝑣 𝑗. Thus, after removing any vertex, +all the remaining vertices on the cycle are still connected, so +all vertices on the cycle are in the same BCC. +□ +B.3 +Proof of Lem. 4.3 +Proof. Assume to the contrary that 𝐶 has at least two CCs in +𝑇. For each CC, we find the shallowest node. Let 𝑢 and 𝑣 be +the shallowest two of these two CCs. WLOG assume 𝑢 is no +deeper than 𝑣. There are two possible positions for 𝑢 and 𝑣 +in the spanning tree 𝑇. We first show that in both cases, 𝑣’s +parent 𝑤 is biconnected with 𝑢. +Case 1: neither 𝑢 nor 𝑣 is the ancestor of the other. Based +on our assumption, there exists at least one path 𝑃 from 𝑢 +to 𝑣 using vertices in 𝐶. Note that no vertices on the tree +path from 𝑢 to 𝑣 are included in 𝑃. We now show that there +are two disjoint paths that connect 𝑤 and 𝑢: 1) the tree path +between𝑤 and𝑢, which does not contain intermediate nodes +from 𝐶, and 2) the path from 𝑤 to 𝑣 then to 𝑢, only using +intermediate nodes from 𝐶. Hence, 𝑤 and 𝑢 are biconnected. +Case 2: 𝑢 is 𝑣’s ancestor. We can similarly show that there +are at least two paths from 𝑤 to the nearest vertex in 𝑢’s +component, one using the tree path while the other using +vertices in 𝐶. Hence, 𝑤 and 𝑢 are biconnected. +In both cases, we can show that 𝑤 and 𝑢 are biconnected. +For any 𝑥 ∈ 𝐶, removing any other vertex 𝑦 ∈ 𝐶, 𝑥 and 𝑤 +are still connected through either 𝑢 or 𝑣. Hence, 𝑤 and 𝐶 are +biconnected, contradicting that 𝑣 is the shallowest tree node +in the BCC (𝑤 is 𝑣’s parent). +□ +B.4 +Proof of Lem. 4.4 +Proof. We first prove that each non-root BCC head is an +articulation point. A BCC head ℎ is the shallowest node +for a BCC 𝐶. Let the 𝑐 be one of ℎ’s children in 𝐶. Assume +to the contrary that ℎ is not an articulation point, then 𝐺 +is still connected after removing ℎ, including 𝑝(ℎ) and 𝑐. +Removing any other vertex other than 𝑐, ℎ, and 𝑝(ℎ) does +not disconnect 𝑐 and ℎ based on the definition of the BCC, +so 𝑐 and 𝑝(ℎ) is also connected using an additional tree edge +ℎ–𝑝(ℎ). Combining both cases, 𝑝(ℎ) is biconnected with 𝑐, +contradicting that ℎ is a BCC head. +We now show an articulation point 𝑎 must be a BCC head. +Assume to the contrary that 𝑎 is not a BCC head, then based +on Lem. 4.3, all 𝑎’s children must be in the same BCC as 𝑝(𝑎). +Since 𝑎 is an articulation point, removing it disconnects +𝐺. However, all 𝑎’s children’s subtrees are still connected, +so as 𝑉 \ 𝑇𝑎 using tree edges. Hence, at least one of 𝑎’s +children, referred to as 𝑐, is disconnected from 𝑝(𝑎), since +otherwise 𝑎 is not an articulation point. In this case, 𝑎 is the +BCC head of the BCC which 𝑎 and 𝑐 is in, contradicting the +assumption. +□ +14 + +B.5 +Proof of Lem. 4.5 +Proof. To prove that the skeleton is generated correctly, we +show that the functions InSkeleton, Fence, and Back work +as expected, and InSkeleton returns true iff. edge 𝑢–𝑣 is a +plain edge or a cross edge. We first prove that a vertex 𝑢 is +𝑣’s ancestor if and only if first[𝑢] ≤ first[𝑣] and last[𝑢] ≥ +last[𝑣], which is exactly Line 14. +On an Euler tour of a spanning tree, each edge appears +exactly twice (one time in each direction) in a DFS order. +first[·]/last[·] stores the time stamps each the vertex first/last +appears on the Euler tour. We first show that ∀𝑣 ∈ 𝑇𝑢, +first[𝑢] ≤ first[𝑣] and last[𝑣] ≤ last[𝑢]. This is because +all the tree edges in 𝑇𝑢 are traversed after 𝑢 have been +traversed, so ∀𝑣 ∈ 𝑇𝑢, first[𝑣] ≥ first[𝑢]; and 𝑢 last ap- +pears when all the tree edges in 𝑇𝑢 have been traversed, +so ∀𝑣 ∈ 𝑇𝑢, last[𝑣] ≤ last[𝑢]. +We show that if 𝑢 is an ancestor of 𝑣, then the function +Line 14 in Alg. 1 returns true. If𝑢 is an ancestor of 𝑣, then 𝑣 ∈ +𝑇𝑢, so that first[𝑢] ≤ first[𝑣] and last[𝑢] ≥ last[𝑣] ≥ first[𝑣]. +Then we will show if 𝑢 is not an ancestor of 𝑣, then the +function Line 14 in Alg. 1 returns false (at least one of the two +conditions is false). If 𝑢 is not an ancestor of 𝑣, there are two +cases for𝑢:𝑢 ∈ 𝑇𝑣 or𝑢 ∉ 𝑇𝑣. If𝑢 ∈ 𝑇𝑣, then first[𝑣] ≤ first[𝑢], +so the first condition in function Line 14 is false. If 𝑢 ∉ 𝑇𝑣, +either last[𝑢] < first[𝑣] and last[𝑣] < first[𝑢]. If last[𝑢] < +first[𝑣], the second condition in function Line 14 is false. If +last[𝑣] < first[𝑢], because first[𝑣] < last[𝑣] < first[𝑢], the +first condition is false. Therefore, 𝑢 is an ancestor of 𝑣 iff. +the function Line 14 in Alg. 1 returns true. This function is +called on edge 𝑢–𝑣 by the function InSkeleton only when +it is a tree edge. Therefore, 𝑢 is a non-parent ancestor of 𝑣. +Therefore, InSkeleton returns true on Line 10 iff. 𝑢–𝑣 is a +cross edge. +We then prove that the function Line 12 in Alg. 1 can +correctly determine whether a tree edge 𝑢–𝑣 is a fence edge. +We first show that if tree edge 𝑢–𝑣 is a fence edge, Line 12 +in Alg. 1 returns true. We just showed that ∀𝑥 ∈ 𝑇𝑢, first[𝑢] ≤ +first[𝑥] and last[𝑢] ≥ last[𝑥] ≥ first[𝑥]. If tree edge 𝑢–𝑣 is +a fence edge, then for all the edges with one endpoint in 𝑇𝑣, +the other endpoint must be in 𝑇𝑢. Recall that low[𝑣] is the +earliest (with the smallest first value) vertex connected to 𝑣’s +subtree. This means that this earliest vertex is also in𝑇𝑢, and +therefore low[𝑣] ≥ first[𝑢]. Similarly, high[𝑣], which is the +latest (with the largest first value) vertex connected to 𝑣’s +subtree, should also be in 𝑇𝑢. Therefore last[𝑢] ≥ high[𝑣]. +Then we show that if tree edge 𝑢–𝑣 is not a fence edge, +Line 12 in Alg. 1 returns false. If tree edge 𝑢–𝑣 is not a fence +edge, then there exists an edge 𝑥 ′–𝑦′, where 𝑥 ′ ∈ 𝑇𝑣 and 𝑦′ ∉ +𝑇𝑢. Because 𝑦′ ∉ 𝑇𝑢, either first[𝑦′] < first[𝑢] or first[𝑦′] > +last[𝑦′]. If first[𝑦′] < first[𝑢], then +low[𝑣] ≤ 𝑤1[𝑥 ′] ≤ first[𝑦′] < first[𝑢] +Then the first condition in Line 12 in Alg. 1 is false. If +first[𝑦′] > last[𝑢], then +high[𝑣] ≥ 𝑤2[𝑥 ′] ≥ first[𝑦′] > last[𝑢] +Then the second condition in Line 12 in Alg. 1 is false. +Therefore, the tree edge 𝑢–𝑣 is a fence edge iff. function +Line 12 in Alg. 1 returns true. +In summary, InSkeleton returns true iff. edge 𝑢–𝑣 is a +plain edge or a cross edge. Therefore, the skeleton 𝐺 ′ can be +determined correctly by InSkeleton. +□ +B.6 +Proof of Thm. 5.1 +Proof. Before we show the analysis, we will first review the +two key techniques that name this algorithm in ConnectIt: +low-diameter decomposition (LDD) [53] and a union-find +structure by Jayanti et al. [47]. A (𝛽,𝑑)-decomposition of a +graph𝐺 = (𝑉, 𝐸) is a partition of𝑉 into subsets𝑉1,𝑉2, · · · ,𝑉𝑘 +such that (1) the diameter of each 𝑉𝑖 is at most 𝑑, and (2) the +number of edges (𝑢, 𝑣) ∈ 𝐸 with endpoints in different sub- +sets, i.e., such that 𝑢 ∈ 𝑉𝑖, 𝑣 ∈ 𝑉𝑗 and 𝑖 ≠ 𝑗, is at most +𝛽𝑚. A parallel (𝛽,𝑂((log𝑛)/𝛽)) decomposition algorithm +is provided by Miller et. al. [53], using 𝑂(𝑛 + 𝑚) work and +𝑂((log2 𝑛)/𝛽) span whp. The high-level idea of LDD is to +start with a single source and search out using BFS. Then in +later rounds, we exponentially add new sources to the fron- +tier and continue BFS processes. By controlling the speed to +add new sources, the entire BFS will finish in 𝑂((log𝑛)/𝛽) +rounds, leaving at most 𝛽𝑚 edges with endpoints from dif- +ferent sources. +Once the LDD is computed, the algorithm will examine all +cross edges (endpoints from different sources) using a union- +find structure by Jayanti et al. [47] to merge different compo- +nents. The algorithm either performs finds naively without +using any path compression or uses a strategy called Find- +Two-Try-Split. Such strategies guarantee provably-efficient +bounds. The original bound is 𝑂(𝑙 ·(𝛼(𝑛,𝑙/(𝑛𝑝))+log(𝑛𝑝/𝑙 + +1))) expected work and 𝑂(log𝑛) PRAM time for a problem +instance with 𝑙 operations on 𝑛 elements on a PRAM with 𝑝 +processors. When translating this bound to the binary fork- +join model, all 𝑙 operations can be in parallel in the worst +case, which leads to the work bound as 𝑂(𝑙 log𝑛). +We note that if we set 𝛽 = 1/log𝑛, the LDD takes 𝑂(𝑛 + +𝑚) work and 𝑂((log3 𝑛)/𝛽) span whp, and the union-find +part takes 𝑂(𝛽𝑚 log𝑛) = 𝑂(𝑚) work and 𝑂(log2 𝑛) span. +Combining the two pieces together gives 𝑂(𝑛 +𝑚) work and +𝑂((log3 𝑛)/𝛽) span whp for the “LDD-UF-JTB” algorithm. +□ +C +Performance Analysis of the Local +Search Optimality +In FAST-BCC, CC is an important primitive used in First-CC +and Last-CC. We optimized the CC implementation using +hash bags and local searches proposed by a recent paper [67]. +These optimizations function as a parallel granularity control. +15 + +0.00 +0.02 +0.04 +YT +0.00 +0.05 +0.10 +OK +0.00 +0.05 +0.10 +LJ +0 +1 +TW +0 +2 +FT +0.00 +0.02 +GG +First CC +Rooting +Tagging +Last CC +0 +2 +SD +0 +10 +20 +CW +0 +20 +HL14 +0 +50 +HL12 +0.000 +0.025 +0.050 +CA +0.0 +0.5 +USA +0.0 +0.2 +0.4 +GE +0.00 +0.05 +0.10 +HH5 +0.0 +0.1 +0.2 +CH5 +0.0 +0.2 +0.4 +GL2 +0.0 +0.5 +1.0 +GL5 +0.0 +0.5 +1.0 +GL10 +Orig. +Opt. +0.0 +0.5 +1.0 +GL15 +Orig. +Opt. +0.0 +0.5 +1.0 +GL20 +Orig. +Opt. +0 +5 +10 +COS5 +Orig. +Opt. +0 +2 +SQR +Orig. +Opt. +0 +2 +REC +Orig. +Opt. +0 +1 +2 +SQR' +Orig. +Opt. +0 +1 +2 +REC' +Orig. +Opt. +0.0 +0.5 +1.0 +Chn7 +Orig. +Opt. +0 +10 +Chn8 +Figure 6. Optimized BCC breakdown. 𝑦-axis is the running time in seconds. "Orig."= our original BCC implementation, "Opt."= our +implementation optimized with hash bags and local search proposed in [67]. +When the frontier (vertices being processed in one round) +is small, the algorithm explores multi-hop neighbors of the +frontier instead of one-hop neighbors to saturate all threads +with sufficient work. It helps reduce the number of total +rounds in a connectivity search, thus reducing the synchro- +nization costs between rounds. It works favorably well on +large-diameter graphs. We measure the improvement from +the optimizations in Fig. 6, where Orig. is the version without +the optimizations and Opt. is the version with the optimiza- +tions. As shown in Fig. 6, on low-diameter graphs, Orig. and +Opt. have similar performance. On large-diameter, Opt. can +be 1.1–4.5× faster than Orig. and is 1.8× faster on average. +D +Performance of the Tarjan-Vishkin +Algorithm and Space Usage +For completeness, we also implemented the faithful Tarjan- +Vishkin algorithm [63] discussed in Appendix A. We ac- +knowledge that some existing papers [28, 37] discussed some +possible optimizations for Tarjan-Vishkin. However, engi- +neering the Tarjan-Vishkin algorithm is not the focus of this +paper, and we mainly use it to measure the space usage and +get a sense on how Tarjan-Vishkin compares to other exist- +ing BCC algorithms. Our conclusions are consistent with the +results drawn from the previous papers [28, 62]. +The running time and space usage are given in Tab. 3 and +Fig. 7. For our implementations, Tarjan-Vishkin and use up +to 11× extra space than FAST-BCC on FT and SD (including +the space to store the input graph). The space overhead is +decided by edge-to-vertex ratio, and for graphs with smaller +ratios (e.g., chain graphs), the overhead is small. Our TV +implementation cannot run on the three largest graphs (CW, +HL14, and HL12) on our machines with 1.5TB memory. We +note that the smallest among them (CW) only takes about +300GB to store the graph, and FAST-BCC uses 572GB mem- +ory to process it. Since all three graphs have relatively large +Ours +GBBS +TV +SEQ +Social +YT +0.030 +0.040 +0.076 +0.175 +OK +0.103 +0.158 +2.10 +3.14 +LJ +0.104 +0.159 +0.859 +1.87 +TW +1.44 +2.83 +25.7 +49.2 +FT +3.10 +6.44 +41.6 +122 +Web +GG +0.029 +0.045 +0.119 +0.255 +SD +3.11 +5.61 +43.0 +92.3 +CW +22.9 +39.7 +N/A +695 +HL14 +31.1 +50.7 +N/A +1011 +HL12 +89.1 +105 +N/A +3027 +Road +CA +0.040 +0.372 +0.079 +0.206 +USA +0.336 +4.64 +0.673 +2.25 +GE +0.267 +2.02 +0.492 +2.88 +𝒌-NN +HH5 +0.073 +0.447 +0.169 +0.509 +CH5 +0.128 +1.44 +0.380 +0.528 +GL2 +0.402 +1.53 +0.771 +2.51 +GL5 +0.472 +2.80 +1.73 +4.03 +GL10 +0.668 +1.64 +3.68 +7.07 +GL15 +0.751 +1.51 +5.06 +8.92 +GL20 +0.861 +1.48 +6.91 +10.2 +COS5 +8.46 +17.5 +50.1 +120 +Synthetic +SQR +1.32 +15.4 +6.79 +24.4 +REC +1.35 +47.0 +6.99 +16.8 +SQR’ +1.31 +12.5 +2.76 +10.6 +REC’ +1.37 +22.4 +2.83 +10.7 +Chn7 +0.278 +81.6 +0.343 +3.33 +Chn8 +3.25 +957 +3.97 +38.9 +Table 3. Tarjan-Vishkin running time (in seconds). “N/A” = +not applicable because of out of memory. +edge-to-vertex ratios, our TV implementations are unlikely +to execute on these graphs for shared-memory machines +in foreseeable future. GBBS is slightly more space-efficient +16 + +YT +OK +LJ +TW +FT +GG +SD +CW +HL14 +HL12 +CA +USA +GE +HH5 +CH5 +GL2 +GL5 +GL10 +GL15 +GL20 +COS5 +SQR +REC +SQR' +REC' +Chn7 +Chn8 +1 +5 +10 +15 +FAST-BCC +GBBS +Tarjan-Vishkin +Figure 7. Space Usage Comparision. 𝑦-axis is the space usage normalized to GBBS. Lower is better. +than FAST-BCC, and takes about 20% less space than us. The +reason is that they need to compute fewer number of tags +than FAST-BCC. +On all graphs, TV is faster than SEQ on 96 cores, but slower +than FAST-BCC. The overhead of TV is due to the cost to +explicitly construct the skeleton. On social, web, and 𝑘-NN +graphs, the speedup for TV on 96 cores over SEQ is only +1.4–3×. TV works well on road and synthetic graphs due +to small edge-to-vertex ratio, so the 𝑂(𝑚) work and space +for generating the skeleton does not dominate the running +time. In this case, polylogarithmic span allows TV to perform +consistently better than GBBS. +17 + diff --git a/b9AzT4oBgHgl3EQfZvxg/content/tmp_files/load_file.txt b/b9AzT4oBgHgl3EQfZvxg/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..cb4a918960c22f20fc798fdf6e2f1dc277366fd9 --- /dev/null +++ b/b9AzT4oBgHgl3EQfZvxg/content/tmp_files/load_file.txt @@ -0,0 +1,2030 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf,len=2029 +page_content='Provably Fast and Space-Efficient Parallel Biconnectivity Xiaojun Dong UC Riverside xdong038@ucr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='edu Letong Wang UC Riverside lwang323@ucr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='edu Yan Gu UC Riverside ygu@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='ucr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='edu Yihan Sun UC Riverside yihans@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='ucr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='edu Abstract Biconnectivity is one of the most fundamental graph prob- lems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' The canonical parallel biconnectivity algorithm is the Tarjan-Vishkin algorithm, which has 𝑂(𝑛 +𝑚) optimal work (number of operations) and polylogarithmic span (longest de- pendent operations) on a graph with 𝑛 vertices and 𝑚 edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' However, Tarjan-Vishkin is not widely used in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We believe the reason is the space-inefficiency (it generates an auxiliary graph with 𝑂(𝑚) edges).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' In practice, existing par- allel implementations are based on breath-first search (BFS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Since BFS has span proportional to the diameter of the graph, existing parallel BCC implementations suffer from poor per- formance on large-diameter graphs and can be even slower than the sequential algorithm on many real-world graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We propose the first parallel biconnectivity algorithm (FAST-BCC) that has optimal work, polylogarithmic span, and is space-efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Our algorithm first generates a skele- ton graph based on any spanning tree of the input graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Then we use the connectivity information of the skeleton to compute the biconnectivity of the original input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' All the steps in our algorithm are highly-parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We carefully analyze the correctness of our algorithm, which is highly non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We implemented FAST-BCC and compared it with exist- ing implementations, including GBBS, Slota and Madduri’s algorithm, and the sequential Hopcroft-Tarjan algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We ran them on a 96-core machine on 27 graphs, including social, web, road, 𝑘-NN, and synthetic graphs, with signif- icantly varying sizes and edge distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' FAST-BCC is the fastest on all 27 graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' On average (geometric means), FAST-BCC is 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='1× faster than GBBS, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='1× faster than the best existing baseline on each graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' CCS Concepts: • Theory of computation → Shared mem- ory algorithms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Graph algorithms analysis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Parallel al- gorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Keywords: Parallel Algorithms, Graph Algorithms, Bicon- nectivity 1 Introduction Computing the biconnected components is one of the most fundamental graph problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Given an undirected graph 𝐺 = (𝑉, 𝐸) with 𝑛 = |𝑉 | vertices and 𝑚 = |𝐸| edges, a connected component (CC) is a maximal subset in 𝑉 such that every two vertices in it are connected by a path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' A biconnected component (BCC) (or blocks) is a maximal subset 𝐶 ⊆ 𝑉 such that 𝐶 is connected and remains con- nected after removing any vertex 𝑣 ∈ 𝐶.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' In this paper, we use BCC (or CC) for both the biconnected (or connected) com- ponent in the graph and the problem to compute all BCCs (or CCs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' BCC has extensive applications such as planarity testing [7, 23, 44], centrality computation [46, 57, 58], and network analysis [6, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Sequentially, the Hopcroft-Tarjan algorithm [43] can com- pute the BCCs of a graph in 𝑂(𝑛 + 𝑚) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' However, this algorithm requires generating a spanning tree of 𝐺 based on the depth-first search (DFS), which is considered hard to be parallelized [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Later, Tarjan and Vishkin proposed the canonical parallel BCC algorithm along with the Euler-tour technique [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' It uses an arbitrary spanning tree (AST) (a spanning tree with any possible shape) of the graph instead of the depth-first tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Tarjan-Vishkin algorithm has𝑂(𝑛+𝑚) optimal work (number of operations) and polylogarithmic span (longest dependent operations), assuming an efficient parallel CC algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Although the Tarjan-Vishkin algorithm is theoretically considered “optimal” in work and span, significant challenges still remain in achieving a high-performance implementa- tion in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' The main issue in Tarjan-Vishkin is space- inefficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Tarjan-Vishkin generates an auxiliary graph 𝐺 ′ = (𝑉 ′, 𝐸′) (which we refer to as the skeleton), where every edge 𝑒 ∈ 𝐸 maps to a vertex in 𝑉 ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Tarjan and Vishkin showed that computing CC on𝐺 ′ gives the BCC on𝐺, and we refer to this step as the connectivity phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' This skeleton- connectivity framework is adopted in many later papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' It first generates a skeleton, which is an auxiliary graph 𝐺 ′ from 𝐺, and then finds the CCs on 𝐺 ′ that reflects BCC information on the input graph 𝐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Unfortunately, in Tarjan- Vishkin, generating the skeleton 𝐺 ′ (which takes 𝑂(𝑚) extra space) and computing CC on 𝐺 ′ greatly increase the memory usage and slow down the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' In practice, most existing parallel BCC implementations also follow the skeleton-connectivity framework but over- come the space issue by using other skeletons based on breadth-first search (BFS) trees [24, 25, 28, 30, 38, 62, 66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' They are based on sparse certificates [26], and more discus- sions are given in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' These algorithms either use skele- tons with 𝑂(𝑛) size [24, 25, 28, 38, 66] or maintain implicit skeletons with 𝑂(𝑛) auxiliary space [30, 62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We say a BCC algorithm is space-efficient if it uses 𝑂(𝑛) auxiliary space (other than the input graph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' However, since computing BFS has span proportional to the graph, these BFS-based algo- rithms can be fast on small-diameter graphs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=', social and web graphs), but have poor performance on large-diameter graphs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=', 𝑘-nn and road graphs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' In our experiments, we observe that existing parallel implementations can even be 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='01356v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content="DS] 3 Jan 2023 Ours GBBS SM'14 SEQ Ours GBBS SM'14 SEQ Social YT 5." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='88 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='36 3.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='60 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='00 Chn7 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='08 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='00 GE 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='77 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='43 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='44 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='00 Chn8 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='06 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='18 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='00 TOTAL MEAN 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='89 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='96 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='00 1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='5 2 4 8 16 32 >32 MEAN = geometric mean n = no support Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' The heatmap of relative speedup for parallel BCC algorithms over the sequential Hopcroft-Tarjan algo- rithm [43] using 96 cores (192 hyper-threads).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Larger/green means better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' The numbers indicate how many times a parallel algorithm is faster than sequential Hopcroft-Tarjan (< 1 means slower).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' The two baseline algorithms are from [30, 62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Complete results are in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' slower than sequential Hopcroft-Tarjan on many real-world graphs (see GBBS [30] and SM’14 [62] in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' In this paper, we give the first space-efficient (𝑂(𝑛) auxiliary space) parallel biconnectivity algorithm that has efficient 𝑂(𝑚 + 𝑛) work and polylogarithmic span.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Our skeleton 𝐺 ′ is based on an arbitrary spanning tree (AST).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Unlike Tarjan-Vishkin, our 𝐺 ′ is a subgraph of 𝐺 and can be maintained implicitly in 𝑂(𝑛) auxiliary space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' The key idea is to carefully identify some fence edges, which indicate the “boundaries” of the BCCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' At a high level, we categorize all graph edges into fence tree edges, plain (non-fence) tree edges, back edges, and cross edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Our skeleton 𝐺 ′ contains the plain tree edges and cross edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Using 𝑂(𝑛) space, we can efficiently determine the category of each edge in 𝐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' When processing the skeleton, we use the input graph 𝐺 but skip the fence and back edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We show that the BCC information of𝐺 can be constructed from the CC information of 𝐺 ′ plus some simple postprocessing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Since our algorithm is based on Fencing an Arbitrary Spanning Tree, we call our algorithm FAST-BCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' More details of FAST-BCC are in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We note that conceptually our algorithm is simple, but the correctness analysis is highly non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We implement our theoretically-efficient FAST-BCC al- gorithm and compare it to the state-of-the-art parallel BFS- based BCC implementations GBBS [30] and SM’14 [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We also compare FAST-BCC to the sequential Hopcroft-Tarjan algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We test 27 graphs, including social, web, road, 𝑘-NN, and synthetic graphs, with significantly varying sizes and edge distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' The details of the graphs and results are given in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We also show the relative running time in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 1, normalized to the sequential Hopcroft-Tarjan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' On a machine with 96 cores, FAST-BCC is the fastest on all tested graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We use the geometric means to compare the “average” performance across multiple graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Due to work- and space-efficiency, our algorithm running on one core is competitive with Hopcroft-Tarjan (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='8× slower on average).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Polylogarithmic span leads to good parallelism for all types of graphs (15–66× self-relative speedup on average).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' On small-diameter graphs (social and web graphs), although GBBS and SM’14 also achieve good parallelism, FAST-BCC is still 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='2–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='1× faster than the best of the two, and is 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='9–39× faster than sequential Hopcroft-Tarjan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' For large-diameter graphs (road, 𝑘-nn, grid, and chain graphs), existing BFS-based implementations can perform worse than Hopcroft-Tarjan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Due to the low span, FAST-BCC is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='7–295× faster than GBBS (10× on average), and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='1–18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='5× faster than sequential Hopcroft-Tarjan (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='2× on average).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' On all graphs, FAST-BCC is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='1× faster on average than the best of the three existing implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Our code is publicly available [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 2 Preliminaries Computational Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We use the work-span (or work- depth) model for fork-join parallelism with binary forking to analyze parallel algorithms [14, 29], which is recently used in many papers on parallel algorithms [3, 9, 10, 12, 13, 15–21, 32– 34, 40, 41, 60, 69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We assume a set of threads that share a common memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' A process can fork two child software threads to work in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' When both children complete, the parent process continues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' The work of an algorithm is the total number of instructions and the span (depth) is the length of the longest sequence of dependent instructions in the computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We say an algorithm is work-efficient if its work is asymptotically the same as the best sequential algo- rithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We can execute the computation using a randomized work-stealing scheduler [5, 22] in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We assume unit- cost atomic operation compare_and_swap(𝑝, 𝑣old, 𝑣new) (or CAS), which atomically reads the memory location pointed to by 𝑝, and write value 𝑣new to it if the current value is 𝑣old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' It returns true if successful and false otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Given an undirected graph 𝐺 = (𝑉, 𝐸), we use 𝑛 = |𝑉 |, 𝑚 = |𝐸|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Let diam(𝐺) be the diameter of 𝐺, and 𝑥–𝑦 be an edge between 𝑥 and 𝑦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' CC and BCC are as defined in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' An articulation point (or cut vertex) is a vertex s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' removing it increases the number of CCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' A bridge (or cut edge) is an edge s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' removing it increases the number of CCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' A spanning tree𝑇 of a connected graph 𝐺 is a spanning subgraph of 𝐺 that contains no cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' The spanning forest is defined similarly if 𝐺 is disconnected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' For simplicity, we as- sume 𝐺 is connected, but our algorithm and implementation work on any graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Given a graph 𝐺 and a rooted spanning tree 𝑇, an edge is a tree edge if it is in 𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' A non-tree edge is a back edge if one endpoint is the ancestor of the other endpoint, and a cross edge otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 2 Step 3 shows an 2 𝐺 = (𝑉, 𝐸) : Input Graph 𝑇 = (𝑉, 𝐸𝑇 ): A spanning tree in 𝐺 𝑎,𝑏,𝑐,𝑢, 𝑣,ℎ,𝑤,𝑥,𝑦,𝑧,𝑢′, 𝑣′,𝑐′ · · · ∈ 𝑉 : Vertices in 𝐺 𝑥–𝑦 ∈ 𝐸 : An edge in 𝐺 𝐶,𝐶𝑖 : A BCC in 𝐺 𝑇𝑢 : 𝑢’s subtree in 𝑇 ℎ𝐶 : The BCC head of 𝐶 𝑝(𝑢) : 𝑢’s parent in 𝑇 𝑥 ~ 𝑦 : A tree path in 𝑇 𝑃 = 𝑥–𝑦–· · · : A path 𝐺′ : The skeleton Fence edge: (𝑝(𝑣), 𝑣) ∈ 𝐸𝑇 , � (𝑥,𝑦) ∈ 𝐸, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 𝑥 ∈ 𝑇𝑣 and 𝑦 ∉ 𝑇𝑝 (𝑣) (no edge from 𝑣’s subtree escapes from 𝑝(𝑣)’s subtree) Plain edge : (𝑝(𝑣), 𝑣) ∈ 𝐸𝑇 , (𝑝(𝑣), 𝑣) is not a fence edge Back edge, Cross edge : Edges in 𝐸 \\ 𝐸𝑇 , defined as usual Skeleton 𝐺′ = (𝑉, 𝐸′) in FAST-BCC : 𝐸′ = {plain & cross edges} Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Notations and terminologies in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' If 𝑇 is a BFS tree, there are no back edges;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' if 𝑇 is a DFS tree, there are no cross edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We use 𝑥 ~𝑦 to denote the tree path between 𝑥 and 𝑦 on 𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We denote the parent of vertex 𝑢 as 𝑝(𝑢), and the subtree of 𝑢 as 𝑇𝑢.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' The notation used in this paper is given in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We use 𝑂(𝑓 (𝑛)) with high probability (whp) in 𝑛 to mean 𝑂(𝑐𝑓 (𝑛)) with probability at least 1 − 𝑛−𝑐 for 𝑐 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Euler tour technique (ETT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' ETT is proposed by Tarjan and Vishkin [63] in their BCC algorithm to root a spanning tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Later, ETT becomes a widely-used primitive in both sequential and parallel settings, including computational ge- ometry [2], graph algorithms [4, 27, 65], maintaining subtree or tree path sums [29], and many others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' ETT is needed in Tarjan-Vishkin because when an arbitrary spanning tree is generated for a graph (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=', from a CC algorithm), it is not rooted, and thus we do not have the parent-child information for the vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Given an unrooted tree 𝑇 with 𝑛 − 1 edges, ETT finds an Euler tour of 𝑇, which is a cycle traversing each edge in 𝑇 exactly twice (once in each direction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' ETT first constructs a linked list on the 2𝑛 − 2 directed tree edges, and runs list ranking on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We refer the audience to the text- books on parallel algorithms [45, 56] for more details on ETT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Using the semisort algorithm from [14, 42] and list ranking from [14], ETT costs 𝑂(𝑛) expected work and 𝑂(log𝑛) span whp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Given𝑇, we can set any vertex as the root of𝑇, and use ETT to determine the directions of the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We can then determine the parent of any vertex, and whether an edge is a tree edge, back edge, or cross edge in 𝑂(1) work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 3 Existing BCC Algorithms This section reviews the existing BCC algorithms and imple- mentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We will use the skeleton-connectivity framework to describe the existing BCC algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' The skeleton phase generates a skeleton 𝐺 ′ from 𝐺, which is an auxiliary graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Then the connectivity phase computes the connectivity on 𝐺 ′ to construct the BCCs of 𝐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Existing BCC algorithms can be categorized by how the skeleton 𝐺 ′ is generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' The Hopcroft-Tarjan algorithm uses DFS-based skeletons;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' the Tarjan-Vishkin Algorithm generates a skeleton based on an arbitrary spanning tree (AST);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' almost all other BCC algo- rithms (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='3) use BFS-based skeletons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='1 The Hopcroft-Tarjan Algorithm Sequentially, Hopcroft-Tarjan BCC algorithm [43] has 𝑂(𝑛 + 𝑚) work using a depth-first search (DFS) tree 𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Based on 𝑇, two tags first[·] and low[·] are assigned to each vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' first[𝑣] is the preorder number of each vertex in 𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' low[𝑣] gives the earliest (smallest preorder) vertex incident on any vertex 𝑢 ∈ 𝑇𝑣 via a non-tree edge and 𝑢 itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' More formally, low[𝑣] = min{𝑤1[𝑢] | 𝑢 ∈ 𝑉 is in the subtree rooted at 𝑣} 𝑤1[𝑢] = min{{first[𝑢]} ∪ {first[𝑢′] | (𝑢,𝑢′) ∉ 𝑇 }} To compute the BCCs, an additional stack is maintained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Each time we visit a new edge, it is pushed into the stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' When an articulation point 𝑝(𝑢) is found by 𝑢 (low[𝑢] ≥ first[𝑝(𝑢)]), edges are popped from the stack until 𝑢–𝑝(𝑢) is popped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' These edges and the relevant vertices form a BCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Conceptually, the skeleton in Hopcroft-Tarjan is the DFS tree without the “fence edges” of 𝑢–𝑝(𝑢) when low[𝑢] ≥ first[𝑝(𝑢)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' This insight also inspires our BCC algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='2 The Tarjan-Vishkin Algorithm Hopcroft-Tarjan uses a DFS tree as the skeleton, but DFS is in- herently serial and hard to be parallelized [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' To parallelize BCC, the Tarjan-Vishkin algorithm [63] uses an arbitrary spanning tree (AST) instead of a DFS tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' This spanning tree 𝑇 can be obtained by any parallel CC algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' The algorithm then uses ETT (which is also proposed in that pa- per) to root the tree 𝑇 (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Then the algorithm builds a skeleton 𝐺 ′ = (𝐸, 𝐸′) and runs a connectivity algorithm on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We describe 𝐺 ′ in more details in Appendix A, and only briefly review it here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' The vertices in 𝐺 ′ correspond to the edges in 𝐺1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' To determine the edges in 𝐺 ′, the algo- rithm uses four tags (first[·], last[·], low[·], and high[·]) for each vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Here first[𝑢] and last[𝑢] are the first and last appearance of vertex 𝑢 in the Euler tour (note that this is not the same first[·] in Hopcroft-Tarjan, but conceptually equivalent).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' low[·] is the same as defined in Hopcroft-Tarjan, and high[·] is defined symmetrically: high[𝑣] = max{𝑤2[𝑢] | 𝑢 ∈ 𝑉 is in the subtree rooted at 𝑣} 𝑤2[𝑢] = max{{first[𝑢]} ∪ {first[𝑢′] | (𝑢,𝑢′) ∉ 𝑇 }} All tags can be computed in 𝑂(𝑛 + 𝑚) expected work and 𝑂(log𝑛) span whp using ETT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Tarjan-Vishkin then finds the CCs on 𝐺 ′ to compute the BCCs of 𝐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' However, 𝐺 ′ in Tarjan- Vishkin can be large, making the algorithm less practical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Assuming an efficient ETT and a parallel CC algorithm, Tarjan-Vishkin uses 𝑂(𝑛 + 𝑚) optimal expected work and polylogarithmic span.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' However, the space-inefficiency ham- pers the practicability of Tarjan-Vishkin since 𝐺 ′ contains 𝑚 vertices and 𝑂(𝑚) edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' In our experiments, Tarjan-Vishkin takes up to 11× extra space than our FAST-BCC or GBBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' On our machine with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='5TB memory, Tarjan-Vishkin ran out of memory when processing the Clueweb graph [52], although 1In a later paper [37], it was shown that the number of vertices in 𝐺′ can be reduced to 𝑂 (𝑛), but |𝐸′| is still 𝑂 (𝑚).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 3 it only takes about 300GB to store the graph (see discus- sions in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' The large space usage forbids running Tarjan-Vishkin on large-scale graphs on most multicore ma- chines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Even for small graphs, high space usage can increase memory footprint and slow down the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Some existing BCC implementations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=', GBBS [30] and TV-filter [28]) were also described as Tarjan-Vishkin algo- rithms, probably since they also use the skeleton-connectivity framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We note that their correctness relies on BFS- based skeletons (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=', sparse certificates [26]), and we catego- rized them below together with a few other algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='3 Other Existing Algorithms / Implementations Before Tarjan-Vishkin, Savage and JáJá [59] showed a par- allel BCC algorithm based on matrix-multiplication with 𝑂(𝑛3 log𝑛) work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Tsin and Chin [64] gave an algorithm that uses an AST-based skeleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' It is quite similar to Tarjan- Vishkin, but uses 𝑂(𝑛2) work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' To achieve space-efficiency, many later parallel BCC algo- rithms use BFS-based skeletons [24, 25, 28, 30, 38, 48, 62, 66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Many of them use the similar idea of sparse certificates [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' BCC is much simpler with a BFS tree—all non-tree edges are cross edges with both endpoints in the same or adja- cent levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Cong and Bader’s TV-filter algorithm [28] uses the skeleton as the BFS tree 𝑇 and an arbitrary spanning tree/forest for 𝐺 \\ 𝑇 (𝑂(𝑛) total size).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Slota and Madduri’s algorithms [62] and Dhulipala et al.’s algorithm [30] use the skeletons as the input graph𝐺 excluding𝑂(𝑛) vertices/edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' The other algorithms [24, 25, 38, 66] use a BFS tree as the skeleton, and compute connectivity dynamically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' All these algorithms are space-efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Their skeleton graphs either have 𝑂(𝑛) size [24, 25, 28, 38, 66] or can be implicitly repre- sented using 𝑂(𝑛) information [30, 62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' However, the span to generate a BFS tree is proportional to the diameter of the graph, which is inefficient for large-diameter graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='4 Space-Efficient BCC Representation Since some vertices (articulation points) appear in multiple BCCs (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 2 as an example), we need a representation of all BCCs in a space-efficient manner (𝑂(𝑛) space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We use a commonly used representation [10, 30, 38] in our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Given a spanning tree 𝑇, we assign a label for each vertex except for the root of 𝑇, indicating which BCC this vertex is in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' For all vertices with the same label, we find another vertex called the component head (see details in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='1) attached to this label, and all these vertices and the component head form a BCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' An example of this representation is given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' It is easy to see that this representation uses 𝑂(𝑛) space since we have 𝑛 − 1 labels for all vertices and at most 𝑛 − 1 component heads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 4 The FAST-BCC Algorithm In this section, we present our FAST-BCC algorithm with analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Our algorithm is the first parallel BCC algorithm that is work-efficient, space-efficient, and has polylogarith- mic span.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Recall that BFS-based algorithms are space-efficient, Algorithm 1: The FAST-BCC algorithm Input: An undirected graph 𝐺 = (𝑉, 𝐸) Output: The labels 𝑙[·] for vertices, and the component head for each BCC 1 Compute the spanning forest 𝐹 of 𝐺 ⊲ First CC 2 Root all trees in 𝐹 using the Euler tour technique ⊲ Rooting 3 Compute tags (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' low,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' high) of each vertex based on the Euler tour ⊲ Tagging 4 Compute the vertex label 𝑙[·] using connectivity on 𝐺 with edges satisfying InSkeleton(𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 𝑣) = true ⊲ Last CC 5 ParallelForEach 𝑢 ∈ 𝑉 with 𝑙[𝑢] ≠ 𝑙[𝑝(𝑢)] 6 Set the component head of 𝑙[𝑢] as 𝑝(𝑢) 7 Function InSkeleton(𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 𝑣)⊲ Decide if 𝑢–𝑣 is in skeleton 𝐺′ 8 if (𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 𝑣) is a tree edge then 9 return ¬ Fence(𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 𝑣) and ¬ Fence(𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='𝑢) 10 else return ¬ Back(𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 𝑣) and ¬ Back(𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='𝑢) 11 Function Fence(𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 𝑣) ⊲ Decide if tree edge is fence edge 12 return first[𝑢] ≤ low[𝑣] and last[𝑢] ≥ high[𝑣] 13 Function Back(𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 𝑣) ⊲ Decide if non-tree edge is back edge 14 return first[𝑢] ≤ first[𝑣] and last[𝑢] ≥ first[𝑣] but BFS itself does not parallelize well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Tarjan-Vishkin is based on AST and is highly parallel, but generating the skele- ton is space-inefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' To achieve both high parallelism and space efficiency, we need novel algorithmic insights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Interestingly, our key idea is to revisit the sequential DFS- based Hopcroft-Tarjan algorithm (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Although DFS is inherently sequential, the insights in Hopcroft-Tarjan in- spire our parallel BCC algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' The (implicit) skeleton in Hopcroft-Tarjan is simple and the skeleton size is small (𝑂(𝑛)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Unlike many later parallel BCC algorithms with the high-level ideas to combine cycles (based on Fact 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='2), the idea in Hopcroft-Tarjan is the “fencing” condition as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' When computing the CC on the skeleton 𝐺 ′ (the DFS tree) and traversing the edge from 𝑣 to 𝑝(𝑣), the CC on𝐺 ′ (BCC on 𝐺) is fenced if low[𝑣] ≥ first[𝑝(𝑣)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' This condition partitions the DFS tree 𝑇 into multiple CCs that correspond to BCCs in 𝐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Note that 𝐺 ′ in Hopcroft-Tarjan only contains edges from the DFS tree, because there are no cross edges in DFS trees and all back edge information is captured by low[·].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Now we try to generalize this idea to an arbitrary span- ning tree (AST).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Directly using the “fencing” condition in Hopcroft-Tarjan does not work since we need to deal with cross edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Note that a fence edge𝑣–𝑝(𝑣) in Hopcroft-Tarjan means that vertices in 𝑢’s subtree do not have an edge that escapes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=', the other endpoint is outside) 𝑝(𝑢)’s subtree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We define our fence edges also based on this condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' More for- mally, we say a tree edge (𝑢, 𝑣) where𝑢 = 𝑝(𝑣) is a fence edge if there is no edge (𝑥,𝑦) ∈ 𝐸 such that 𝑥 ∈ 𝑇𝑣 and 𝑦 ∉ 𝑇𝑢.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' In- tuitively, it means 𝑣’s subtree𝑇𝑣 is “isolated” from other parts outside 𝑝(𝑣)’s subtree, and only interacts with the outside world through 𝑝(𝑣).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' To get an equivalent condition for an AST, we borrow the idea from Tarjan-Vishkin and also com- pute four axillary arrays first[·], last[·], low[·], and high[·].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 4 Input Graph 𝑮: contains 3 BCCs {s, u}, {r, s, t, v, w, x}, {t, y, z}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Step 1: First CC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Find the CCs of 𝐺 and a spanning tree (forest).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' r s v w u x y z t r s u v w x y z t Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='2: Set any vertex as the root and run list ranking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' The result implies the tree edge directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Step 3: Tagging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='1: Find the CCs of the skeleton 𝐺′ only with cross and plain edges in 𝐺 (solid edges in Step 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Ignore the root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='2: Assign the component head to each CC in 𝐺′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Each CC in 𝐺′ with its component head is a BCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' r s u v w x y z t r s u v w x y z t Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='1: Based on the spanning tree 𝑇 of 𝐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Create the linked list of the Euler tour of 𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Step 2: Rooting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Generate rooted spanning trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' r s v w u x y z t r s u v w x y z t Fence edge Plain edge Back edge Cross edge Compute tags (first/last/low/high/…) for each vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Use these tags to identify fence, plain, cross, and back edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' BCC1 {s, u}: Head s + {u} BCC2 {s, t, v, w, x}: Head r + {s, t, v, w, x} BCC3 {t, y, z}: Head t + {y, z} Tree edge: Non-tree edge: Step 4: Last CC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Run CC on the skeleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' The outline of the FAST-BCC algorithm and a running example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' The four steps are explained in detail in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' The “fencing” condition then becomes low[𝑣] ≥ first[𝑝(𝑣)] and high[𝑣] ≤ last[𝑝(𝑣)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' A non-fence tree edge is referred to as a plain edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Note that the information for back edges is already captured by the low[·] and high[·] arrays, which will also be used to decide fence edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Our algorithm will ignore back edges as in Hopcroft-Tarjan, and our skeleton 𝐺 ′ contains plain tree edges and cross edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Since the main approach in our algorithm is Fencing an Arbitrary Spanning Tree, we call our algorithm FAST-BCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We note that the high-level idea of fencing (find some special edges on the spanning tree) is also used in some existing work [10, 30, 62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Our design of the skeleton and the fencing condition is the first to achieve work-efficiency, polylogarithmic span, and space-efficiency for the BCC problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' The outline of the algorithm is given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 2, and the pseudocode is in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Although our fencing algorithm is simple, we note that formally proving the correctness (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='2) is highly non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='1 Algorithmic Details Our FAST-BCC algorithm has four steps: First-CC (gener- ate spanning trees), Rooting (root the spanning trees using ETT), Tagging (compute first[·], last[·], 𝑤1[·], 𝑤2[·], low[·], high[·], 𝑝[·]), and Last-CC (run CC on the skeleton and post- processing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' In the skeleton-connectivity framework, the first three steps are the skeleton phase (compute the skele- ton 𝐺 ′), and the last step is the connectivity phase (run CC on 𝐺 ′ to find all BCCs in 𝐺).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' First-CC (Step 1 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 2, Line 1 in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' This step finds all CCs in 𝐺 and generates a spanning forest 𝐹 of 𝐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' For simplicity, in the following, we focus on one CC and its spanning tree 𝑇, which is unrooted at this moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' If 𝐺 contains multiple CCs, they are simply processed in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Running CC only requires 𝑂(𝑛) auxiluary space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Rooting (Step 2 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 2, Line 2 in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We use the Euler tour technique (ETT) in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 2 to root 𝑇, which implies the tree edge directions (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 2, Step 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' ETT requires 𝑂(𝑛) space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Tagging (Step 3 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 2, Line 3 in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' This step generates the tags used in the algorithm, including 𝑤1[·], 𝑤2[·], low[·], high[·], first[·], last[·] (same as in Tarjan-Vishkin, see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 3) and the parent array 𝑝[·].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' low[·] and high[·] values are com- puted by looping over all edges and getting arrays 𝑤1 and 𝑤2, and applying 𝑛 1D range-minimum queries (RMQ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' This step takes in 𝑂(𝑛 + 𝑚) work and 𝑂(log𝑛) span [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' These tags will help to decide the four edge types (see details below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' All the tag arrays have size 𝑂(𝑛).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Last-CC (Step 4 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 2, Line 4–6 in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' As men- tioned, our skeleton graph 𝐺 ′ contains plain tree edges and cross edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' To achieve space efficiency, we do not explic- itly store 𝐺 ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Since 𝐺 ′ is a subgraph of 𝐺, we can directly use 𝐺 but skip the fence edges and back edges, which can be determined using the tags generated in Step 3 (Line 7– 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Then we compute the CCs on the skeleton 𝐺 ′ (Line 4), which assigns a label 𝑙[𝑣] to each vertex (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 2, Step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' In Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='11, we show that if two vertices are connected in 𝐺 ′, they must be biconnected on the input graph 𝐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We then assign the head to each label (Lines 5 and 6) by looping over all fence edges (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 2, Step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' For a fence edge 𝑢–𝑝(𝑢), if 𝑢 and 𝑝(𝑢) have different labels (Line 5), 𝑝(𝑢) (intuitively) isolates vertices below 𝑢 with the other parts in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Thus, we assign 𝑝(𝑢) as the component head of 𝑢’s CC in 𝐺 ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We prove the correctness of this step in Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='9 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' This step also only requires 𝑂(𝑛) auxiluary space, which is needed by running CC on 𝐺 but skip certain edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='2 Correctness for the FAST-BCC Algorithm We now prove the correctness of our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Note that our algorithm will identify the spanning forest in the first step and deal with each CC respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' For simplicity, through- out the section, we focus on one CC in 𝐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' In the following, when we use the concepts about a span- ning tree of the graph (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=', root, parent, child, and subtree), we refer to the specific spanning tree identified in Step 1 of our algorithm, and use 𝑇 to represent it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Recall that 𝑇𝑢 denotes the subtree rooted at vertex 𝑢, and 𝑢 ~𝑣 denotes the tree path on 𝑇 from 𝑢 to 𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Some other notation is given in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' In a spanning tree, we say a node 𝑢 is shallower (deeper) than 𝑣 if 𝑢 is closer (farther) to the root than 𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We use node and vertex interchangeably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 5 Algorithm 1 is correct Fact 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='1 Two BCCs 𝐶1 ∩ 𝐶2 ≤ 1 Fact 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='2 Cycle ⇒ BCC Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='3 A BCC is connected on 𝑇 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='4 BCC head ⟺ articulation point Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='6 Property of plain tree edge Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='5 The skeleton 𝐺′ is generated correctly Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='8 (inductive) Non-head vertices in a BCC get the same label Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='9 BCC head is identified as component head Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='7 Every BCC in 𝐺 is identified by Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 1 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='11 (constructive) vertices with the same label are biconnected Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='12 Component head for label 𝑙 is biconnected with all vertices with label 𝑙 Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='10 Every BCC identified by Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 1 is biconnected Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' The structure of the correctness proof for Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We note that although Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 1 is simple, the correctness proof is sophisticated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We show the relationship of facts, lemmas, and theorems in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' The proofs for Fact 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='2 and Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='3 to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='5 are given in Appendix B, and here we mainly focus on the proofs that reflect some key ideas in our new algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We first show some facts for BCCs based on the definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Fact 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Two BCCs share at most one common vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Fact 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' For a cycle in a graph, all vertices on the cycle are in the same BCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Given a graph 𝐺, vertices in each BCC 𝐶 ⊆ 𝑉 must also be connected in an arbitrary spanning tree 𝑇 for 𝐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Since each BCC 𝐶 must be connected in the spanning tree in 𝑇, there must exist a unique shallowest node in this BCC on 𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We call this shallowest node the BCC head of the BCC 𝐶, and denote it as ℎ𝐶.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Each non-root BCC head is an articulation point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' An articulation point must be a BCC head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' The function InSkeleton (Line 7) in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 1 can correctly skip the fence and back edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Next, we show a useful property of the plain tree edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' For a plain tree edge 𝑥–𝑦 where 𝑥 is the parent of 𝑦, let 𝑧 be 𝑥’s parent, then 𝑥,𝑦,𝑧 are biconnected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Since 𝑥–𝑦 is not a fence edge, there must be an edge 𝑎–𝑏, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 𝑎 ∈ 𝑇𝑦 and 𝑏 ∉ 𝑇𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' The cycle 𝑦 ~𝑎–𝑏 ~𝑧–𝑥–𝑦 then contains 𝑥, 𝑦, and 𝑧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Due to Fact 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='2, 𝑥, 𝑦, and 𝑧 are in the same BCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' □ Next, we show that Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 1 can correctly identify all BCCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We will show two directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' First, if two vertices 𝑢 and 𝑣 are biconnected, Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 1 must put them in a BCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Second, for any two vertices 𝑢 and 𝑣 in a BCC found by Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 1, they must be biconnected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' For 𝑢, 𝑣 ∈ 𝑉 , if they are biconnected, Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 1 assigns them to the same BCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' To prove Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='7, we discuss two cases: 1) one of 𝑢 and 𝑣 is a BCC head, and 2) neither of them is a BCC head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' For a BCC 𝐶 and two vertices 𝑢, 𝑣 ∈ 𝐶 \\ {ℎ𝐶}, they are connected in the skeleton 𝐺 ′ and will get the same label in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' If all tree edges connecting 𝐶 \\ {ℎ𝐶} are plain tree edges, 𝑢 and 𝑣 are already connected in 𝐺 ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Next, we show that the two endpoints of every fence edge are also connected in 𝐺 ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' To do so, we first sort (only conceptually) all vertices in 𝐶 \\ {ℎ𝐶} by their depth in 𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Then we inductively show from bottom up (deep to shallow) that, given a vertex 𝑣 ∈ 𝐶, 𝑇𝑣 ∩ 𝐶 (𝑣’s subtree in 𝐶) is connected in 𝐺 ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' The base case is the deepest vertices in𝐶\\{ℎ𝐶}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' In this case, their subtree contains only one vertex so they are connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We now consider the inductive step—if for all vertices with depth ≥ 𝑑, their subtrees in 𝐶 are connected in 𝐺 ′, then for all vertices with depth 𝑑 − 1, their subtrees in 𝐶 are also connected in 𝐺 ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Consider a vertex 𝑢 ∈ 𝐶 \\ {ℎ𝐶} with depth 𝑑 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' If 𝑢 has only one child 𝑣 in 𝐶, then 𝑢–𝑣 is a plain tree edge since otherwise 𝑣’s subtree cannot escape 𝑢’s subtree and 𝑢 is an articulation point (disconnecting 𝑣 and 𝑝(𝑢)), contradicting Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Assume 𝑢 has multiple children 𝑐1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' ,𝑐𝑘 in 𝐶.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Let 𝑢–𝑣 be a fence edge that is not in 𝐺 ′, where 𝑣 = 𝑐𝑖 is a child of 𝑢.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We will show that 𝑢 and 𝑣 are still connected in 𝐺 ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Since 𝑢 is not a BCC head, 𝑝(𝑢) must also be in 𝐶.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Based on the definition of BCC, if we remove 𝑢, 𝑣 and 𝑝(𝑢) are still connected 𝐶.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Let the path be 𝑃 = 𝑣–𝑥1–𝑥2–.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='–𝑥𝑘–𝑝(𝑢) where 𝑥𝑖 ∈ 𝐶 and 𝑥𝑖 ≠ 𝑢.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We will construct a path in 𝐺 ′ from 𝑃 that connects 𝑣 and 𝑢.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Let 𝑥𝑗+1 be the first vertex on path 𝑃 that is not in 𝑇𝑢.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We will use the path 𝑣 = 𝑥0–𝑥1–𝑥2–.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='–𝑥𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' All nodes in this path have depths ≥ 𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Due to the induction hypothesis, if some of the edges are back or fence edges, we can replace them with the paths in 𝐺 ′, and denote this path as 𝑃 ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Then, since 𝑥𝑗+1 ∉ 𝑇𝑢 is connected to 𝑥𝑗 ∈ 𝑇𝑢, all edges on tree path 𝑥𝑗 ~𝑢 are plain tree edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' As a result, 𝑢 and 𝑣 are connected in 𝐺 ′ using the path 𝑃 ′ from 𝑣 to 𝑥𝑗, and the tree path from 𝑥𝑗 to 𝑢 (all edges are in 𝐺 ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' By the induction, all vertices in 𝐶 \\ {ℎ𝐶} are connected in 𝐺 ′, and hence get the same label after Line 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Any BCC head will be correctly identified as a component head in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Consider a BCC 𝐶 and its BCC head ℎ𝐶.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Among all the children of ℎ𝐶, a subset 𝑆 of them are in the same BCC 𝐶.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Consider any 𝑐 ∈ 𝑆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We will show that the edge 𝑐–ℎ𝐶 must be identified correctly in Line 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We first show that 𝑐–ℎ𝐶 must be a fence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' If ℎ𝐶 is the root of 𝑇, and in this case, all tree edges connecting to ℎ𝐶 are fence edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Otherwise, this can be inferred from the contra- positive of Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' If 𝑐–ℎ𝐶 is a plain tree edge, 𝑐, ℎ𝐶, and 𝑝(ℎ𝐶) must be biconnected, which means 𝑝(ℎ𝐶) is also in the BCC 𝐶.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' This contradicts the assumption that ℎ𝐶 is the shallowest node (BCC head) in the BCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We then show that after we run the CC on the skeleton 𝐺 ′ (Line 4), ℎ𝐶 and 𝑐 have different labels (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=', ℎ𝐶 and 𝑐 are not 6 connected in 𝐺 ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Assume to the contrary that there exists a path 𝑃 from 𝑐 to ℎ𝐶 on 𝐺 ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Consider the last node 𝑡 on the path before ℎ𝐶.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Because ℎ𝐶–𝑐 is a fence edge and is ignored in 𝐺 ′, 𝑐 ≠ 𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We discuss three cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' (1) 𝑡 is not in the ℎ𝐶’s subtree 𝑇ℎ𝐶.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Consider the first edge 𝑥–𝑦 on the path 𝑃 such that 𝑥 ∈ 𝑇ℎ𝐶 and 𝑦 ≠ 𝑇ℎ𝐶.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Since 𝑥–𝑦 escapes ℎ𝐶’s subtree, the tree path 𝑃 ′ = 𝑥 ~ℎ𝐶 only contains plain tree edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Let 𝑐′ be ℎ𝐶’s child on the path 𝑃 ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' From Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='6, 𝑐′, ℎ𝐶, and 𝑝(ℎ𝐶) are biconnected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' In this case, ℎ𝐶–𝑐 ~𝑥 ~𝑐′–ℎ𝐶 is a cycle, and Fact 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='2 shows that 𝑐′, ℎ𝐶 and 𝑐 are biconnected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' The contrapositive of Fact 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='1 indicates that 𝑐′, ℎ𝐶, 𝑐, and 𝑝(ℎ𝐶) are all biconnected, contradicting the assumption that ℎ𝐶 is the BCC head (the shallowest node in the BCC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' (2) 𝑡 ∈ 𝑇ℎ𝐶, but 𝑡 is not ℎ𝐶’s child.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' This is impossible because 𝑡–ℎ𝐶 is a back edge, which is not in 𝐺 ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' (3) 𝑡 is a child of ℎ𝐶.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' This case is similar to (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' By replacing 𝑐′ in the previous proof by 𝑡, we can get the same contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Combining all cases proves that there is no path in 𝐺 ′ between ℎ𝐶 and its children in 𝐶, so 𝑙[ℎ𝐶] is different from the labels of its children in 𝐶.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' □ Combining Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='8 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='9, we can prove Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We then show the other direction—all the BCCs computed by Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 1 are indeed biconnected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' If two vertices 𝑢 and 𝑣 are identified as in the same BCC by Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 1, they must be biconnected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Similar to the previous proof, we consider two cases: (1) none of the two vertices is a component head (they are con- nected in 𝐺 ′), proved in Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='11, and (2) one of them is identified as a component head in Line 6, proved in Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' If two vertices 𝑢 and 𝑣 are connected in the skeleton 𝐺 ′, they are biconnected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Since 𝑢 and 𝑣 are connected in 𝐺 ′, there exists a path 𝑃 from 𝑢 to 𝑣 only using edges in 𝐺 ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Let 𝑃 be 𝑢 = 𝑝0–𝑝1–.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='– 𝑝𝑘−1–𝑝𝑘 = 𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We will show that after removing any vertex 𝑝𝑖 where 1 ≤ 𝑖 < 𝑘 on 𝑃, 𝑝𝑖−1 and 𝑝𝑖+1 are still connected, meaning that 𝑢 and 𝑣 are biconnected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We summarize all possible local structures in three cases, based on whether 𝑝𝑖−1 (and 𝑝𝑖+1) is a child of 𝑝𝑖 in 𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Case 1: both 𝑝𝑖−1 and 𝑝𝑖+1 are 𝑝𝑖’s children.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Since 𝑝𝑖−1–𝑝𝑖 is not a fence edge, there must be an edge 𝑥–𝑦 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 𝑥 ∈ 𝑇𝑝𝑖−1 and 𝑦 ∉ 𝑇𝑝𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Similarly, for 𝑝𝑖–𝑝𝑖+1, there exists an edge (𝑥 ′,𝑦′) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 𝑥 ′ ∈ 𝑇𝑃𝑖+1 and 𝑦′ ∉ 𝑇𝑃𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Hence, without using 𝑝𝑖, 𝑝𝑖−1 and 𝑝𝑖+1 are still connected by the path 𝑝𝑖−1 ~𝑥–𝑦 ~𝑦′–𝑥 ′ ~𝑝𝑖+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Here since 𝑦,𝑦′ ∉ 𝑇𝑝𝑖, 𝑦 ~𝑦′ does not contain 𝑝𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Case 2: one of 𝑝𝑖−1 and 𝑝𝑖+1 is 𝑝𝑖’s child.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' WLOG, assume 𝑝𝑖−1 is the child.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Since 𝑝𝑖−1–𝑝𝑖 is not a fence edge, there must be an edge 𝑥–𝑦 such that 𝑥 ∈ 𝑇𝑝𝑖−1 and 𝑦 ∉ 𝑇𝑝𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Also, since 𝑝𝑖+1 is either the parent of 𝑝𝑖 or connected to 𝑝𝑖 using a cross edge, 𝑝𝑖+1 ∉ 𝑇𝑝𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Hence, without using 𝑝𝑖, 𝑝𝑖−1 and 𝑝𝑖+1 are still connected using the path 𝑝𝑖−1 ~𝑥–𝑦 ~𝑝𝑖+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Case 3: neither 𝑝𝑖−1 nor 𝑝𝑖+1 is a child of 𝑝𝑖, and neither of them is in 𝑇𝑝𝑖 (otherwise they are connected by a back edge).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Without using 𝑝𝑖, 𝑝𝑖−1 and 𝑝𝑖+1 are still connected using the tree path 𝑝𝑖−1 ~𝑝𝑖+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Since removing any vertex on the path 𝑃 does not discon- nect the path, all vertices in the same CC of the skeleton are biconnected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' If Line 6 in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 1 assigns ℎ as the component head of a connected component (CC) 𝐶 in the skeleton 𝐺 ′, then ℎ is biconnected with 𝐶.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' First of all, assume ℎ is assigned as the component head because of its child 𝑐, where ℎ–𝑐 is a fence edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We will show that the connected component 𝐶 in 𝐺 ′ containing 𝑐 is biconnected with ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' There are two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Case 1: 𝐶 only contains vertices in 𝑇𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' This means that no vertices in 𝑇𝑐 have a cross edge to another vertex outside 𝑇𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Therefore, either all edges incident on 𝑐′ ∈ 𝑇𝑐 do not escape from 𝑇𝑐, or some node 𝑐′ ∈ 𝑇𝑐 is connected to nodes outside 𝑇𝑐 via back edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' In the former case, all the edges connecting 𝑐 and its children are fence edges, and thus 𝐶 only contains 𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' In this case, ℎ is trivially biconnected with 𝐶.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' In the latter case, assume 𝑥 ∈ 𝑇𝑐 ∩𝐶 has a back edge connected to 𝑦 ∉ 𝑇𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Note that 𝑦 can only be ℎ—if 𝑦 is ℎ’s ancestor, then edge 𝑥–𝑦 escapes 𝑇ℎ, so ℎ–𝑐 is a plain tree edge (contradiction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Therefore, we can find a cycle ℎ–𝑐 ~𝑥–ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' From Fact 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='2, ℎ,𝑐,𝑥 are biconnected, and ℎ is in the same BCC as 𝑐 and 𝑥, and thus all vertices in 𝐶 (Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='11 and Fact 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Case 2: 𝐶 contains both vertices in 𝑇𝑐 and some vertices in 𝑇ℎ \\𝑇𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Hence, there exists a cross edge 𝑥–𝑦, where 𝑥 ∈ 𝑇𝑐 and 𝑦 ∉ 𝑇𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We can find a cycle ℎ,~𝑥–𝑦 ~ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' From Fact 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='2, ℎ,𝑐,𝑢 are biconnected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' ℎ is in the same BCC as 𝑐 and 𝑢.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' □ Combining Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='11 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='12 proves Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='7 shows that if two vertices are put in the same BCC by Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 1, they are biconnected in𝐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='10 indicates that two vertices biconnected in 𝐺 will be put in the same BCC by Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='5 indications back edges and fence edges are identified correctly by Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Combining them together indicates that Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 1 is correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='3 Cost Bounds for the FAST-BCC Algorithm We now analyze the cost bounds of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 1 computes the BCCs of a graph𝐺 with𝑛 vertices and 𝑚 edges using 𝑂(𝑛 +𝑚) expected work, 𝑂(log3 𝑛) span whp, and 𝑂(𝑛) auxiliary space (other than the input).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' The first and last steps compute the graph connectiv- ity twice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Graph connectivity can be computed in 𝑂(𝑛 + 𝑚) expected work and 𝑂(log3 𝑛) span whp [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' In Step 2, ETT can be performed 𝑂(𝑛) expected work and 𝑂(log𝑛) span whp (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' In Step 3, computing low[·] and high[·] ar- rays based on RMQ takes 𝑂(𝑚) work and 𝑂(log𝑛) span [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Adding all pieces together gives the work and span bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' For the space, all arrays for the tags have size 𝑂(𝑛).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' As mentioned, we do not generate the skeleton explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' In the last step, we try all the edges in 𝐺 but skip the back and fence edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' In all, the auxiliary space needed is 𝑂(𝑛).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' □ 7 5 Implementation Details We discuss some implementation details of FAST-BCC in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Connectivity is used twice in FAST-BCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' The only existing parallel CC implementation with good theoretical guarantee we know of is the SDB algorithm [61] (an initial version of GBBS is based on this algorithm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' A recent paper by Dhulipala et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' [31] gave 232 parallel CC implementations, many of which outperformed the SDB algorithm, but no analysis of work-efficiency was given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' A more recent version of GBBS uses the UF-Async algorithm in [31] to compute CC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' To achieve efficiency both in theory and in practice, FAST-BCC uses the LDD-UF-JTB algorithm from [31] and we provide a new analysis that this algorithm is indeed theoretically-efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' LDD-UF-JTB consists of two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' It first runs a low- diameter decomposition (LDD) algorithm [53] to find a de- composition (partition of vertices) of the graph such that each component has a low diameter and the number of edges crossing different components is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' The second step is to use a union-find structure by Jayanti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' [47] to union components connected by cross-component edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We now show the bounds of this algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' The LDD-UF-JTB algorithm computes the CCs of a graph 𝐺 with 𝑛 vertices and 𝑚 edges using 𝑂(𝑛 + 𝑚) expected work and 𝑂(log3 𝑛) span whp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Therefore, using LDD-UF-JTB for CC preserves the cost bounds in Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We prove Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='1 in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We optimized LDD-UF-JTB using the hash bag and local search techniques proposed from [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' These optimizations are only used in computing CCs in our algorithm, and we do not claim them as contributions of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' In our tests, using these optimizations improves the performance of FAST- BCC by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='5× on average (up to 5×).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Some results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We note that among all 232 CC algorithms in [31], no one is constantly faster, and the relative performance is decided by the input graph properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' In FAST-BCC, we currently use the same CC algorithm for all graphs, and we acknowledge that using the fastest CC algorithm on each graph can further improve the performance of FAST-BCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We choose LDD-UF-JTB mainly because it is theoretically- efficient, and also can generate CC as a by-product efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Spanning Forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' The spanning forest of 𝐺 is obtained as a by-product of Step 1, which saves all edges to form the CCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We then re-order the vertices in the compressed sparse row (CSR) format to let each CC be contiguous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Euler Tour Technique (ETT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We use the standard ETT to root the spanning trees (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We replicate each undirected edge in 𝑇 into two directed edges and semisort them [42], so edges with the same first endpoint are con- tiguous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Then we construct a circular linked list as the Euler circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Assume a vertex 𝑣 has 𝑘 in-coming neighbors 𝑢1, 𝑢2, · · , 𝑢𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' For every incoming edge of 𝑣 except for the last one, we link it to its next outgoing edge (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=', 𝑢𝑖–𝑣 is linked to 𝑣–𝑢𝑖+1 for 1 ≤ 𝑖 < 𝑘).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' For the last incoming edge, we link it to the first outgoing edge of 𝑣 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=', 𝑢𝑘–𝑣 is linked to 𝑣–𝑢1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' After we obtain the Euler circuit of the tree, we flatten the linked list to an array by list ranking, and acquire the Euler tour order of each vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' For list ranking, we coarsen the base cases by sampling √𝑛 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We start from these nodes in parallel, with each node sequentially following the pointers until it visits the next sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Then we compute the offsets of each sample by prefix sum, pass the offsets to other nodes by chasing the pointers from the samples, and scatter all nodes into a contiguous array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Computing Tags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We use several tags 𝑤1, 𝑤2, first, last, low, and high for each vertex, defined the same as Tarjan- Vishkin [63] (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We use CAS operations to compute first and last as they represent the first and last appearances of a vertex in the Euler tour order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' For each tree edge (𝑢, 𝑣), if first[𝑢] < first[𝑣], we set 𝑝(𝑣) = 𝑢, or vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Computing low and high are similar, so we only discuss low here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We first initialize 𝑤1[𝑣] with first[𝑣] for each 𝑣 ∈ 𝑉 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Then it traverses all non-tree edges𝑢–𝑣 and updates 𝑤1[𝑢] and 𝑤1[𝑣] with the minimum of first[𝑢] and first[𝑣].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We build a parallel sparse table [14] on 𝑤1 to support range minimum queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Note that first[𝑣] and last[𝑣] reflect the range of 𝑣’s subtree in the Euler tour order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Thus, low[𝑣] can be computed by finding the minimum element in 𝑤1[·] in the range between first[𝑣] and last[𝑣].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' high[·] can be computed similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 6 Experiments Setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We run our experiments on a 96-core (192 hyper- threads) machine with four Intel Xeon Gold 6252 CPUs, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='5 TB of main memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We implemented all algorithms in C++ using ParlayLib [11] for fork-join parallelism and some parallel primitives (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=', sorting).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We use numactl -i all in experiments with more than one thread to spread the memory pages across CPUs in a round-robin fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We run each test for 10 times and report the median.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We tested on 27 graphs, including social networks, web graphs, road graphs, 𝑘-NN graphs, and synthetic graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' The information of the graphs is given in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' In addi- tion to commonly-used benchmarks of social, web and, road graphs, we also use 𝑘-NN graphs and synthetic graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 𝑘- NN graphs are widely used in machine learning algorithms (see discussions in [68]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' In 𝑘-NN graphs, each vertex is a multi-dimensional data point and has 𝑘 edges pointing to its 𝑘-nearest neighbors (excluding itself).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We also create six synthetic graphs, including two grids (SQR and REC), two sampled grids (SQR’ and REC’, each edge is created with probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='6), and two chains (Chn7 and Chn8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' SQR and SQR’ have sizes 104 ×104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' REC and REC’ have sizes 103 ×105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Each row and column in grid graphs are circular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Chn7 and Chn8 have sizes 107 and 108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' The tested graphs cover a wide range of sizes and edge distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 8 𝒏 𝒎 𝑫 #BCC |BCC1|% Ours GBBS SM’ SEQ 𝑻best Notes par.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='9 957 307 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='320 703∗ 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='9 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='0 Chain of size 108 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Graph information, running times (in seconds), and speedups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 𝑇best/ours (highlighted in yellow) is the fastest time of the other implementations / our time, both using all cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' “𝑛” = number of vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' “𝑚” = number of edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' “𝐷” = approximate diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' “#BCC” = number of BCCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' “|BCC1|%” = percentage of the largest BCCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' “GBBS” = GBBS’s implementation [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' “SM’14” = Slota and Madduri’s algorithm [62] (the faster of the two proposed algorithms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Since SM’14 has scalability issues (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 4), we report the 16-core time if it is faster, and denote as (∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' “SEQ” = Hopcroft-Tarjan BCC algorithm [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Details about the baselines are introduced in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' The fastest runtime for each graph is underlined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Red numbers are parallel runtime slower than the sequential algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' “par.” = parallel running time (on 192 hyper-threads).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' “seq.” = sequential running time (on 1 thread).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' “spd.” = self-relative speedup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' “n” = no support, because SM’14 only works on connected graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' For real-world directed graphs, we symmetrize them to test BCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We call the social and web graphs low-diameter graphs as they have smaller diameters (mostly within a few hundreds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We call the road, 𝑘-NN, and synthetic graphs large-diameter graphs as their diameters are mostly more than a thousand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' When comparing the average running times across multiple graphs, we always take the geometric mean of the numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Baseline Algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We call all existing algorithms that we compare to the baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We implement the highly- optimized sequential Hopcroft-Tarjan [43] algorithm for comparison, referred to as SEQ or the sequential baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We compare the number of BCCs reported by each algorithm with SEQ to verify correctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We also compare to two most recent available BCC im- plementations GBBS [30], and Slota and Madduri [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We use SM’14 to denote the better of the two BCC algorithms in Slota and Madduri [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' On many graphs, we observe that SM’14 is faster on 16 threads than using all 192 threads, in which case we report the lower time of 16 and 192 threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Through correspondence with the authors, we understand that SM’14 requires the input graph to be connected, so we only report the running time when it gives the correct an- swers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' As few graphs we tested are entirely connected, we focus on comparisons with GBBS and SEQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We also compare our breakdown and sequential running times with GBBS since GBBS can process most of the tested graphs2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Unfortunately, we cannot find any existing implementa- tions for Tarjan-Vishkin to compare with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We are aware of 2GBBS updated a new version after this paper was accepted, so we also updated the numbers using their latest version (Nov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Some new features in the latest version greatly improved their BCC performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 9 1248 24 96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content="2 1 5 20 TW 1248 24 96 SD 1248 24 96 USA 1248 24 96 GL5 1248 24 96 REC FAST-BCC GBBS SM'14 Figure 4." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Scalability curves for different BCC algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' In each plot, 𝑥-axis is core counts (last data point is 96 core with hyperthreading) and 𝑦-axis is speedups normalized to SEQ (the sequential Hopcroft-Tarjan algorithm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Higher is better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' SEQ is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' two papers that implemented Tarjan-Vishkin [28, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Ed- wards and Vishkin’s implementation [37] is on the XMT architecture and they did not release their code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Cong and Bader’s code [28] is released, but it was written in 2005 and uses some system functions that are no longer supported on our machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' For a full comparison, we implemented a faithful Tarjan-Vishkin from the original paper [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' As en- gineering Tarjan-Vishkin is not the main focus of this paper, we mainly use it to evaluate the memory usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We note that the implementations for both GBBS and SM’14 exclude the postprocessing to compute the actual BCCs, but only report the number of BCCs at the end of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We include this step in FAST-BCC, although this postprocessing only takes at most 2% of the total running time in all our tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='1 Overall Performance We present the running time of all algorithms in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Our FAST-BCC is faster than all baselines on all graphs, mainly due to the theoretical efficiency—work- and space- efficiency enables competitive sequential times over the Hopcroft-Tarjan sequential algorithm, and polylogarithmic span ensures good speedup for all graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Sequential Running Time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We first compare the sequen- tial running time of SEQ, GBBS, and FAST-BCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' SEQ and FAST-BCC use 𝑂(𝑛 + 𝑚) work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' To enable parallelism, both FAST-BCC and GBBS traverse all edges multiple times (run- ning CC twice in Steps 1 and 4, and computing low/high for the skeleton in Step 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We describe more details about GBBS implementation in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' On average, our sequential time is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='8× slower than SEQ, but is 10% faster than GBBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Scalability and Parallelism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' To measure parallelism, we report the scalability curves for FAST-BCC, GBBS and SM’14 on some representative graphs (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' For fair comparison, the speedup numbers in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 4 are normalized to the running time of SEQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' On these five graphs, FAST-BCC is the only algorithm that scales to all processors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' It also outperforms GBBS and SM’14 on all graphs with all numbers of threads (expect REC on 2 cores).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We noticed that SM’14 suffers from scalability issues, and the best performance can be achieved at around 16 threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Hence, we report SM’14’s better run- ning time of 16 and 192 threads in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' GBBS has similar issues on a few graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' However, as GBBS’s performance does not drop significantly as core count increases, we con- sistently report GBBS’s time on 192 threads in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Comparing the self-relative speedup with GBBS, our aver- age self-relative speedups on both low-diameter graphs and large-diameter graphs are 36×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' On large-scale low-diameter graphs with sufficient parallelism, the self-relative speedup can be up to 66×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Even on large-diameter graphs, FAST-BCC achieves up to 47× self-relative speedup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' In comparison, the self-relative speedup of GBBS’s BFS-based algorithm is 29× on low-diameter graphs and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='7× on large-diameter graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' This makes GBBS only 11% faster than SEQ on large- diameter graphs (and can be slower on some graphs), while ours is 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='1–18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='5× better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Overall, our parallel running time is 10× faster on large-diameter graphs and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='6× faster on low-diameter graphs than GBBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' On some graphs, SM’14 achieves better performance than GBBS, but FAST-BCC is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='7–11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='1× faster than SM’14 on all the graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' To verify that GBBS’s performance is bottlenecked by BFS, we created 𝑘-NN graphs GL2–20 from the set of points but with different values of 𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' When increasing 𝑘 over 5, the graphs have more edges but smaller diameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' For both FAST-BCC and SEQ, the running times increase when 𝑘 grows due to more edges (and thus more work), but the trend of GBBS’s running time is decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' This indicates that the BFS is the dominating part of running time for GBBS, and the performance on GBBS is bottlenecked by the 𝑂(Diam(𝐺) log𝑛) span.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='2 Performance Breakdown To understand the performance gain of FAST-BCC over prior parallel BFS-based BCC algorithms, we compare our perfor- mance breakdown with GBBS in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We choose GBBS because it can process all graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Since GBBS is also in the skeleton-connectivity framework, we use the same four step names for GBBS as in FAST-BCC, but there are a few dif- ferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' (1) For First-CC, FAST-BCC generates a spanning forest while GBBS only finds all CCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' (2) For Rooting, FAST- BCC uses ETT to root the tree while GBBS applies BFS on all CCs to find the spanning trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' (3) The task for Tagging is almost the same, but GBBS computes fewer tags than FAST- BCC since it is based on BFS trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' FAST-BCC uses 1D RMQ queries that are theoretically-efficient, while GBBS uses a bottom-up traversal on the BFS tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' (4) For Last-CC, both algorithms run CC algorithms on the skeletons to find BCCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We start with discussing the two steps First-CC and Last- CC that use connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' GBBS can be faster than our al- gorithm in First-CC on some graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' The reason is that our algorithm also constructs the spanning forest in First-CC, while GBBS has to run BFS in Rooting to generate the BFS spanning forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' In Last-CC, the two algorithms achieves similar performance, and in many cases, FAST-BCC is faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We note that the CC algorithm is independent with the BCC 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='04 YT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='1 OK 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='1 LJ 0 1 2 TW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='0 FT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='04 GG First CC Rooting Tagging Last CC 0 2 4 SD 0 20 40 CW 0 20 40 HL14 0 50 100 HL12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='2 CA 0 2 4 USA 0 1 2 GE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='4 HH5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content="0 CH5 0 1 GL2 0 2 GL5 0 1 GL10 GBBS Ours 0 1 GL15 GBBS Ours 0 1 GL20 GBBS Ours 0 10 COS5 GBBS Ours 0 10 SQR GBBS Ours 0 20 40 REC GBBS Ours 0 5 10 SQR' GBBS Ours 0 10 20 REC' GBBS Ours 0 50 Chn7 GBBS Ours 0 500 1000 Chn8 Figure 5." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' BCC breakdown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 𝑦-axis is the running time in seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' algorithm itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Both the CC algorithm used in our imple- mentation and GBBS are based on algorithms in an existing paper [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' As mentioned, based on the results in [31], the “best” CC algorithm can be very different for different types of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' One can also plug in any CC algorithms to FAST- BCC or GBBS BCC algorithm to achieve better performance for specific input graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' In the Rooting step (generate rooted spanning trees, the red bar), FAST-BCC is significantly faster than GBBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' GBBS is based on a BFS tree, and after computing the CCs of in- put graph 𝐺, it has to run BFS on 𝐺 again, which results in 𝑂(𝑚 + 𝑛) work and 𝑂(Diam(𝐺) log𝑛) span.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' In comparison, FAST-BCC obtains the spanning trees from the First-CC step, and only uses ETT in the Rooting step with 𝑂(𝑛) expected work and 𝑂(log𝑛) span whp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 5, this step for GBBS is the dominating cost for large-diameter graphs, and this is likely the case for other parallel BCC algorithms using BFS-based skeletons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' FAST-BCC almost entirely saves the cost in this step (13× faster on average on large-diameter graphs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' For low-diameter graphs, the two algorithms per- form similarly—FAST-BCC is about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='1× faster in this step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' In the Tagging step (the green bars), both FAST-BCC and GBBS compute the tags such as low and high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Since FAST- BCC uses an AST, the values of the arrays are computed using 1D range-minimum query (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='1) with 𝑂(log𝑛) span.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' GBBS computes them by a bottom-up traversal on the BFS tree, with 𝑂(Diam(𝐺) log𝑛) span.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Hence, on large- diameter, GBBS also consumes much time on this step, and FAST-BCC is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='2–830× faster than GBBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' On low-diameter graphs, GBBS also gets sufficient parallelism, and the per- formance for both algorithms are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' In summary, on all graphs, FAST-BCC is faster than GBBS mainly due to the efficiency in the Rooting and Tagging step, and the reason is that our algorithm has polylogarithmic span, while GBBS relies on the BFS spanning tree and re- quires 𝑂(Diam(𝐺) log𝑛) span.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='3 The Tarjan-Vishkin Algorithm Although engineering the Tarjan-Vishkin (TV) Algorithm is not the focus of this paper, for completeness, we also im- plemented the faithful Tarjan-Vishkin algorithm [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We mainly use it to measure the space usage and get a sense on how Tarjan-Vishkin compares to other existing BCC algo- rithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We report the relative space usage of FAST-BCC, TV, and GBBS in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 7, normalized to the most space-efficient implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Because of the space inefficiency, our TV implementation cannot run on the three largest graphs (CW, HL14, and HL12) on our machines with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='5TB memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We note that the smallest among them (CW) only takes about 300GB to store the graph, and our algorithm uses 572GB memory to process it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' GBBS is slightly more space-efficient than FAST-BCC, and takes about 20% less space than us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' The reason is that they need to compute fewer number of tags than FAST-BCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Regarding running time, we take the aver- age of the running times for each algorithm on each category of the graph instances, and normalize to FAST-BCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Ours GBBS Our-TV SEQ Social 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='67 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='1 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='2 Web 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='69 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='61 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='3 Road 1 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='89 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='94 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='18 𝑘-NN 1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='58 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='13 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='68 Synthetic 1 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='96 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='0 Due to the cost to explicitly construct the skeleton, TV performs slowly on small-diameter graphs, and is slower than GBBS even on 𝑘-NN graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' On all these graphs, the speedup for TV on 96 cores over SEQ is only 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='4–3×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' This is consistent with the findings in prior BCC papers [28, 62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' TV works well on road and synthetic graphs due to small edge- to-vertex ratio, so the 𝑂(𝑚) work and space for generating the skeleton does not dominate the running time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' In this case, polylogarithmic span allows TV to perform consistently better than GBBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' On all graphs, TV is faster than SEQ on 96 cores, but slower than FAST-BCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 11 7 Conclusion In this paper, we propose the FAST-BCC (Fencing on Arbi- trary Spanning Tree) algorithm for parallel biconnectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' FAST-BCC has 𝑂(𝑚 + 𝑛) expected optimal work, polylog- arithmic span (high parallelism), and uses 𝑂(𝑛) auxiliary space (space-efficient).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' The theoretical efficiency also en- ables high performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' On our machine with 96 cores and a variety of graph types, FAST-BCC outperforms all existing BCC implementations on all tested graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Acknowledgement This work is supported by NSF grant CCF-2103483 and IIS- 2227669, and UCR Regents Faculty Fellowships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We thank anonymous reviewers for the useful feedbacks.' metadata={'source': 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+page_content=' [10] Naama Ben-David, Guy E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Blelloch, Jeremy T Fineman, Phillip B Gib- bons, Yan Gu, Charles McGuffey, and Julian Shun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Implicit Decomposition for Write-Efficient Connectivity Algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' In IEEE International Parallel and Distributed Processing Symposium (IPDPS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' [11] Guy E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Blelloch, Daniel Anderson, and Laxman Dhulipala.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' ParlayLib-a toolkit for parallel algorithms on shared-memory multi- core machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' In ACM Symposium on Parallelism in Algorithms and Architectures (SPAA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 507–509.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} 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Wang, and Xing Xie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Learning transportation mode from raw gps data for geographic applications on the web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' In International World Wide Web Conference (WWW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 247–256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' A More details and discussion for the Tarjan-Vishkin Algorithm To parallelize BCC, the Tarjan-Vishkin algorithm [63] uses an arbitrary spanning tree (AST)𝑇 instead of a DFS tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' The spanning tree can be obtained by any parallel CC algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 13 The first is the famous Euler tour technique (ETT) that effi- ciently roots a tree (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Our algorithm also uses ETT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' The second technique is to build the skeleton 𝐺 ′ based on an AST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Unfortunately, 𝐺 ′ in Tarjan-Vishkin is very large, making the algorithm less practical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Given the input graph 𝐺 = (𝑉, 𝐸) and an AST 𝑇, 𝐺 ′ = (𝐸, 𝐸′) where 𝐸′ consists of (𝑒1,𝑒2) (𝑒1,𝑒2 ∈ 𝐸 are edges in 𝐺) iff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' one of the following conditions hold: 𝑒1 = (𝑢, 𝑝(𝑢)), 𝑒2 = (𝑢, 𝑣) in 𝐺 \\𝑇, and 𝑢, 𝑣 ∈ 𝑉, first[𝑣] < first[𝑢].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 𝑒1 = (𝑢, 𝑝(𝑢)), 𝑒2 = (𝑣, 𝑝(𝑣)), and (𝑢, 𝑣) is a cross edge in 𝐺 \\𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 𝑒1 = (𝑢, 𝑣), where 𝑣 = 𝑝(𝑢) is not the root in 𝑇, and 𝑒2 = (𝑣, 𝑝(𝑣)), and there exists a non-tree edge (𝑥,𝑦) such that 𝑥 ∈ 𝑇𝑢 and 𝑦 ∉ 𝑇𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' The above relationships can be determined by using the four axillary arrays first[·], last[·], low[·], and high[·] as mentioned in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' In fact, we can prove that FAST-BCC is equivalent to Tarjan-Vishkin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' However, the analysis for Tarjan-Vishkin is also quite involved (we refer to JáJá’s textbook for a good reference [45]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Hence, we give a standalone analysis for FAST-BCC in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='2, since we feel that understanding the analysis of Tarjan-Vishkin (correctness and cost bounds) and the analysis in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='2 is in a similar level of difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' In addition, we believe that the five algorithms we de- scribed and tested experimentally (Hopcroft-Tarjan, Tarjan- Vishkin, FAST-BCC, GBBS, SM’14) are similar, once we put them in the skeleton-connectivity framework and explicitly specify what the skeleton graph 𝐺 ′ is in each algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Thus, the skeleton-connectivity framework brings in a different angle to understand parallel BCC algorithms, and eventually helps us come up with FAST-BCC that is simple and efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' B Additional Proofs The proofs here are not very complicated, and should have been shown previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We provide them here mainly for completeness since the proofs of other lemmas in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='2 use them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='1 Proof of Fact 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='1 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Assume to the contrary that two BCCs𝐶1 and𝐶2 share at least two common vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' After removing an arbitrary vertex from 𝐶1 ∪ 𝐶2, there is at least one common vertex remaining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' WLOG, we assume 𝑢 is a remaining common vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Because 𝐶1 and 𝐶2 are BCCs and 𝑢 is in both of them, all the remaining vertices in 𝐶1 ∪ 𝐶2 are connected to 𝑢, so they remain in the same CC as 𝑢.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Therefore, 𝐶1 ∪𝐶2 is a BCC, which contradicts with that 𝐶1 and 𝐶2 are two BCCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' □ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='2 Proof of Fact 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='2 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We first rewrite the cycle starts and ends with 𝑣𝑖 as 𝑣0–𝑣1–𝑣2–.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='–𝑣 𝑗–.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='–𝑣𝑘 (𝑣0 = 𝑣𝑘 = 𝑣𝑖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' All the other vertices on the cycle appear exactly once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Then, there exists at least two disjoint paths that connect 𝑣𝑖 and 𝑣 𝑗: one is 𝑣0–.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='–𝑣 𝑗, the other one is𝑣 𝑗–.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='–𝑣𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Removing any vertex other than𝑣𝑖 and 𝑣 𝑗 disconnects at most one of the two paths, while the other path still connects 𝑣𝑖 and 𝑣 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Thus, after removing any vertex, all the remaining vertices on the cycle are still connected, so all vertices on the cycle are in the same BCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' □ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='3 Proof of Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='3 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Assume to the contrary that 𝐶 has at least two CCs in 𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' For each CC, we find the shallowest node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Let 𝑢 and 𝑣 be the shallowest two of these two CCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' WLOG assume 𝑢 is no deeper than 𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' There are two possible positions for 𝑢 and 𝑣 in the spanning tree 𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We first show that in both cases, 𝑣’s parent 𝑤 is biconnected with 𝑢.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Case 1: neither 𝑢 nor 𝑣 is the ancestor of the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Based on our assumption, there exists at least one path 𝑃 from 𝑢 to 𝑣 using vertices in 𝐶.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Note that no vertices on the tree path from 𝑢 to 𝑣 are included in 𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We now show that there are two disjoint paths that connect 𝑤 and 𝑢: 1) the tree path between𝑤 and𝑢, which does not contain intermediate nodes from 𝐶, and 2) the path from 𝑤 to 𝑣 then to 𝑢, only using intermediate nodes from 𝐶.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Hence, 𝑤 and 𝑢 are biconnected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Case 2: 𝑢 is 𝑣’s ancestor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We can similarly show that there are at least two paths from 𝑤 to the nearest vertex in 𝑢’s component, one using the tree path while the other using vertices in 𝐶.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Hence, 𝑤 and 𝑢 are biconnected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' In both cases, we can show that 𝑤 and 𝑢 are biconnected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' For any 𝑥 ∈ 𝐶, removing any other vertex 𝑦 ∈ 𝐶, 𝑥 and 𝑤 are still connected through either 𝑢 or 𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Hence, 𝑤 and 𝐶 are biconnected, contradicting that 𝑣 is the shallowest tree node in the BCC (𝑤 is 𝑣’s parent).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' □ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='4 Proof of Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='4 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We first prove that each non-root BCC head is an articulation point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' A BCC head ℎ is the shallowest node for a BCC 𝐶.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Let the 𝑐 be one of ℎ’s children in 𝐶.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Assume to the contrary that ℎ is not an articulation point, then 𝐺 is still connected after removing ℎ, including 𝑝(ℎ) and 𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Removing any other vertex other than 𝑐, ℎ, and 𝑝(ℎ) does not disconnect 𝑐 and ℎ based on the definition of the BCC, so 𝑐 and 𝑝(ℎ) is also connected using an additional tree edge ℎ–𝑝(ℎ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Combining both cases, 𝑝(ℎ) is biconnected with 𝑐, contradicting that ℎ is a BCC head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We now show an articulation point 𝑎 must be a BCC head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Assume to the contrary that 𝑎 is not a BCC head, then based on Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='3, all 𝑎’s children must be in the same BCC as 𝑝(𝑎).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Since 𝑎 is an articulation point, removing it disconnects 𝐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' However, all 𝑎’s children’s subtrees are still connected, so as 𝑉 \\ 𝑇𝑎 using tree edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Hence, at least one of 𝑎’s children, referred to as 𝑐, is disconnected from 𝑝(𝑎), since otherwise 𝑎 is not an articulation point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' In this case, 𝑎 is the BCC head of the BCC which 𝑎 and 𝑐 is in, contradicting the assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' □ 14 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='5 Proof of Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='5 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' To prove that the skeleton is generated correctly, we show that the functions InSkeleton, Fence, and Back work as expected, and InSkeleton returns true iff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' edge 𝑢–𝑣 is a plain edge or a cross edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We first prove that a vertex 𝑢 is 𝑣’s ancestor if and only if first[𝑢] ≤ first[𝑣] and last[𝑢] ≥ last[𝑣], which is exactly Line 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' On an Euler tour of a spanning tree, each edge appears exactly twice (one time in each direction) in a DFS order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' first[·]/last[·] stores the time stamps each the vertex first/last appears on the Euler tour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We first show that ∀𝑣 ∈ 𝑇𝑢, first[𝑢] ≤ first[𝑣] and last[𝑣] ≤ last[𝑢].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' This is because all the tree edges in 𝑇𝑢 are traversed after 𝑢 have been traversed, so ∀𝑣 ∈ 𝑇𝑢, first[𝑣] ≥ first[𝑢];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' and 𝑢 last ap- pears when all the tree edges in 𝑇𝑢 have been traversed, so ∀𝑣 ∈ 𝑇𝑢, last[𝑣] ≤ last[𝑢].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We show that if 𝑢 is an ancestor of 𝑣, then the function Line 14 in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 1 returns true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' If𝑢 is an ancestor of 𝑣, then 𝑣 ∈ 𝑇𝑢, so that first[𝑢] ≤ first[𝑣] and last[𝑢] ≥ last[𝑣] ≥ first[𝑣].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Then we will show if 𝑢 is not an ancestor of 𝑣, then the function Line 14 in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 1 returns false (at least one of the two conditions is false).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' If 𝑢 is not an ancestor of 𝑣, there are two cases for𝑢:𝑢 ∈ 𝑇𝑣 or𝑢 ∉ 𝑇𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' If𝑢 ∈ 𝑇𝑣, then first[𝑣] ≤ first[𝑢], so the first condition in function Line 14 is false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' If 𝑢 ∉ 𝑇𝑣, either last[𝑢] < first[𝑣] and last[𝑣] < first[𝑢].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' If last[𝑢] < first[𝑣], the second condition in function Line 14 is false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' If last[𝑣] < first[𝑢], because first[𝑣] < last[𝑣] < first[𝑢], the first condition is false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Therefore, 𝑢 is an ancestor of 𝑣 iff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' the function Line 14 in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 1 returns true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' This function is called on edge 𝑢–𝑣 by the function InSkeleton only when it is a tree edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Therefore, 𝑢 is a non-parent ancestor of 𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Therefore, InSkeleton returns true on Line 10 iff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 𝑢–𝑣 is a cross edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We then prove that the function Line 12 in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 1 can correctly determine whether a tree edge 𝑢–𝑣 is a fence edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We first show that if tree edge 𝑢–𝑣 is a fence edge, Line 12 in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 1 returns true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We just showed that ∀𝑥 ∈ 𝑇𝑢, first[𝑢] ≤ first[𝑥] and last[𝑢] ≥ last[𝑥] ≥ first[𝑥].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' If tree edge 𝑢–𝑣 is a fence edge, then for all the edges with one endpoint in 𝑇𝑣, the other endpoint must be in 𝑇𝑢.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Recall that low[𝑣] is the earliest (with the smallest first value) vertex connected to 𝑣’s subtree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' This means that this earliest vertex is also in𝑇𝑢, and therefore low[𝑣] ≥ first[𝑢].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Similarly, high[𝑣], which is the latest (with the largest first value) vertex connected to 𝑣’s subtree, should also be in 𝑇𝑢.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Therefore last[𝑢] ≥ high[𝑣].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Then we show that if tree edge 𝑢–𝑣 is not a fence edge, Line 12 in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 1 returns false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' If tree edge 𝑢–𝑣 is not a fence edge, then there exists an edge 𝑥 ′–𝑦′, where 𝑥 ′ ∈ 𝑇𝑣 and 𝑦′ ∉ 𝑇𝑢.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Because 𝑦′ ∉ 𝑇𝑢, either first[𝑦′] < first[𝑢] or first[𝑦′] > last[𝑦′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' If first[𝑦′] < first[𝑢], then low[𝑣] ≤ 𝑤1[𝑥 ′] ≤ first[𝑦′] < first[𝑢] Then the first condition in Line 12 in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 1 is false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' If first[𝑦′] > last[𝑢], then high[𝑣] ≥ 𝑤2[𝑥 ′] ≥ first[𝑦′] > last[𝑢] Then the second condition in Line 12 in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 1 is false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Therefore, the tree edge 𝑢–𝑣 is a fence edge iff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' function Line 12 in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 1 returns true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' In summary, InSkeleton returns true iff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' edge 𝑢–𝑣 is a plain edge or a cross edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Therefore, the skeleton 𝐺 ′ can be determined correctly by InSkeleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' □ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='6 Proof of Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='1 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Before we show the analysis, we will first review the two key techniques that name this algorithm in ConnectIt: low-diameter decomposition (LDD) [53] and a union-find structure by Jayanti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' A (𝛽,𝑑)-decomposition of a graph𝐺 = (𝑉, 𝐸) is a partition of𝑉 into subsets𝑉1,𝑉2, · · · ,𝑉𝑘 such that (1) the diameter of each 𝑉𝑖 is at most 𝑑, and (2) the number of edges (𝑢, 𝑣) ∈ 𝐸 with endpoints in different sub- sets, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=', such that 𝑢 ∈ 𝑉𝑖, 𝑣 ∈ 𝑉𝑗 and 𝑖 ≠ 𝑗, is at most 𝛽𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' A parallel (𝛽,𝑂((log𝑛)/𝛽)) decomposition algorithm is provided by Miller et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' [53], using 𝑂(𝑛 + 𝑚) work and 𝑂((log2 𝑛)/𝛽) span whp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' The high-level idea of LDD is to start with a single source and search out using BFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Then in later rounds, we exponentially add new sources to the fron- tier and continue BFS processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' By controlling the speed to add new sources, the entire BFS will finish in 𝑂((log𝑛)/𝛽) rounds, leaving at most 𝛽𝑚 edges with endpoints from dif- ferent sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Once the LDD is computed, the algorithm will examine all cross edges (endpoints from different sources) using a union- find structure by Jayanti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' [47] to merge different compo- nents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' The algorithm either performs finds naively without using any path compression or uses a strategy called Find- Two-Try-Split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Such strategies guarantee provably-efficient bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' The original bound is 𝑂(𝑙 ·(𝛼(𝑛,𝑙/(𝑛𝑝))+log(𝑛𝑝/𝑙 + 1))) expected work and 𝑂(log𝑛) PRAM time for a problem instance with 𝑙 operations on 𝑛 elements on a PRAM with 𝑝 processors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' When translating this bound to the binary fork- join model, all 𝑙 operations can be in parallel in the worst case, which leads to the work bound as 𝑂(𝑙 log𝑛).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We note that if we set 𝛽 = 1/log𝑛, the LDD takes 𝑂(𝑛 + 𝑚) work and 𝑂((log3 𝑛)/𝛽) span whp, and the union-find part takes 𝑂(𝛽𝑚 log𝑛) = 𝑂(𝑚) work and 𝑂(log2 𝑛) span.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Combining the two pieces together gives 𝑂(𝑛 +𝑚) work and 𝑂((log3 𝑛)/𝛽) span whp for the “LDD-UF-JTB” algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' □ C Performance Analysis of the Local Search Optimality In FAST-BCC, CC is an important primitive used in First-CC and Last-CC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We optimized the CC implementation using hash bags and local searches proposed by a recent paper [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' These optimizations function as a parallel granularity control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='04 YT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='10 OK 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='10 LJ 0 1 TW 0 2 FT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='02 GG First CC Rooting Tagging Last CC 0 2 SD 0 10 20 CW 0 20 HL14 0 50 HL12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='050 CA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='0 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='0 GL5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='0 GL10 Orig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='0 GL20 Orig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 0 5 10 COS5 Orig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 0 2 SQR Orig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 0 2 REC Orig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=" 0 1 2 SQR' Orig." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=" 0 1 2 REC' Orig." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='0 Chn7 Orig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 0 10 Chn8 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Optimized BCC breakdown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 𝑦-axis is the running time in seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' "Orig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' "= our original BCC implementation, "Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' "= our implementation optimized with hash bags and local search proposed in [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' When the frontier (vertices being processed in one round) is small, the algorithm explores multi-hop neighbors of the frontier instead of one-hop neighbors to saturate all threads with sufficient work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' It helps reduce the number of total rounds in a connectivity search, thus reducing the synchro- nization costs between rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' It works favorably well on large-diameter graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We measure the improvement from the optimizations in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 6, where Orig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' is the version without the optimizations and Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' is the version with the optimiza- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 6, on low-diameter graphs, Orig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' and Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' have similar performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' On large-diameter, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' can be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='1–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='5× faster than Orig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' and is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='8× faster on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' D Performance of the Tarjan-Vishkin Algorithm and Space Usage For completeness, we also implemented the faithful Tarjan- Vishkin algorithm [63] discussed in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We ac- knowledge that some existing papers [28, 37] discussed some possible optimizations for Tarjan-Vishkin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' However, engi- neering the Tarjan-Vishkin algorithm is not the focus of this paper, and we mainly use it to measure the space usage and get a sense on how Tarjan-Vishkin compares to other exist- ing BCC algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Our conclusions are consistent with the results drawn from the previous papers [28, 62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' The running time and space usage are given in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 3 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' For our implementations, Tarjan-Vishkin and use up to 11× extra space than FAST-BCC on FT and SD (including the space to store the input graph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' The space overhead is decided by edge-to-vertex ratio, and for graphs with smaller ratios (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=', chain graphs), the overhead is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Our TV implementation cannot run on the three largest graphs (CW, HL14, and HL12) on our machines with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='5TB memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' We note that the smallest among them (CW) only takes about 300GB to store the graph, and FAST-BCC uses 572GB mem- ory to process it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Since all three graphs have relatively large Ours GBBS TV SEQ Social YT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='030 0.' metadata={'source': 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seconds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' “N/A” = not applicable because of out of memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' edge-to-vertex ratios, our TV implementations are unlikely to execute on these graphs for shared-memory machines in foreseeable future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=" GBBS is slightly more space-efficient 16 YT OK LJ TW FT GG SD CW HL14 HL12 CA USA GE HH5 CH5 GL2 GL5 GL10 GL15 GL20 COS5 SQR REC SQR' REC' Chn7 Chn8 1 5 10 15 FAST-BCC GBBS Tarjan-Vishkin Figure 7." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Space Usage Comparision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 𝑦-axis is the space usage normalized to GBBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' Lower is better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' than FAST-BCC, and takes about 20% less space than us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' The reason is that they need to compute fewer number of tags than FAST-BCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' On all graphs, TV is faster than SEQ on 96 cores, but slower than FAST-BCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' The overhead of TV is due to the cost to explicitly construct the skeleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' On social, web, and 𝑘-NN graphs, the speedup for TV on 96 cores over SEQ is only 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content='4–3×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' TV works well on road and synthetic graphs due to small edge-to-vertex ratio, so the 𝑂(𝑚) work and space for generating the skeleton does not dominate the running time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' In this case, polylogarithmic span allows TV to perform consistently better than GBBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} +page_content=' 17' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfZvxg/content/2301.01356v1.pdf'} diff --git a/c9E4T4oBgHgl3EQfQAyD/content/tmp_files/2301.04978v1.pdf.txt b/c9E4T4oBgHgl3EQfQAyD/content/tmp_files/2301.04978v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..fe98b3ed8759835614a53a3bd822bb51aacbee7b --- /dev/null +++ b/c9E4T4oBgHgl3EQfQAyD/content/tmp_files/2301.04978v1.pdf.txt @@ -0,0 +1,469 @@ +Fourier coefficients of the net-baryon number density +Christian Schmidt𝑎,∗ +𝑎Fakultät für Physik, Universität Bielefeld, +Universitätsstraße 25, Bielefeld, Germany +E-mail: schmidt@physik.uni-bielefeld.de +We calculate Fourier coefficients of the net-baryon number as a function of a purely imaginary +chemical potential. The asymptotic behavior of these coefficients is governed by the singularity +structure of the QCD partition function and thus encodes information on phase transitions. For +the calculation of the Fourier coefficients from lattice data of the Bielefeld-Parma collaboration +we use a novel Filon-type quadrature, designed for highly oscillatory integrals. We find sensitivity +to chiral scaling in a narrow temperature interval below the Roberge-Weiss transition temperature. +Scaling fits yield reasonable values for the position of the Lee-Yang edge singularity in the complex +chemical potential plane. Our lattice data has been obtained from simulations with (2+1)-flavors +of highly improved staggered quarks (HISQ) at imaginary chemical potential on 𝑁𝜏 = 4, 6 and 8 +lattices at physical quark masses. +The 39th International Symposium on Lattice Field Theory, +8th-13th August, 2022, +Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, Germany +∗Speaker +© Copyright owned by the author(s) under the terms of the Creative Commons +Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). +https://pos.sissa.it/ +arXiv:2301.04978v1 [hep-lat] 12 Jan 2023 + +Fourier coefficients of the net-baryon number density +Christian Schmidt +1. +Introduction +A detailed calculation of the QCD phase diagram at nonzero temperature and density from first +principles is an unsolved open issue. Unfortunately, lattice QCD calculations are hindered by the +infamous sign problem as soon as a non-vanishing chemical potential 𝜇 > 0 is introduced. In order +to alleviate or circumvent the sign problem, many numerical methods have been developed, which +include the Taylor expansion about 𝜇 = 0 as well as calculations at purely imaginary chemical +potential 𝜇 = 𝑖𝜇𝐼, combined with an analytic continuation to real 𝜇 values. +Since the QCD +partition function is periodic in 𝜇𝐼/𝑇 ≡ ˆ𝜇𝐼 [1], with periodicity 2𝜋/𝑁𝑐, where 𝑁𝑐 denotes the +number of colors, it is quite natural to analyze data that is obtained from lattice QCD calculations +with imaginary chemical potential in terms of a Fourier expansion. +In particular, the Fourier +decomposition of the net baryon number density, +𝜒𝐵 +1 (𝑇, ˆ𝜇𝐵) = 𝑛𝐵(𝑇, ˆ𝜇𝐵) +𝑇3 += +1 +𝑉𝑇3 +𝜕 +ˆ𝜇𝐵 +ln 𝑍𝐺𝐶(𝑇, ˆ𝜇𝐵), +(1) +where 𝑍𝐺𝐶(𝑇, ˆ𝜇𝐵) is the grand canonical partition function1, ˆ𝜇𝐵 = 𝜇𝐵/𝑇 is the reduced baryon +chemical potential, and 𝑇 the temperature, is straightforwardly accessible from lattice QCD data +and has been the starting point of many recent studies [2–7]. As 𝜒𝐵 +1 exhibits the same periodicity +as the partition function and in addition is an odd function of 𝜇𝐼 +𝐵, we can expand Im𝜒𝐵 +1 as a Fourier +sine series +Re[𝜒𝐵 +1 (𝑇, 𝑖 ˆ𝜇𝐼 +𝐵)] = 0, +Im[𝜒𝐵 +1 (𝑇, 𝑖 ˆ𝜇𝐼 +𝐵)] = +∞ +∑︁ +𝑘=1 +𝑏𝑘(𝑇) sin +� +𝑘 ˆ𝜇𝐼 +𝐵 +� +. +(2) +Note that the real part of 𝜒𝐵 +1 vanishes identically at ˆ𝜇𝐵 = 𝑖 ˆ𝜇𝐼 +𝐵. The set of coefficients {𝑏𝑘(𝑇)}∞ +𝑘=1 +encode – up to an unimportant integration constant – the complete information on the QCD partition +function and thus on the thermodynamic properties of QCD matter in the region of the phase diagram +where the series, Eq. (2), converges. This is similar to the set of Taylor expansion coefficients of +the dimensionless pressure {𝑐2𝑘(𝑇)}∞ +𝑘=0, defined as +𝑝(𝑇, ˆ𝜇𝐵) +𝑇4 += +1 +𝑉𝑇3 ln 𝑍𝐺𝐶(𝑇, ˆ𝜇𝐵) = +∞ +∑︁ +𝑘=0 +𝑐2𝑘(𝑇) ˆ𝜇2𝑘 +𝐵 . +(3) +The calculation of the coefficients 𝑐2𝑘(𝑇) is numerically very demanding and currently only 𝑐2𝑘(𝑇) +for 𝑘 ≤ 4 are known from direct calculations at 𝜇𝐵 = 0. Moreover, statistical and systematical +errors for 𝑐6(𝑇) and 𝑐8(𝑇) are still very large, for recent results see [8]. It might thus be tempting to +verify or even complement the information from the known Taylor coefficients by a calculation of +the first few Fourier coefficients 𝑏𝑘(𝑇). Unfortunately, the calculation of the coefficients 𝑏𝑘(𝑇) is +equally difficult. However, just as the Taylor coefficients have a physical interpretation as cumulants +of the net baryon charge 𝐵, the coefficients 𝑏𝑘(𝑇) bear some physical meaning as well. The Fourier +expansion can formally be seen as a fugacity expansion. The set of available {𝑏𝑘(𝑇)} can thus be +used to determine the canonical partition functions 𝑍𝐶(𝑇,𝑉, 𝑁) [6, 7]. For the same reason the +Fourier expansion of the net baryon number density can be understood as a relativistic extension +of Mayer’s cluster expansion in fugacities [3]. In that spirit, the first coefficient 𝑏1(𝑇) is given by +1The dependence on the volume V is suppressed for simplicity. +2 + +Fourier coefficients of the net-baryon number density +Christian Schmidt +the partial pressure of the (non interacting) |𝐵| = 1 sector, whereas 𝑏2(𝑇) parametrizes the leading +order of the baryon-baryon interaction. Based on the cluster expansion, the authors of Ref. [3] +have introduced a model (CEM) that can predict the coefficients {𝑏𝑘(𝑇)}∞ +𝑘=2, based on the first two +coefficients 𝑏1(𝑇) and 𝑏2(𝑇). While this model verifies lattice results for the coefficients 𝑏3(𝑇) +and 𝑏4(𝑇), it exhibits an exponential decay of 𝑏𝑘(𝑇) for 𝑘 → ∞ at fixed 𝑇 and thus does not +incorporate critical behavior. The asymptotic behavior of the model was adjusted to a power-law +decay in Ref. [4] (RFM), without spoiling the agreement with the lattice data. A more thorough +investigation of the asymptotic behavior of the {𝑏𝑘(𝑇)} in terms of the universal 𝑂(4)-critical +scaling was performed in Ref. [5]. +We are aiming at an analysis of the analytic structure of the QCD phase diagram, by means +of Lee-Yang zeros [9]. Zeros of the partition function will manifest as poles of the baryon number +density 𝜒𝐵 +1 (𝑇) in the complex ˆ𝜇𝐵 plane. A new method for the analytic continuation of 𝜒𝐵 +1 (𝑇) +was introduced in Ref. [10] and is based on a multi-point Padé analysis. Since this method yields +a rational function approximation to the lattice data, it is straightforward to determine poles of the +observable in the complex 𝜇𝐵 plane. The closest pole might be associated with the Lee-Yang edge +singularity in QCD, which exhibits a well defined universal scaling and can be used to determine +various non-universal parameters, including the location of the critical point. Recent results on +Lee-Yang edge singularities and their scaling have been presented on this conference [11, 12]. The +verification of this method has been demonstrated by considering the universal scaling in the vicinity +of the Roberge-Weiss transition in QCD [10, 12, 13]. Here we will introduce a new method for the +numerical calculation of the Fourier coefficients {𝑏𝑘(𝑇)} and verify expected signals on universal +scaling in their large 𝑘 behavior. +2. +Determination of the Fourier coefficients +In many applications the Fourier coefficients are calculated by the conventional Discrete- +Fourier-Transform (DFT) or the popular Fast-Fourier-Transformation (FFT) algorithms. However, +as our input data stems from lattice QCD calculations and we are interested in the large 𝑘 behavior +of {𝑏𝑘(𝑇)}, these algorithms have two crucial drawbacks for our purpose. Firstly, the number of +detectable frequencies is directly related to the sampling rate of the function. However, lattice QCD +calculations are expensive and we want to keep the number of sampling points 𝑁 low. Secondly, +the numerical (root-mean-square) error of the DFT and FFT algorithm increases at least as ∼ 𝑁1/2 +[14]. It thus seems advantageous to first perform a numerical interpolation of the lattice data +before calculating the Fourier coefficients. Furthermore, it is obvious that the calculation of 𝑏𝑘(𝑇) +demands solving a highly oscillatory integral, we have +𝑏𝑘(𝑇) = 2 +𝜋 +𝜋 +∫ +0 +Im +� +𝜒𝐵 +1 +� +𝑇, 𝑖 ˆ𝜇𝐼 +𝐵 +�� +sin +� +𝑘 ˆ𝜇𝐼 +𝐵 +� +d ˆ𝜇𝐼 +𝐵. +(4) +A popular numerical method for oscillatory integrals is the Filon-type quadrature, which simply +makes use of the interpolating polynomial for the non oscillatory part of the integrand (here +Im[𝜒𝐵 +1 (𝑇, 𝑖 ˆ𝜇𝐼 +𝐵)]), whereas the oscillator (here sin�𝑘 ˆ𝜇𝐼 +𝐵 +�) is treated analytically. This method has +the advantage that it is asymptotic in the sense that its error decreases with increasing frequency 𝑘. +3 + +Fourier coefficients of the net-baryon number density +Christian Schmidt +Figure 1: Comparison of the Hermite interpolation of the net baryon number density Im[𝜒𝐵 +1 (𝑇, 𝑖 ˆ𝜇𝐼 +𝐵)] as +function of 𝜇𝐼 +𝐵 with a [11/8] rational approximation. The lattice data is obtained from a calculation on +a 363 × 6 lattice at 𝑇 = 190 MeV using SIMULATeQCD [16]. The inlay on the left shows the second +baryon number cumulant Re[𝜒𝐵 +2 (𝑇, 𝑖 ˆ𝜇𝐼 +𝐵)], together with the first derivative of the rational approximation of +Im[𝜒𝐵 +1 (𝑇, 𝑖 ˆ𝜇𝐼 +𝐵)]. The figure on the right is a zoom into the peak region. +In Ref. [15] a Filon-type quadrature has been constructed which uses in addition to the interpolating +data also derivatives, i.e. the interpolating polynomial is taken to be a piece-wise polynomial, +matching values and derivatives to order 𝑠 (Hermite interpolation) at the boundaries. The asymptotic +analysis performed in [15] shows that for this method, and a general oscillator of the form 𝑒𝑖𝜔𝑔(𝑥), +the error decreases as O(𝜔−𝑠−2). +In Fig. 1 we show the 2𝑛𝑑 order Hermite interpolation to Im[𝜒𝐵 +1 (𝑇, 𝑖 ˆ𝜇𝐼 +𝐵)], which continuously +matches 𝜒𝐵 +1 , 𝜒𝐵 +2 and 𝜒𝐵 +3 . The green error band is obtained by assuming independent, normally +distributed errors at the simulation points and bootstrapping. For comparison we show a [11/8] +rational polynomial obtained with the multi-point Padé method [10]. We see that both methods +always agree within errors, even in the peak region where the differences are most pronounced. +Once we have an analytic expression for an interpolating function at hand, it is easy to calculate the +integral Eq. (4) analytically. In particular, for the Hermite interpolation we can split the integration +over the interval [0, 𝜋] to a sum over the intervals defined by the 𝑁 data points. We have +𝑏𝑘(𝑇) = +𝑁 −1 +∑︁ +𝑗=1 +ˆ𝜇( 𝑗+1) +𝐵 +∫ +ˆ𝜇( 𝑗) +𝐵 +𝑝 𝑗(𝑥) sin(𝑘𝑥) d𝑥, +with +0 = ˆ𝜇1 +𝐵 < ˆ𝜇2 +𝐵 · · · < ˆ𝜇𝑁 +𝐵 = 𝜋 +(5) +denoting the 𝑁 locations of the data points and 𝑝 𝑗(𝑥) the interpolating polynomial of Im[𝜒𝐵 +1 (𝑇, 𝑖𝑥)] +for 𝑥 ∈ [ ˆ𝜇( 𝑗) +𝐵 , ˆ𝜇( 𝑗+1) +𝐵 +]. The results of the analytic integration for both types of interpolations are +shown in Fig. 2. The left (right) panel shows the results for 𝑇 = 190 (𝑇 = 180) MeV. Error bars for +the Hermite interpolation are again obtained from bootstrapping. We find that both interpolations +yield consistent results at least up to frequency 𝑘 ≲ 10. +4 + +Im Xi +.B +0.35 +- +[11/8] rational approximation +0.38 +0.30 +0.37 +0.25 +0.36 +0.20 - +0.35 +0.25 +0.15 - +Re X2 +B +0.00 +0.34 +0.25 +0.10 +0.33 +0.50 +0.32 +0.05 +0.75 +1.00 +0.31 +0.00 +0 +i +2 +3 +0.30 + +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +1.8 +2.0 +2.2 +2.4 +2.6 +2.8Fourier coefficients of the net-baryon number density +Christian Schmidt +Figure 2: Preliminary calculation of Fourier coefficients 𝑏𝑘 (𝑇) of the net baryon number density Im[𝜒𝐵 +1 ] +as a function of the frequency 𝑘 obtained from an analytic integration based on two different interpolating +functions (Hermite and rational). The left (right) panel shows results for 𝑇 = 190 (𝑇 = 180) MeV. The +normalizing factor is 𝑘(−1)𝑘+1 (left) and 𝑘2(−1)𝑘+1 (right). The calculation is based on lattice data of 𝜒𝐵 +1 , +𝜒𝐵 +2 and 𝜒𝐵 +3 from 363 × 6 lattices using SIMULATeQCD [16]. A Fit to ansatz Eq. (7) to the data at 𝑇 = 180 +MeV is also shown. +3. +Universal scaling +Finally we discuss the expected asymptotic behavior of the Fourier coefficients 𝑏𝑘(𝑇) in the +vicinity of a phase transition [5]. For 𝑂(𝑁) and 𝑍(2) symmetric spin models in 3d, it is well known +that the order parameter 𝑀 ∼ 𝜕 ln 𝑍(𝑇, ℎ)/𝜕ℎ, where ℎ is the symmetry breaking field, exhibits +branch-cuts in the complex ℎ-plane. The position of the branch-cut singularity is identical to the +Lee-yang edge (LYE) singularity, defined as the point where the linear density of the Lee-Yang +zeros diverges in the continuum limit. For the analysis here, we estimate the leading singular +behavior of the net baryon number density 𝜒𝐵 +1 . This is particularly easy in case of the Roberge- +Weiss transition, where we find Im[𝜒𝐵 +1 ] ∼ 𝑀 and ℎ ∼ 𝜇𝐼 +𝐵. For fixed 𝑇 = 𝑇𝑅𝑊 we thus assume +Im[𝜒𝐵 +1 ] ∼ (𝜋 − ˆ𝜇𝐼 +𝐵)1/𝛿, where 𝛿 refers to a a critical exponent of the 3d Z(2) universality class. For +the Fourier coefficients one thus obtains +𝑏𝑘 ∼ +𝜋 +∫ +0 +d ˆ𝜇𝐼 +𝐵 (𝜋 − ˆ𝜇𝐼 +𝐵)1/𝛿 sin +� +𝑘 ˆ𝜇𝐼 +𝐵 +� +∼ (−1)𝑘+1 +𝑘1+1/𝛿 . +(6) +The analysis is similar but more involved in the case of the chiral O(4) transition in presence of an +explicit symmetry breaking quark mass (crossover). In essence one finds [5] +𝑏𝑘 ∼ 𝑒−𝑘 ˆ𝜇𝑅 +𝐿𝑌 𝐸 +𝑘2−𝛼 +� +sin +� +𝑘 ˆ𝜇𝐼 +𝐿𝑌 𝐸 − 𝛼𝜋/2 +� ++ 𝑅± sin +� +𝑘 ˆ𝜇𝐼 +𝐿𝑌 𝐸 + 𝛼𝜋/2 +�� +, +(7) +for 𝑇𝑐𝑒𝑝 < 𝑇 < 𝑇𝑅𝑊 . Here 𝑇𝑐𝑒𝑝 denotes the temperature of the QCD critical point, the branch-cut +singularity is located at ˆ𝜇𝐿𝑌 𝐸 = ˆ𝜇𝑅 +𝐿𝑌 𝐸 + 𝑖 ˆ𝜇𝐼 +𝐿𝑌 𝐸, and 𝛼 ≈ −0.21 and 𝑅± ≈ 1.85 denote universal +quantities from the O(4) universality class. Hence, the behavior resembles a damped oscillation +were the exponential suppression relates to the real part of the LYE and the period of the oscillation +to the imaginary part of the LYE. +5 + +100 +T = 190 MeV +rational +T = 180 MeV +rational +T +Hermite +0.4 - +chiral fit +10-1 +T +Hermite +0.3 - +10-2 +)k+1 +10-3. +0.2 - +10-4 +0.1 - +10-5 +0.0 +10-6 +k +k +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 +17.5 +20.0 +0.0 +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 +17.5 +20.0Fourier coefficients of the net-baryon number density +Christian Schmidt +In Fig. 2 we show examples for both of these scenarios. However, a clear oscillatory behavior, +which indicates sensitivity to the chiral O(4) transition could only be found for two of our temper- +atures, 𝑇 = 180 and 𝑇 = 185 MeV. For 𝑇 < 185 MeV, the suppression due to the real part is so +large that the oscillations are hidden in the noise. Fits to the asymptotic behavior of 𝑏𝑘(𝑇), for +𝑇 = 180 and 𝑇 = 185 MeV with ansatz Eq. (7) yield locations for the LYE which are consistent with +results from the poles of the multi-point Padé [12]. In fact, the real parts are in good agreement, the +imaginary parts come out slightly lower. The fit shown in Fig. 2 yields ˆ𝜇𝐿𝑌 𝐸 = 0.97(6) +3.123(3)𝑖. +4. +Summary, conclusion and outlook +We have presented a preliminary calculation of Fourier coefficients {𝑏𝑘(𝑇)} of the net baryon +number Im[𝜒𝐵 +1 (𝑇, 𝑖 ˆ𝜇𝐼 +𝐵)]. The calculation is based on lattice data from the Bielefeld-Parma collabo- +ration [12] and uses a novel Filon-type quadrature. With this method we were able to obtain Fourier +coefficients for frequencies of 𝑘 ≲ 10. Through the asymptotic behavior of these coefficients one +might identify branch-cut singularities in the complex chemical potential plane. However, sensi- +tivity to the chiral O(4) transition was only found in a narrow temperature interval 𝑇 ∈ [180, 185] +MeV. For temperatures below 𝑇 = 180 MeV, the exponential suppression with is associated with +the real part of the LYE seems too strong. To alleviate this problem in future calculation we might +improve the numerical quadrature further by investigate adaptive Filon-type methods and go to +lighter than physical quark masses. The latter will reduce the real part of the LYE and thus lift the +exponential suppression. +Acknowledgments +This work was supported in part by the Deutsche Forschungsgemeinschaft (DFG) through the +grant 315477589-TRR 211 and "NFDI 39/1" for the PUNCH4NFDI consortium and the grant EU +H2020-MSCA-ITN-2018-813942 (EuroPLEx) of the European Union. +References +[1] A. Roberge and N. Weiss, Gauge Theories With Imaginary Chemical Potential and the +Phases of QCD, Nucl. Phys. B 275 (1986) 734. +[2] V. Vovchenko, A. Pasztor, Z. Fodor, S. D. Katz and H. Stoecker, Repulsive baryonic +interactions and lattice QCD observables at imaginary chemical potential, Phys. Lett. B 775 +(2017) 71 [1708.02852]. +[3] V. Vovchenko, J. Steinheimer, O. Philipsen and H. Stoecker, Cluster Expansion Model for +QCD Baryon Number Fluctuations: No Phase Transition at 𝜇𝐵/𝑇 < 𝜋, Phys. Rev. D 97 +(2018) 114030 [1711.01261]. +[4] G. A. Almasi, B. Friman, K. Morita, P. M. Lo and K. Redlich, Fourier coefficients of the +net-baryon number density and chiral criticality, Phys. Rev. D 100 (2019) 016016 +[1805.04441]. +6 + +Fourier coefficients of the net-baryon number density +Christian Schmidt +[5] G. A. Almási, B. Friman, K. Morita and K. Redlich, Fourier coefficients of the net baryon +number density and their scaling properties near a phase transition, Phys. Lett. B 793 (2019) +19 [1902.05457]. +[6] V. G. Bornyakov, D. L. Boyda, V. A. Goy, A. V. Molochkov, A. Nakamura, A. A. Nikolaev +et al., New approach to canonical partition functions computation in 𝑁 𝑓 = 2 lattice QCD at +finite baryon density, Phys. Rev. D 95 (2017) 094506 [1611.04229]. +[7] V. G. Bornyakov, N. V. 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Di Renzo, J. Goswami, G. Nicotra et al., +Determination of Lee-Yang edge singularities in QCD by rational approximations, PoS +(LATTICE2022) (2023) 164 [2301.03952]. +[13] C. Schmidt, D. A. Clarke, P. Dimopoulos, J. Goswami, G. Nicotra, F. Di Renzo et al., +Detecting critical points from Lee-Yang edge singularities in lattice QCD, Acta Phys. Pol. B +Proc. Suppl. 16 (2023) A52 [2209.04345]. +[14] J. C. Schatzman, Accuracy of the discrete fourier transform and the fast fourier transform, +SIAM Journal on Scientific Computing 17 (1996) 1150 +[https://doi.org/10.1137/S1064827593247023]. +[15] A. Iserles and S. P. NØrsett, On quadrature methods for highly oscillatory integrals and their +implementation, BIT Numerical Mathematics 44 (2004) 755. +[16] D. Bollweg, L. Altenkort, D. A. Clarke, O. Kaczmarek, L. Mazur, C. Schmidt et al., HotQCD +on multi-GPU Systems, PoS LATTICE2021 (2022) 196 [2111.10354]. +7 + diff --git a/c9E4T4oBgHgl3EQfQAyD/content/tmp_files/load_file.txt b/c9E4T4oBgHgl3EQfQAyD/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..879280db215451d9b863da3081e9967f7eb773c5 --- /dev/null +++ b/c9E4T4oBgHgl3EQfQAyD/content/tmp_files/load_file.txt @@ -0,0 +1,335 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf,len=334 +page_content='Fourier coefficients of the net-baryon number density Christian Schmidt𝑎,∗ 𝑎Fakultät für Physik, Universität Bielefeld, Universitätsstraße 25, Bielefeld, Germany E-mail: schmidt@physik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='uni-bielefeld.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='de We calculate Fourier coefficients of the net-baryon number as a function of a purely imaginary chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' The asymptotic behavior of these coefficients is governed by the singularity structure of the QCD partition function and thus encodes information on phase transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' For the calculation of the Fourier coefficients from lattice data of the Bielefeld-Parma collaboration we use a novel Filon-type quadrature, designed for highly oscillatory integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' We find sensitivity to chiral scaling in a narrow temperature interval below the Roberge-Weiss transition temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' Scaling fits yield reasonable values for the position of the Lee-Yang edge singularity in the complex chemical potential plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' Our lattice data has been obtained from simulations with (2+1)-flavors of highly improved staggered quarks (HISQ) at imaginary chemical potential on 𝑁𝜏 = 4, 6 and 8 lattices at physical quark masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' The 39th International Symposium on Lattice Field Theory, 8th-13th August, 2022, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, Germany ∗Speaker © Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='0 International License (CC BY-NC-ND 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' https://pos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='sissa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='it/ arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='04978v1 [hep-lat] 12 Jan 2023 Fourier coefficients of the net-baryon number density Christian Schmidt 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' Introduction A detailed calculation of the QCD phase diagram at nonzero temperature and density from first principles is an unsolved open issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' Unfortunately, lattice QCD calculations are hindered by the infamous sign problem as soon as a non-vanishing chemical potential 𝜇 > 0 is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' In order to alleviate or circumvent the sign problem, many numerical methods have been developed, which include the Taylor expansion about 𝜇 = 0 as well as calculations at purely imaginary chemical potential 𝜇 = 𝑖𝜇𝐼, combined with an analytic continuation to real 𝜇 values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' Since the QCD partition function is periodic in 𝜇𝐼/𝑇 ≡ ˆ𝜇𝐼 [1], with periodicity 2𝜋/𝑁𝑐, where 𝑁𝑐 denotes the number of colors, it is quite natural to analyze data that is obtained from lattice QCD calculations with imaginary chemical potential in terms of a Fourier expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' In particular, the Fourier decomposition of the net baryon number density, 𝜒𝐵 1 (𝑇, ˆ𝜇𝐵) = 𝑛𝐵(𝑇, ˆ𝜇𝐵) 𝑇3 = 1 𝑉𝑇3 𝜕 ˆ𝜇𝐵 ln 𝑍𝐺𝐶(𝑇, ˆ𝜇𝐵), (1) where 𝑍𝐺𝐶(𝑇, ˆ𝜇𝐵) is the grand canonical partition function1, ˆ𝜇𝐵 = 𝜇𝐵/𝑇 is the reduced baryon chemical potential, and 𝑇 the temperature, is straightforwardly accessible from lattice QCD data and has been the starting point of many recent studies [2–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' As 𝜒𝐵 1 exhibits the same periodicity as the partition function and in addition is an odd function of 𝜇𝐼 𝐵, we can expand Im𝜒𝐵 1 as a Fourier sine series Re[𝜒𝐵 1 (𝑇, 𝑖 ˆ𝜇𝐼 𝐵)] = 0, Im[𝜒𝐵 1 (𝑇, 𝑖 ˆ𝜇𝐼 𝐵)] = ∞ ∑︁ 𝑘=1 𝑏𝑘(𝑇) sin � 𝑘 ˆ𝜇𝐼 𝐵 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' (2) Note that the real part of 𝜒𝐵 1 vanishes identically at ˆ𝜇𝐵 = 𝑖 ˆ𝜇𝐼 𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' The set of coefficients {𝑏𝑘(𝑇)}∞ 𝑘=1 encode – up to an unimportant integration constant – the complete information on the QCD partition function and thus on the thermodynamic properties of QCD matter in the region of the phase diagram where the series, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' (2), converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' This is similar to the set of Taylor expansion coefficients of the dimensionless pressure {𝑐2𝑘(𝑇)}∞ 𝑘=0, defined as 𝑝(𝑇, ˆ𝜇𝐵) 𝑇4 = 1 𝑉𝑇3 ln 𝑍𝐺𝐶(𝑇, ˆ𝜇𝐵) = ∞ ∑︁ 𝑘=0 𝑐2𝑘(𝑇) ˆ𝜇2𝑘 𝐵 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' (3) The calculation of the coefficients 𝑐2𝑘(𝑇) is numerically very demanding and currently only 𝑐2𝑘(𝑇) for 𝑘 ≤ 4 are known from direct calculations at 𝜇𝐵 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' Moreover, statistical and systematical errors for 𝑐6(𝑇) and 𝑐8(𝑇) are still very large, for recent results see [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' It might thus be tempting to verify or even complement the information from the known Taylor coefficients by a calculation of the first few Fourier coefficients 𝑏𝑘(𝑇).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' Unfortunately, the calculation of the coefficients 𝑏𝑘(𝑇) is equally difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' However, just as the Taylor coefficients have a physical interpretation as cumulants of the net baryon charge 𝐵, the coefficients 𝑏𝑘(𝑇) bear some physical meaning as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' The Fourier expansion can formally be seen as a fugacity expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' The set of available {𝑏𝑘(𝑇)} can thus be used to determine the canonical partition functions 𝑍𝐶(𝑇,𝑉, 𝑁) [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' For the same reason the Fourier expansion of the net baryon number density can be understood as a relativistic extension of Mayer’s cluster expansion in fugacities [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' In that spirit, the first coefficient 𝑏1(𝑇) is given by 1The dependence on the volume V is suppressed for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' 2 Fourier coefficients of the net-baryon number density Christian Schmidt the partial pressure of the (non interacting) |𝐵| = 1 sector, whereas 𝑏2(𝑇) parametrizes the leading order of the baryon-baryon interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' Based on the cluster expansion, the authors of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' [3] have introduced a model (CEM) that can predict the coefficients {𝑏𝑘(𝑇)}∞ 𝑘=2, based on the first two coefficients 𝑏1(𝑇) and 𝑏2(𝑇).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' While this model verifies lattice results for the coefficients 𝑏3(𝑇) and 𝑏4(𝑇), it exhibits an exponential decay of 𝑏𝑘(𝑇) for 𝑘 → ∞ at fixed 𝑇 and thus does not incorporate critical behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' The asymptotic behavior of the model was adjusted to a power-law decay in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' [4] (RFM), without spoiling the agreement with the lattice data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' A more thorough investigation of the asymptotic behavior of the {𝑏𝑘(𝑇)} in terms of the universal 𝑂(4)-critical scaling was performed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' We are aiming at an analysis of the analytic structure of the QCD phase diagram, by means of Lee-Yang zeros [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' Zeros of the partition function will manifest as poles of the baryon number density 𝜒𝐵 1 (𝑇) in the complex ˆ𝜇𝐵 plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' A new method for the analytic continuation of 𝜒𝐵 1 (𝑇) was introduced in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' [10] and is based on a multi-point Padé analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' Since this method yields a rational function approximation to the lattice data, it is straightforward to determine poles of the observable in the complex 𝜇𝐵 plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' The closest pole might be associated with the Lee-Yang edge singularity in QCD, which exhibits a well defined universal scaling and can be used to determine various non-universal parameters, including the location of the critical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' Recent results on Lee-Yang edge singularities and their scaling have been presented on this conference [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' The verification of this method has been demonstrated by considering the universal scaling in the vicinity of the Roberge-Weiss transition in QCD [10, 12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' Here we will introduce a new method for the numerical calculation of the Fourier coefficients {𝑏𝑘(𝑇)} and verify expected signals on universal scaling in their large 𝑘 behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' Determination of the Fourier coefficients In many applications the Fourier coefficients are calculated by the conventional Discrete- Fourier-Transform (DFT) or the popular Fast-Fourier-Transformation (FFT) algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' However, as our input data stems from lattice QCD calculations and we are interested in the large 𝑘 behavior of {𝑏𝑘(𝑇)}, these algorithms have two crucial drawbacks for our purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' Firstly, the number of detectable frequencies is directly related to the sampling rate of the function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' However, lattice QCD calculations are expensive and we want to keep the number of sampling points 𝑁 low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' Secondly, the numerical (root-mean-square) error of the DFT and FFT algorithm increases at least as ∼ 𝑁1/2 [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' It thus seems advantageous to first perform a numerical interpolation of the lattice data before calculating the Fourier coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' Furthermore, it is obvious that the calculation of 𝑏𝑘(𝑇) demands solving a highly oscillatory integral, we have 𝑏𝑘(𝑇) = 2 𝜋 𝜋 ∫ 0 Im � 𝜒𝐵 1 � 𝑇, 𝑖 ˆ𝜇𝐼 𝐵 �� sin � 𝑘 ˆ𝜇𝐼 𝐵 � d ˆ𝜇𝐼 𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' (4) A popular numerical method for oscillatory integrals is the Filon-type quadrature, which simply makes use of the interpolating polynomial for the non oscillatory part of the integrand (here Im[𝜒𝐵 1 (𝑇, 𝑖 ˆ𝜇𝐼 𝐵)]), whereas the oscillator (here sin�𝑘 ˆ𝜇𝐼 𝐵 �) is treated analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' This method has the advantage that it is asymptotic in the sense that its error decreases with increasing frequency 𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' 3 Fourier coefficients of the net-baryon number density Christian Schmidt Figure 1: Comparison of the Hermite interpolation of the net baryon number density Im[𝜒𝐵 1 (𝑇, 𝑖 ˆ𝜇𝐼 𝐵)] as function of 𝜇𝐼 𝐵 with a [11/8] rational approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' The lattice data is obtained from a calculation on a 363 × 6 lattice at 𝑇 = 190 MeV using SIMULATeQCD [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' The inlay on the left shows the second baryon number cumulant Re[𝜒𝐵 2 (𝑇, 𝑖 ˆ𝜇𝐼 𝐵)], together with the first derivative of the rational approximation of Im[𝜒𝐵 1 (𝑇, 𝑖 ˆ𝜇𝐼 𝐵)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' The figure on the right is a zoom into the peak region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' [15] a Filon-type quadrature has been constructed which uses in addition to the interpolating data also derivatives, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' the interpolating polynomial is taken to be a piece-wise polynomial, matching values and derivatives to order 𝑠 (Hermite interpolation) at the boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' The asymptotic analysis performed in [15] shows that for this method, and a general oscillator of the form 𝑒𝑖𝜔𝑔(𝑥), the error decreases as O(𝜔−𝑠−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' 1 we show the 2𝑛𝑑 order Hermite interpolation to Im[𝜒𝐵 1 (𝑇, 𝑖 ˆ𝜇𝐼 𝐵)], which continuously matches 𝜒𝐵 1 , 𝜒𝐵 2 and 𝜒𝐵 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' The green error band is obtained by assuming independent, normally distributed errors at the simulation points and bootstrapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' For comparison we show a [11/8] rational polynomial obtained with the multi-point Padé method [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' We see that both methods always agree within errors, even in the peak region where the differences are most pronounced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' Once we have an analytic expression for an interpolating function at hand, it is easy to calculate the integral Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' (4) analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' In particular, for the Hermite interpolation we can split the integration over the interval [0, 𝜋] to a sum over the intervals defined by the 𝑁 data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' We have 𝑏𝑘(𝑇) = 𝑁 −1 ∑︁ 𝑗=1 ˆ𝜇( 𝑗+1) 𝐵 ∫ ˆ𝜇( 𝑗) 𝐵 𝑝 𝑗(𝑥) sin(𝑘𝑥) d𝑥, with 0 = ˆ𝜇1 𝐵 < ˆ𝜇2 𝐵 · · · < ˆ𝜇𝑁 𝐵 = 𝜋 (5) denoting the 𝑁 locations of the data points and 𝑝 𝑗(𝑥) the interpolating polynomial of Im[𝜒𝐵 1 (𝑇, 𝑖𝑥)] for 𝑥 ∈ [ ˆ𝜇( 𝑗) 𝐵 , ˆ𝜇( 𝑗+1) 𝐵 ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' The results of the analytic integration for both types of interpolations are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' The left (right) panel shows the results for 𝑇 = 190 (𝑇 = 180) MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' Error bars for the Hermite interpolation are again obtained from bootstrapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' We find that both interpolations yield consistent results at least up to frequency 𝑘 ≲ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' 4 Im Xi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='35 [11/8] rational approximation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='20 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='15 - Re X2 B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='00 0 i 2 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='30 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='8Fourier coefficients of the net-baryon number density Christian Schmidt Figure 2: Preliminary calculation of Fourier coefficients 𝑏𝑘 (𝑇) of the net baryon number density Im[𝜒𝐵 1 ] as a function of the frequency 𝑘 obtained from an analytic integration based on two different interpolating functions (Hermite and rational).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' The left (right) panel shows results for 𝑇 = 190 (𝑇 = 180) MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' The normalizing factor is 𝑘(−1)𝑘+1 (left) and 𝑘2(−1)𝑘+1 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' The calculation is based on lattice data of 𝜒𝐵 1 , 𝜒𝐵 2 and 𝜒𝐵 3 from 363 × 6 lattices using SIMULATeQCD [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' A Fit to ansatz Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' (7) to the data at 𝑇 = 180 MeV is also shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' Universal scaling Finally we discuss the expected asymptotic behavior of the Fourier coefficients 𝑏𝑘(𝑇) in the vicinity of a phase transition [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' For 𝑂(𝑁) and 𝑍(2) symmetric spin models in 3d, it is well known that the order parameter 𝑀 ∼ 𝜕 ln 𝑍(𝑇, ℎ)/𝜕ℎ, where ℎ is the symmetry breaking field, exhibits branch-cuts in the complex ℎ-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' The position of the branch-cut singularity is identical to the Lee-yang edge (LYE) singularity, defined as the point where the linear density of the Lee-Yang zeros diverges in the continuum limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' For the analysis here, we estimate the leading singular behavior of the net baryon number density 𝜒𝐵 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' This is particularly easy in case of the Roberge- Weiss transition, where we find Im[𝜒𝐵 1 ] ∼ 𝑀 and ℎ ∼ 𝜇𝐼 𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' For fixed 𝑇 = 𝑇𝑅𝑊 we thus assume Im[𝜒𝐵 1 ] ∼ (𝜋 − ˆ𝜇𝐼 𝐵)1/𝛿, where 𝛿 refers to a a critical exponent of the 3d Z(2) universality class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' For the Fourier coefficients one thus obtains 𝑏𝑘 ∼ 𝜋 ∫ 0 d ˆ𝜇𝐼 𝐵 (𝜋 − ˆ𝜇𝐼 𝐵)1/𝛿 sin � 𝑘 ˆ𝜇𝐼 𝐵 � ∼ (−1)𝑘+1 𝑘1+1/𝛿 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' (6) The analysis is similar but more involved in the case of the chiral O(4) transition in presence of an explicit symmetry breaking quark mass (crossover).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' In essence one finds [5] 𝑏𝑘 ∼ 𝑒−𝑘 ˆ𝜇𝑅 𝐿𝑌 𝐸 𝑘2−𝛼 � sin � 𝑘 ˆ𝜇𝐼 𝐿𝑌 𝐸 − 𝛼𝜋/2 � + 𝑅± sin � 𝑘 ˆ𝜇𝐼 𝐿𝑌 𝐸 + 𝛼𝜋/2 �� , (7) for 𝑇𝑐𝑒𝑝 < 𝑇 < 𝑇𝑅𝑊 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' Here 𝑇𝑐𝑒𝑝 denotes the temperature of the QCD critical point, the branch-cut singularity is located at ˆ𝜇𝐿𝑌 𝐸 = ˆ𝜇𝑅 𝐿𝑌 𝐸 + 𝑖 ˆ𝜇𝐼 𝐿𝑌 𝐸, and 𝛼 ≈ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='21 and 𝑅± ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='85 denote universal quantities from the O(4) universality class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' Hence, the behavior resembles a damped oscillation were the exponential suppression relates to the real part of the LYE and the period of the oscillation to the imaginary part of the LYE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' 5 100 T = 190 MeV rational T = 180 MeV rational T Hermite 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='4 - chiral fit 10-1 T Hermite 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='3 - 10-2 )k+1 10-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='2 - 10-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='1 - 10-5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='0 10-6 k k 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='5 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='5 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='0Fourier coefficients of the net-baryon number density Christian Schmidt In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' 2 we show examples for both of these scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' However, a clear oscillatory behavior, which indicates sensitivity to the chiral O(4) transition could only be found for two of our temper- atures, 𝑇 = 180 and 𝑇 = 185 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' For 𝑇 < 185 MeV, the suppression due to the real part is so large that the oscillations are hidden in the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' Fits to the asymptotic behavior of 𝑏𝑘(𝑇), for 𝑇 = 180 and 𝑇 = 185 MeV with ansatz Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' (7) yield locations for the LYE which are consistent with results from the poles of the multi-point Padé [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' In fact, the real parts are in good agreement, the imaginary parts come out slightly lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' The fit shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' 2 yields ˆ𝜇𝐿𝑌 𝐸 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='97(6) +3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content='123(3)𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' Summary, conclusion and outlook We have presented a preliminary calculation of Fourier coefficients {𝑏𝑘(𝑇)} of the net baryon number Im[𝜒𝐵 1 (𝑇, 𝑖 ˆ𝜇𝐼 𝐵)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' The calculation is based on lattice data from the Bielefeld-Parma collabo- ration [12] and uses a novel Filon-type quadrature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' With this method we were able to obtain Fourier coefficients for frequencies of 𝑘 ≲ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' Through the asymptotic behavior of these coefficients one might identify branch-cut singularities in the complex chemical potential plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' However, sensi- tivity to the chiral O(4) transition was only found in a narrow temperature interval 𝑇 ∈ [180, 185] MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' For temperatures below 𝑇 = 180 MeV, the exponential suppression with is associated with the real part of the LYE seems too strong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' To alleviate this problem in future calculation we might improve the numerical quadrature further by investigate adaptive Filon-type methods and go to lighter than physical quark masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' The latter will reduce the real part of the LYE and thus lift the exponential suppression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' Acknowledgments This work was supported in part by the Deutsche Forschungsgemeinschaft (DFG) through the grant 315477589-TRR 211 and "NFDI 39/1" for the PUNCH4NFDI consortium and the grant EU H2020-MSCA-ITN-2018-813942 (EuroPLEx) of the European Union.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' References [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' Roberge and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E4T4oBgHgl3EQfQAyD/content/2301.04978v1.pdf'} +page_content=' Weiss, Gauge Theories With Imaginary Chemical Potential and the Phases of QCD, Nucl.' metadata={'source': 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a/edE0T4oBgHgl3EQf5gLh/content/tmp_files/2301.02753v1.pdf.txt b/edE0T4oBgHgl3EQf5gLh/content/tmp_files/2301.02753v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..6709ddde188afa80816ffcc031053db44e883951 --- /dev/null +++ b/edE0T4oBgHgl3EQf5gLh/content/tmp_files/2301.02753v1.pdf.txt @@ -0,0 +1,1894 @@ +1 +Planning and Tracking Control of Full +Drive-by-Wire Electric Vehicles in Unstructured +Scenario +Guoying Chen, Min Hua, Wei Liu, Jinhai Wang, Shunhui Song, Changsheng Liu +Abstract—Full drive-by-wire electric vehicles (FDWEV) with +X-by-wire technology can achieve independent driving, braking, +and steering of each wheel, providing a good application platform +for autonomous driving technology. Path planning and tracking +control, in particular, are critical components of autonomous +driving. However, It is challenging to comprehensively design +an robust control algorithm by integrating vehicle path planning +in a complicated unstructured scenario for FDWEV. To address +the above issue, this paper first proposes the artificial potential +field (APF) method for path planning in the prescribed park +with different static obstacles to generate the reference path +information, where speed planning is incorporated considering +kinematics and dynamic constraints. Second, two tracking control +methods, curvature calculation (CC-based) and model predictive +control (MPC-based) methods with the lateral dynamics model, +are proposed to track the desired path under different driving +conditions, in which a forward-looking behavior model of the +driver with variable preview distance is designed based on +fuzzy control theory. CarSim-AMESim-Simulink co-simulation is +conducted with the existence of obstacles. The simulation results +show that the proposed two control approaches are effective +for many driving scenarios and the MPC-based path-tracking +controller enhances dynamic tracking performance and ensures +good maneuverability under high-dynamic driving conditions. +Index Terms—Full drive-by-wire electric vehicles, path plan- +ning, tracking control, curvature calculation, model predictive +control +I. INTRODUCTION +V +EHICLE driving safety has always been an important +research field for vehicle manufacturing [1]–[4]. With the +increasingly intelligent, electrified, wire-controlled, and net- +worked vehicles [5], the intelligent full drive-by-wire electric +vehicle possesses more advantages than traditional vehicles, +such as independent driving, braking, and steering control [6], +[7]. In addition, each wheel in the steering system can achieve +positive and negative 180-degree rotation around the z-axis +Guoying Chen is with the State Key Laboratory of Automotive Simulation +and Control, Jilin University, Changchun, 130022, P.R. China. (e-mail: cgy- +011@163.com), Corresponding author. +Min Hua is with the School of Engineering, University of Birmingham, +Birmingham, B15 2TT, UK. (e-mail: mxh623@student.bham.ac.uk). +Wei Liu is with the School of Electrical and Computer Engineering, Purdue +University, West Lafayette, IN 47907, USA (e-mail: liu3044@purdue.edu). +Jinhai Wang is with the School of Automotive Engineering, Wuhan +University of Technology, Wuhan, 430070, P.R. China (e-mail: wangjin- +hai@whut.edu.cn). +Shunhui Song is with the School of Automotive Studies, Tongji University, +Shanghai, 218074, P.R. China (e-mail: songshunhui@gmail.com). +Changsheng Liu is with the College of Computer Science and Technol- +ogy, Zhejiang University, Hangzhou, 310027, P.R. China. (e-mail: chang- +shengliu23@gmail.com). +of the tire for various steering modes, e.g., crab, traverse, +diagonal travel, and steer in place [8]. Controllable degrees of +freedom can improve the mobility and flexibility of the vehicle +at low speeds to achieve automatic parking and other functions +in low-speed and narrow spaces, as well as the stability +and safety of the vehicle when driving at high speeds [9]. +Furthermore, due to the application of X-by-wire technology, +four-wheel motor torque, and other parameters can be obtained +in real time, which will provide accurate information for the +advanced dynamics control system and the intelligent chassis +integration control. With the introduction of the additional +sensing signals and position information, the lane change, +overtaking and car-following, and a series of autonomous +operations will be achieved [10]. +Autonomous driving(AD) integrates the information from +the surroundings and the vehicle’s states via advanced sensing +systems (e.g., LIDAR, cameras, GPS, and IMU) [11]–[15], and +communication networks [16] to replace or assist the driver. In +the last decade, with the development and advances in sensors +and computer technologies, AD-related research have become +hot topics in both path planning and autonomous tracking +control, and have been conducted to ensure vehicle safety, +comfort, and energy efficiency [17], [18]. +Many studies have been dedicated to solving the collision- +free path planning problem to meet the constraints of col- +lision avoidance, steering speed, and road curvature under +continuous conditions [19]. State lattice is a graph theory- +based approach that has emerged in recent years, where +kinematic and dynamical constraints can be regarded through +state space rules and repeatable sampling in autonomous +driving. Furthermore, unstructured road environments give +more possibilities for employing this method. Lattice edges +are computed offline to achieve real-time online planning using +look-up tables to reduce the computational burden. However, +several problems exist for urban areas with highly variable +structured environments, such as the discretization of heading +angles that may lead to oscillations between two oriented +samples [20], [21]. A spline-based search tree utilizes a search +tree to generate a path that is tangent to all objects and +assumes that the optimal path is oriented straight ahead or +touches at least one object. Each path segment from one +object to the next is defined by a constant acceleration, which +is effective in certain scenarios. However, transient lateral +acceleration and sudden changes make this approach infeasible +for vehicle robust control [22]. Alternatively, model predictive +control (MPC) [23] has been widely applied in the field +arXiv:2301.02753v1 [eess.SY] 7 Jan 2023 + +2 +of automotive, where the optimization goal is to find the +optimal value of a series of curvature changes to minimize +the steering effort defined by the road curvature in a collision +scenario, along a prediction domain from a fixed sampling +time [24]. A potential field-based path planner is presented +to provide obstacle avoidance with an adaptive multi-speed +scheduler using a fuzzy system [25]. In [26], a time-horizon +based MPC method is proposed to reduce the calculation load +of vehicle speed predictive control. And the performance is +verified through real road simulation experiments. +Nowadays, the vehicle models adopted for autonomous +tracking control are divided into three types: geometric, +kinematic, and dynamic. The mainstream control algorithms +include PID control, sliding mode control (SMC), neural +network control, and MPC [27] to obtain the corresponding +control parameters: the steering wheel angle, throttle opening, +and braking strength [28]. There is a certain delay in the +actual vehicle control when executing the control command +immediately, which is the biggest problem of PID algorithm +in autonomous control [29]; in addition, advanced control +algorithms, such as SMC and neural networks, are also widely +employed [30], [31]. However, these algorithms greatly depend +on parameter sensitivities and environment information, and +different vehicle constraints need to be considered when the +vehicle model is embedded; there are great limitations for +these optimization algorithms with constraints, although MPC +is capable of solving the constraints problem with high com- +putation cost. Therefore, various tracking control algorithms +need to be traded off in all aspects. The main challenge with +current tracking control algorithms is a reasonable simplifica- +tion of the vehicle dynamics modeling to improve the real- +time performance of the algorithm while ensuring tracking +accuracy. +Unfortunately, most algorithms mainly attempt to find +collision-free paths with no guarantee of feasibility in the real +world due to poor quality and small turns. To mitigate the +above limitations in the harsh condition, in this paper, a com- +plicated path with considerable curvatures will be created, and +the precise and independent control of four-wheel angles and +motor torque can be fully utilized to achieve obstacle-avoiding +tracking control based on the multi-degree-of-freedom control +platform of the FDWEV [32]. The autonomous tracking con- +trol of the FDWEV can provide a theoretical and practical +basis for developing AD technology. The main contribution +of this paper is fourfold. +1) Artificial potential field (APF) for path planning through +comprehensive comparison is optimized to address the poten- +tial unreachable and local optimal problems with the improved +repulsive force function and the direction of repulsive force. +2) For the high maneuverability of the FDWEV, the relevant +kinematics and dynamic constraints are considered to be inte- +grated into the vehicle speed planning. Then interpolation and +piecewise fitting are conducted to provide achievable position +information for tracking control. +3) Based on fuzzy control theory, a forward-looking driver +behavior model with variable preview distance is designed. +The curvature calculation is proposed based on the vehicle +lateral dynamics and kinematic models. For the shortcoming of +the tracking control algorithm based on curvature calculation, +the MPC algorithm is designed by considering the vehicle +constraints, real-time prediction, and feedback optimization of +the vehicle states. +4) Based on the path planning and vehicle speed planning in +the prescribed park, the autonomous tracking controls are ver- +ified based on the co-simulation of Simulink/Carsim/Amesim +platform under different conditions with speeds of 30 km/h and +50 km/h in a harsh environment with considerable curvatures. +The rest of this paper is organized as follows: Section 2 +formulates the lateral dynamics model. In section 3, path +planning with APF method is proposed with the generated +reachable points of vehicle tracking control. In Section 4, +tracking control based on the curvature method and MPC +algorithm are proposed according to the kinematic and dy- +namics model. Section 5 conducts simulation experiments and +analyzes the results for validation and evaluation, followed by +the conclusions in Section 6. +II. LATERAL DYNAMICS MODEL +The lateral dynamics model is established firstly to design +the tracking controller. The global coordinate system XY and +the vehicle coordinate system xy are defined as shown in Fig. +1. +Fig. 1. Schematic diagram for lateral dynamics model. +In detail, x is along the vehicle’s longitudinal direction, and +y is along the lateral direction, a fixed coordinate system x′y′ +is defined, where x′ is defined in the direction of the tangential +velocity V and y′ is pointing at the rotation center O of the +vehicle from the position of the mass center c.g of the vehicle; +the yaw angle of the x-direction of the vehicle coordinate +system with respect to the global X-axis direction is denoted +by the ϕ; V is the tangential velocity of the vehicle, and Vx +(called longitudinal velocity) is set as the x-axis component of +V; δf is the front wheel steering angle, ϕdes is the desired yaw +angle of the vehicle, and ∆ϕ is the yaw angle deviation with +respect to the desired path. In addition, the lateral position +error from the vehicle center of mass along the y-direction to + +R +Y +20 +Ay12 +R +12 +sin( +y +V +eff +x +V +des +X3 +the corresponding desired path point is ∆y, ∆y, also known as +the current deviation [33]. Assuming that the desired path has +a constant curvature and the vehicle has a constant speed in the +longitudinal direction. Based on the two-degree-of-freedom +model derived above, the lateral motion control at a constant +speed ¨y = ˙Vy, ˙y = Vy, are expressed as follows: +¨y = −(Cf + Cr +Vxm +) ˙y + (−Vx − Cflf − Crlr +Vxm +) ˙ϕ + Cf +m δf +(1) +¨ϕ = −(Cflf − Crlr +IzVx +) ˙y − ( +Cfl2 +f + Crl2 +r +IzVx +) ˙ϕ + Cflf +Iz +δf +(2) +∆ϕ = ϕdes − ϕ +(3) +˙∆ϕ = ˙ϕdes − ˙ϕ +(4) +˙∆y = − ˙y + Vx∆ϕ +(5) +The preview deviation yL2 is the lateral position deviation +of the vehicle at the effective preview distance Leff from +the vehicle center of mass, as shown in Fig. 1. The preview +deviation yL2 can be considered as the sum of three different +distances, which is expressed as follows: +yL2 = ∆yL2 + ∆y + Leff sin(∆ϕ) +(6) +where ∆y is the current lateral deviation, ∆ϕ is the heading +angle error, and ϕ is the yaw rate. And the only variable yL2 +depends on the change of the future path. The current lateral +position deviation ∆y depends on the current lateral position +of the vehicle, and the third term of Eq. (6) depends on the +heading angle of the path and the current heading angle of +the vehicle. For small angles ∆ϕ, the above equation can be +approximated as: +yL2 ≈ ∆yL2 + ∆y + Leff sin(∆ϕ) +(7) +To calculate the preview deviation precisely, it is necessary +to find the relationship between the variable yL2 and the +other variables. If the vehicle travels with a constant tangential +velocity V on a circular path with radius R, the relationship +with the ideal yaw rate ˙ϕdes is expressed as: +˙ϕdes = V +R +(8) +It can also be seen from Fig. 1 that the relationship between +the radius of R and the effective preview distance Leff is: +sin(2θ) = Leff +R +(9) +Where θ is the angle between the current driving direction +of the vehicle and the point (Leff, ∆y′ +L2) defined in the +coordinate system x′y′. When combining Eq. (8) and Eq. (9) +and θ = α tan( ∆y′ +L2 +Leff ) (as can be seen in Fig. 1), the following +relationship is derived as follows: +˙ϕdes = V sin(2θ) +Leff += V sin(2 · α tan(∆y′ +L2/Leff)) +Leff +≈ 2V ∆y′ +L2 +L2 +eff +(10) +Then +∆y′ +L2 ≈ +L2 +eff ˙ϕdes +2V +(11) +where the distance ∆y′ +L2 is not equal to ∆yL2, since there is +the lateral angle α between the driving direction of the vehicle +body and the movement direction of the actual wheels due to +the side slip. Therefore, it can be approximated as follows: +∆y′ +L2 ≈ ∆yL2 − Leff tan(α) +(12) +As for the small side slip, the lateral angle α can be +expressed as: +α ≈ tan(α) = dy +dx = dy +dt · dt +dx = +˙y +Vx +(13) +In addition, the preview deviation yL2 needs to be compen- +sated for the side slip. The compensated preview deviation is +described as: +∆yL2Slipcomp ≈ ∆yL2 − Leff · +˙y +Vx +≈ ∆yL2 − +˙y +δfVx +Leff · δf +≈ ∆yL2 − β · Leff · δf +(14) +β = Gδf , ˙y(0, Vx) = +˙y +δfVx += +CfVx(−lfmV 2 +x + CrL2 +r + Crlflr) +−CfmV 2 +x lf + CrmV 2 +x lr + CfCrl2 +f + 2CfCrlflr + CfCrl2r +(15) +When the vehicle is in steady state ¨y = 0, ¨ϕ = 0, the gain of +the above equation can be derived based on the two-freedom +model and the expression for the preview deviation considering +the vehicle side slip can be obtained as: +yL2Slipcomp = ∆yL2Slipcomp + ∆y + Leff∆ϕ += +L2 +eff ˙ϕdes +2V +− β · Leff · δf + ∆y + Leff∆ϕ +(16) +The current lateral deviation ∆y, the lateral velocity y, the +yaw angle deviation ∆ϕ, and the yaw rate ϕ are set as the state +variables x; the input variables u are the front wheel angle δf +and the desired yaw rate ϕdes, and then the output variables +are current lateral deviation ∆y, the yaw angle deviation ∆ϕ, +and the preview deviation yL2slipcomp considering vehicle side +slip. Assuming V ≈ Vx. Thus, the states equations can be +described as follows: +d +dt +� +��� +∆y +˙y +∆ϕ +˙ϕ +� +��� += +� +���� +0 +−1 +Vx +0 +0 +− Cf +Cr +Vxm +0 +−Vx − Cf lf −Crlr +Vxm +0 +0 +0 +−1 +0 +− Cf lf −Crlr +IzVx +0 +− +Cf l2 +f +Crl2 +r +IzVx +� +���� +� +��� +∆y +˙y +∆ϕ +˙ϕ +� +��� ++ +� +��� +0 +0 +Cf +m +0 +0 +1 +Cf lf +Iz +0 +� +��� +� +δf +˙ϕdes +� +(17) + +4 +� +��� +∆y +∆ϕ +yL2slipcomp +˙ϕ +� +��� = +� +��� +1 +0 +0 +0 +0 +0 +1 +0 +1 +0 +Leff +0 +0 +0 +0 +1 +� +��� +� +��� +∆y +˙y +∆ϕ +˙ϕ +� +��� ++ +� +��� +0 +0 +0 +0 +−Leffβ +L2 +eff +2Vx +0 +0 +� +��� +� +δf +˙ϕdes +� +(18) +The lateral dynamics model is established and used to +design the tracking controllers, followed by path planning to +provide the tracking input information. +III. PATH PLANNING +This section focuses on the path planning of the FDWEV +in the presence of static obstacles to provide path point infor- +mation for tracking control by avoiding obstacles reasonably. +Because of the varying sizes and shapes of the various barriers, +it is necessary to project the obstacles onto the ground to +create a two-dimensional environment. Then, an external circle +is utilized to simplify the obstacles so that they fit within a +circle. Within the confines of a predetermined park, the vehicle +is initially simplified to drive in the form of a circle, with the +origin being the mass center of the vehicle. The corresponding +diagram is presented in Fig. 2. +Fig. 2. +Diagram of the simplified obstacle avoidance in a complicated +unstructured scenario +A. Path generator +The artificial potential field (APF) method is widely used for +the path generator. However, it currently suffers from the local +optimal solution problem and target unreachability. Based on +it, some scholars have proposed some improved algorithms +[34], [35]. In this paper, modifying the repulsive force function +and repulsive direction have been presented. +(1) Repulsive force function optimization +When the moving vehicle, the target point, and the obstacle +are all in the same line and the target point is close to the +obstacle, the attractive force of the target point to the moving +vehicle decreases while the repulsive force of the obstacle +from the moving vehicle increases. As a result, the moving +vehicle cannot reach the target point since the repulsive force +does not vary as the driving vehicle gets closer to the target +point in this scenario. Therefore, modifying the repulsive force +function decreases the repulsive force as the moving vehicle +gets closer to the target point. In this way, the repulsive and +the attractive force are equal to zero at the target point, and the +vehicle can maintain the target point. The improved repulsive +field function is described as follows: +Urep,j = +� +� +� +1 +2ηj( +1 +dj(x) − 1 +Q∗ +j +)2d2(x, xgoal) +dj(x) ≤ Q∗ +j +0 +dj(x) > Q∗ +j +(19) +where Urep,j denotes the improved repulsive force function, +ηj is the coefficient constant of the repulsive force function. +dj(x) is the shortest distance between the moving vehicle and +the jth obstacle. Q∗ +j is the maximum range of the influence +for jth obstacle on the moving vehicle. When the shortest +distance between the moving vehicle and the obstacle is less +than Q∗ +j, the repulsive force function will affect the moving +vehicle. d(x, xgoal) denotes the shortest distance between the +moving vehicle and the target point. +The corresponding repulsive force function is: +Frep = −grad(Urep,j) = +� +Frep1N1 + Frep2N2 +dj(x) ≤ Q∗ +j +0 +dj(x) > Q∗ +j +(20) +where: +Frep1 = ηj( +1 +dj(x) − 1 +Q∗ +j +)d2(x, xgoal) +d2 +j(x) +(21) +Frep2 = ηj( +1 +dj(x) − 1 +Q∗ +j +)2d(x, xgoal) +(22) +Frep1 and Frep2 are the two splitting forces of Frep, where the +direction of Frep1 is from the center of the obstacle pointing +to the moving vehicle, the direction of Frep2 and the attractive +force are the same. +This improved method incorporates an adjustment factor +d(x, xgoal) so that the repulsive force decreases while the +moving vehicle is approaching the target point. The repulsive +force is zero, and the attractive force is also zero when +reaching the target point. +(2) Repulsive force direction reformation +Altering the repulsive force described above is insufficient +to solve either of the difficulties. As a result, based on the +repulsive force function, modifying the direction of repulsive +force Frep1 has been conducted. Frep1 and Frep2 are the two +splitting forces of the repulsive force, and the direction of +Frep1 is from the obstacle to the moving vehicle, and the +direction of Frep2 is from the moving vehicle to the target +point. When the angle between Frep2 and the attractive force +is larger than 90◦, this may nevertheless result in a locally +optimal solution. Therefore, modifying the direction of the +repulsive force is to define Frep1 as a tangent direction along +the influencing range of the obstacle and the angle between it +and the attractive force, which is less than or equal to 90◦. +B. Speed planning +(1) Vehicle dynamics constraints + +40 +30 +Influence range of obstacle +Simplified obstacle +20 +Irregular obstacle +10 +Start point +Target point +-10 +Circle-Simplified Vehicle +-20 +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +2005 +According to the vehicle dynamics theory analysis, the +under-steering gain can be defined as K = +m +(lf +lr)2 ( lf +Cαr − +lr +Cαf ). Then, steering gain is the ratio of the yaw rate to the +front wheel angle, which is +ϕ +δf = +Vx/(lf +lr) +1+KV 2 +x +, ensuring the +vehicle’s stability and safety. [36]. +� +� +� +˙ϕ +δ = +Vx +lf + lr +Vx = ˙ϕR +(23) +In the vehicle coordinate system, the vehicle lateral accel- +eration can be expressed as: +αy = lim +∆t→0 +∆Vy + Vx · ∆θ +∆t +(24) +To avoid obstacles, the vehicle is steering in such a way +as to satisfy steady-state steering and to make certain that +the lateral side slip angle of the center of mass is as small +as possible. Assuming the velocity of the vehicle along the +longitudinal axis remains constant, the lateral acceleration can +be represented as follows: +αy = +V 2 +x +lf + lr +· δ +(25) +To make the tires in the linear area, the lateral acceleration +should be no more than 0.4 g, and the front wheel needs to +be satisfied as follows: +δ ≤ 0.4g · lf + lr +V 2 +x +(26) +Therefore, the vehicle’s steering would be simultaneously +impacted by the velocity and steering radius. And the front +wheel steering angle should meet smoothly and securely: +δ = min(0.4g · lf + lr +V 2 +x +, lf + lr +Rmin +) +(27) +During steering, the longitudinal speed is far too high to +have any impact on the safety of the vehicle, and the vehicle +speed at any given curve can be satisfied: +Vlim = +� +gµ/ρd +(28) +Eq. 28 the limited speed under ideal conditions, it is required +to adjust the vehicle speed by inserting an adjustment factor +λd by considering the real instantaneous scenario: +Vlim = λd +� +gµ/ρd +(29) +It can be seen that the only factors that influence the +limited speed are the radius of the road, denoted by λd, +and the maximum adhesion coefficient, denoted by µ. Then +the longitudinal reference speed should also consider the +maximum legal speed restriction. Therefore, the following can +be obtained [37] : +Vref = min(Vset, Vlim) +(30) +where Vset is the desired target speed. +(2) Vehicle kinematic constraints +The front wheels need to be in accordance with the ideal +Ackermann geometry steering, and the vehicle kinematic +model can be described as follows. +� +� +� +� +� +� +� +� +� +˙X = V sin θ +˙Y = V cos θ +θ = V tan δf +lf + lr +(31) +The incomplete constraint can be expressed as: +� ˙X = ˙X sin θ − ˙Y cos θ +˙Y = ˙X sin(δf + θ) − ˙Y cos(δf + θ) − (lf + lr)θ cos δf +(32) +where θ is the angle between the vehicle and the X-axis +direction. Based on the above constraints, it is clear that the +kinematic constraints include a speed limit Vlim without side +slip, with different curvatures: +Vlim = +� +µg +� +(1 + (lf + lr)2ρ2 +d) · +� +R2 − (lf + lr)2 +ρd = 1 +R +(33) +The reference speed Vref will be selected as the smallest +speed value. +IV. TRACKING CONTROL +The tracking control algorithms are designed to accurately +follow the path and the corresponding vehicle speed to avoid +static obstacles. Firstly, the fuzzy control is used to obtain +the preview time, combined with speed planning to get the +variable preview distance. Furthermore, the vehicle kinematics +and dynamics models are considered respectively with vehicle +constraints. Thus, a tracking control model based on curvature +calculation (CC-based) and model prediction control (MPC- +based) methods are established. +A. Design of tracking controller based on curvature calcula- +tion +In the lateral dynamics model, the defined preview deviation +yL2 contains three parts, ∆yL2 indicating the future path +change, ∆y representing the current lateral position deviation, +and ∆ϕ indicating the current heading angle deviation. +(1) Curvature calculation based on current lateral deviation +∆y +To calculate and correct the deviation between the current +path and the reference path, PID controllers, as the most +common feedback controllers used in industrial fields, were +first employed due to no representation of the internal system +and the straightforward application to determine the ideal +curvature κ∆y based on the present deviance. The following +is an equation representing the PID controller. +κ∆y(k) = Kp∆y(k)+Ki +k +� +j=0 +∆y(j)+Kd[∆y(k)−∆y(k−1)] +(34) +The Kp is the proportional coefficient, the Ki is the integral +coefficient, and the Kd is the differential coefficient. The I + +6 +controller provides the most effective results by modifying the +three parameters in the repeated simulations. As a result, the +I controller is used to determine the ideal curvature. Then, +the sensor measurement has a certain delay in the real-world +case, which results in a delay in the acquired current lateral +deviation ∆y. So a delayed module esTDelay needs to be +introduced to provide a more accurate control in Fig. 3. +Fig. 3. Block diagram of the current deviation control loop +(2) Curvature calculation based on preview deviation yL2 +For the control loop of the yL2, the data delay is first +considered to get the delayed preview deviation yL2delayed, +and then the derivation from the lateral dynamics model is +given: +˙ϕdes = V sin (2θ) +Leff +≈ 2V ∆y′ +L2 +L2eff +(35) +The preview deviation distance can therefore be converted +into the ideal curvature as shown below: +κyL2 = ˙ϕdes +Vx +≈ 2∆yL2 +L2eff +(36) +The conversion of the preview deviation into the corre- +sponding curvature value by gain 2/L2 +eff is conducted. Then +a parameter Gout can be added between the two distances, the +range of which is set to 0 < Gout < 1 to make the steering +less aggressive. Since the vehicle is subject to side slip, a +compensation amount is considered to calculate the preview +distance deviation to obtain yL2slipcomp, which is: +yL2slipcomp = yL2 − dy +dxLeff +(37) +where +dy +dx = dy +dt +dt +dx = +˙y +δwheel +1 +Vx +δf = Gδwheel, ˙y(0, Vx) +Vx +δsteeringwheel +SteerRatio +(38) +The compensated preview distance deviation is brought +into the curvature calculation module as the actual preview +deviation, which is yL2slipcomp = yL2. The control block +diagram can be obtained as shown in Fig. 4. +Fig. 4. Block diagram of the preview deviation control loop +Finally, the ideal curvature κ∆y based on the current de- +viation and the ideal curvature κyL2 based on the preview +distance are transformed into the corresponding steering angle +in different cases, the conversion between the steering angle +and the ideal curvature is derived as follows: +Gcurvature,δf (0, Vx) = δf +κ = δfVx +˙ϕ += G−1 +δf, ˙Ψ(0, Vx) · Vx += +−CfmV 2 +x lf + CrmV 2 +x lr + CfCrl2 +f + 2CfCrlflr + CfCrl2 +r +CfCrVx(lf + lr) +Vx += +−CfmV 2 +x lf + CrmV 2 +x lr + CfCrl2 +f + 2CfCrlflr + CfCrl2 +r +CfCr(lf + lr) +(39) +The effective preview time can be obtained from the fuzzy +controller and the speed limit obtained from the vehicle +dynamics and kinematics. An effective preview distance can +be obtained by multiplying the speed with the effective pre- +view time. The CC-based tracking controller framework can +therefore be obtained as shown in Fig. 5. Gout is an adaptive +parameter, and LocalOffsetLimit is set to 0.2m, and the +logic pseudo-code for the judgment is: +Algorithm 1 The logic pseudo-code for the judgment +if ALatLimit ≤ 0.4g then +Gout = 0.8 +else +Gout = 0.65 +end if +if |∆y| > LocalOffsetLimit then +DisableLocalOffsetIntegrator = True +else +DisableLocalOffsetIntegrator = False +end if +B. Design of tracking controller based on model predictive +control +(1) Optimization objectives +The aim is not only to require the vehicle to produce the +smallest lateral deviation but also to achieve the smallest +steering wheel angle variation based on the consideration of +the stability and safety [38], [39]. Therefore, the optimization +objective function is defined as follows: +V (k) = +Hp +� +i=1 +∥z(k + i|k) − r(k + i)∥2 +Q(i) ++ +Hp +� +i=1 +∥u(k + i|k)∥2 +Ru(i) + +Hp +� +i=1 +∥∆u(k + i|k)∥2 +Q(i) +(40) +The reference path information is defined as: +R(k) = [r(k + 1|k) · · · r(k + Hp|k)]T +(41) +Normalization is required due to different magnitudes of +three terms for the objective function; then the objective +function can be expressed as: +V (k) = ∥Z(k) − R(k)∥2 +Q + ∥U(k)∥2 +Ru + ∥∆U(k)∥2 +R +(42) +The vehicle lateral tracking deviation is defined as: +ε(k) = R(k) − Hx(k) − Pu(k − 1) +(43) + +Current position +coordinates +(X,Y.9) +Desired position +Ay +Ayaike +sTpeloy +KNY +coordinates +e +KT/ +(X.Y.0)Current position +coordinates +(x,Y.0) +Desired position +coordinates +Yi2 +sTDeday +V12delared +(x.Y.p) +e +JL2 +Effectivepreview distance +Ls7 +Fig. 5. Block diagram of CC-based control strategy design +Bring the lateral tracking deviation into the objective func- +tion as follows: +V (k) = ∥S∆U(k) − ε(k)∥2 +Q + ∥U(k)∥2 +Ru + ∥∆U(k)∥2 +R += +� +∆U(k)T ST − ε(k)T � +Q [S∆U(k) − ε(k)] ++ +� +u(k − 1)T ΓT + ∆u(k)T ΛT � +Ru [Γu(k − 1) + Λ∆u(k)] ++ ∆U(k)T R∆U(K) +(44) +The above equation is simplified as follows: +V (k) = C − ∆U(k)T M + ∆U(k)T N∆U(K) +M = 2ST Qε(k) − 2ΛT RuΓu(k − 1) +N = ΛT RuΛ + R + ST QS +(45) +It is easy to see that M and N are irrelevant to ∆U(k). +Then the dimensions of the input, output, and state variables +in the system are assumed to be l, m, and n; therefore, the +dimensions of the matrix can be obtained as shown in Table +1. +TABLE I +TABLE OF MATRIX DIMENSION +Matrix +Dimension +Γ +lHu × l +Λ +lHu × lHu +Q +mHp × mHp +Ru +lHu × lHu +R +lHu × lHu +H +mHp × n +P +mHp × l +S +mHp × lHu +ε +m × 1 +M +lHu × 1 +N +lHu × lHu +(2) Constraints +To ensure the safety and stability of the vehicle, limitations +on the steering wheel angle, steering wheel angle rate, lateral +acceleration, lateral deviation, and actuator capacity should be +considered when designing the control algorithms. Firstly, the +constraint of the front wheel angle as the model input is set +as −25◦ − 25◦, which is the extreme value of the steering +angle. Thus, the constraints on the front wheel angle and the +corresponding variation are designed as follows: +− 25◦ ≤ δf ≤ 25◦ +(46) +− 0.47◦ ≤ ∆δf ≤ 0.47◦ +(47) +Then the lateral acceleration is αy ≤ µg in the theoretical +case, and this paper sets the limit as αy ≤ 0.85µg for safety. +The yaw rate can also be obtained as follows: +ψ · Vx ≤ 0.85µg +˙ψ ≤ 0.85µg +Vx +(48) +Finally, the side slip angle cannot be exceeded to 5◦, +within which there is a linear relationship between the lateral +force and side slip angle according to the tyre characteristics. +Therefore, the constraint of the side slip angle of the front +wheel is: +− 2.5◦ ≤ α ≤ 2.5◦ +(49) +(3) Optimization with constraints +The above constraints can first be expressed as linear +inequalities as: +E +� +∆U(k) +1 +� +≤ 0 +(50) +F +� +U(k) +1 +� +≤ 0 +(51) +G +� +Z(k) +1 +� +≤ 0 +(52) +where the matrix F is denoted as f = [F1, F2, · · · , FHu, f], +Fi is the matrix of q × m, f is the matrix of q × 1, it can be +described as: +Hu +� +i=1 +Fiu(k + i − 1|k) + f ≤ 0 +(53) +Because +u(k + i − 1|k) = u(k − 1) + +i−1 +� +j=1 +∆u(k + j|k) +(54) + +PID controller +Data +preprocessing +Current position +coordinates +Current +Desire path information +(X,Y,0) +deviation +Desired curvature +Desiredposition +Preview +Comprehensives +Steering +Vehiclemodel +Gain model +coordinates +deviation +control module +angle +in Carsim +udge +between curvature +of intelligentfull +(x.Y.) +and s teering angle +G (0,V) +drive-by-wire +Speed +Fuzzy +electricvehicle +controller +Vehicle current state +Efficientpreview +Desired curvature +time and distance +Vehicle kinematic +Data +model +preprocessing8 +Bring this equation into Eq. (53): +Hu +� +j=1 +Fj∆u(k|k) + +Hu +� +j=2 +Fj∆u(k + 2|k) + · · · ++ FHu∆u(k + Hu − 1|k) + +Hu +� +j=1 +Fju(k − 1) + f ≤ 0 +(55) +where Fi = �Hu +j=1 Fj +F∆U(k) ≤ −F1u(k − 1) − f +(56) +Thus Eq. (51) can be transformed into a linear constraint +on ∆U(k). Similarly, Eq. (52) can be transformed into a +constraint on ∆U(k). +G +� +Hx(k|k) + Pu(k − 1) + S∆U(K) +1 +� +≤ 0 +(57) +where G = [T, g], g is the last column of G, since it is +obtained: +T [Hx(k|k) + Pu(k − 1)] + TS∆U(k) + g ≤ 0 +(58) +Furthermore, it can be described as: +TS∆U(k) ≤ −T [Hx(k|k) + Pu(k − 1)] − g +(59) +Finally, it can be summarized as: +W∆U(k) ≤ w +(60) +Combining with three equations, it can be described as: +� +� +E +TS +W +� +� ∆U(k) ≤ +� +� +−E1u(k − 1) − e +−T [Hx(k|k) + Pu(k − 1)] − g +w +� +� +(61) +Thus, the optimization problem can be expressed as follows: +min +− ∆U(k)T M + ∆U(k)T N∆U(k) +sub +λ∆U(k) ≤ κ +(62) +The optimization solution is obtained by solving the gradi- +ent function for the objective function and making it equal to +0. +∇∆U(k)V = −M + 2N∆U(k) = 0 +(63) +So the optimal solution is obtained as follows: +∆U(k)opt = 1 +2N −1M +(64) +V. SIMULATION AND RESULTS +The Simulink/Carsim/Amesim co-simulation is developed +under the vehicle speed of 30 km/h and 50 km/h, respectively. +Based on the path planning of the tracking control module +in the static environment, the target steering angle and vehicle +speed are obtained, which will be utilized as the inputs for full +drive-by-wire electric vehicles chassis control [6]. Following +the output of the hierarchical chassis control, the ideal torque +of four wheels and four angles will be produced, which +are utilized as the inputs of Carsim. This paper will verify +the path-tracking control strategies under various working +situations. +A. Results of planning and tracking control algorithm based +on curvature calculation +A practicable path within a specific range is designed for the +FDWEV, with various obstacles, the start point, and a target +point to offer correct position information for the tracking +control module. After obtaining the path point information, the +generated points will be interpolated with a three-order spline. +Then the points are processed using a five-order polynomial +fitting for every 10 path points to make them meet the vehicle +driving requirements and generate a reasonable path [40]. +Then simulation settings with vehicle speeds of 30 km/h and +50 km/h are created in a specific scenario with considerable +curvatures to validate the effectiveness of the tracking control +algorithm. +(1) Vehicle speed is 30 km/h +According to the various deviations and curvature inputs, the +output corresponding to the preview time Tp can be generated +using the fuzzy controller, as illustrated in Fig. 6 (a). Based +on the speed limitations for the curvature of the road, in +this paper, we use the safety factor, which is defined as the +adjustment factor λd = 0.65, and the adhesion coefficient +µ = 0.85. As a result, the reference speed curve with the +restriction can be obtained, and the speed tracking results show +good performance by the chassis integrated control strategy, +as shown in Fig. 6 (b). +Fig. 6 (c) depicts the path tracking results at a speed +of 30 km/h working condition, and Fig. 6 (d) depicts the +corresponding lateral deviation. To determine the present +curvature, a delay is introduced with TDelay = 0.065s, and +the PID controller computes a curvature value. By repeated +simulations, Ki = 1.5 is eventually determined, implying that +the I controller is employed to regulate the deviation directly. +A delay with TDelay = 0.065s is also added to determine +the preview deviation, the preview time determined by the +fuzzy controller can then be multiplied by the reference speed +to obtain the variable preview distance and the gain is set +as Gout = 0.8, which would produce a curvature value with +LocalOffsetLimit = 0.2m. As a result, as illustrated in +Fig. 6 (e), the steering wheel angle can be calculated, where +the transmission ratio is chosen as 16. Further, as illustrated +in Fig. 6 (f) with the lateral acceleration, it shows that the +vehicle is mostly in the stable area and the steering curvature +is significant, and near to 0.1, the lateral acceleration reaches +its maximum, increasing the lateral force and allowing it to +negotiate the curve successfully. +(2) Vehicle speed is 50 km/h +Fig. 7 (a) depicts the obtained preview time Tp when the +vehicle speed increases to 50 km/h, where the road curvature +is large, the preview time and distance are both reduced, and +the majority of the preview time is one second. Because the +maximum adhesion given by the tires is regarded as the major +speed restriction and the vehicle’s limit in speed planning, +when the speed is raised, however, the vehicle speed cannot be +instantly lowered if the curvature is larger than 0.1. Therefore, +it is necessary to slow down the vehicle earlier by establishing +a lower speed threshold. The vehicle’s speed planning and +tracking results are illustrated in Fig. 7 (b). The tracking + +9 +(a) +(b) +(c) +(d) +(e) +(f) +Fig. 6. (a) Tp change curve; (b) Speed tracking results; (c) Path tracking results; (d) Lateral deviation results; (e) Corresponding steering wheel angle results; +(f) Lateral acceleration results. The setting with CC-based method under 30 km/h. +path and accompanying lateral deviation are determined by +lowering the vehicle speed immediately before approaching +the curve and allowing the vehicle to finish the steering +operation for varied curvatures, as shown in Figs. 7 (c) and 7 +(d). It creates a greater deviation than 30 km/h since increasing +the speed makes the vehicle more prone to side slip during the +steering, and even if the vehicle is in a stable zone in advance, +by reducing the speed, the actuators will have a delay in taking +action. As a result, although the curvature calculation (CC- +based) method is straightforward to be calculated, it does not +take into account different restrictions and real-time feedback +correction according to the vehicle’s condition. Therefore, it +will generate larger lateral deviations at higher vehicle speeds +when the road curvature is rather considerable. +Comparing the tracking results under different speeds, the +CC-based tracking controller performs better when the vehicle +speed is relatively lower in such a complicated environment. +However, when the vehicle speed increases, it is not enough +to reduce the speed to a relatively low level in advance where +the curvature is significant. This is mostly because this method +does not have real-time feedback correction of the vehicle’s +state and considers different limits. As a result, it is not ideally +suitable for an environment with greater operational difficulty. +On the other hand, this approach is straightforward and does +not require significant computation, so it is more applicable +under typical working conditions. +B. Results of tracking control algorithm based on MPC +The simulation settings of vehicle speed 30 km/h and 50 +km/h have been specified separately to validate the effec- +tiveness of the tracking controller based on the MPC, with +the control goal and the process of continuous feedback +optimization. +(1) Vehicle speed is 30 km/h +Based on the established vehicle model with both lateral and +yaw directions in MPC, the road surface adhesion coefficient +is µ = 0.85, and the MPC controller is written using the +S-function in Simulink with parameters: sample period T = +0.05s, prediction step Np = 25, control step Nc = 10, and +the objective function weights are set as follows: +Q = +� +��� +2000 +0 +0 +0 +0 +1000 +0 +0 +0 +0 +1000 +0 +0 +0 +0 +1000 +� +��� , R = 1.5 × 105 (65) +The slack factor is taken as 1000, the output deviation +constraint is [-0.5, 0.5], and the front wheel steering angle con- +straint is [−20◦, 20◦]. Therefore, the results can be obtained +and presented in Fig. 8. In Fig. 8 (a), the vehicle can maintain +the longitudinal speed; the tracking control precision is within +0.3 m, which has a strong tracking capability in extremely +complicated environments. From the lateral acceleration, the +vehicle is in a rather stable area; only when the curvature is at +its greatest extent will the lateral acceleration reach its greatest +value. This is because there will be a greater requirement to +increase the lateral force. + +400 +300 +I angle (degree) +200 +100 + wheel +-100 +200 +Steering +-300 +-400 +-500 +0 +5 +10 +15 +20 +25 +30 +Time (s)0.4 +0.2 + acceleration +-0.2 +Lateral +-0.4 +-0.6 +1 +5 +10 +15 +20 +25 +30 +0 +Time (s)1.4 +1.3 +1.2 +1.1 +1.0 +P0.9 +0.8 +0.7 +0.6 +0.5 +5 +10 +15 +20 +25 +0 +30 +Time (s)35 +-: Actual Speed +- - Set Speed +30 +Vehicle Speed (km/h) +25 +20 +15 +10 +5 +5 +10 +15 +20 +25 +30 +0 +Time (s)- -Actual Path +:-. Desired Path0.8 +0.6 +0.4 +Lateral deviation (m) +0.2 +0.0 +0.2 +-0.4 +-0.6 +-0.8 +0 +5 +10 +15 +20 +25 +30 +Time (s)10 +(a) +(b) +(c) +(d) +(e) +(f) +Fig. 7. (a) Tp change curve; (b) Speed tracking results; (c) Path tracking results; (d) Lateral deviation results; (e) Corresponding steering wheel angle results; +(f) Lateral acceleration results. The setting with CC-based method under 50 km/h. +(a) +(b) +(c) +(d) +(e) +(f) +Fig. 8. (a) Tp change curve; (b) Speed tracking results; (c) Path tracking results; (d) Lateral deviation results; (e) Corresponding steering wheel angle results; +(f) Lateral acceleration results. The setting with MPC-based method under 30 km/h. + +1.4 +1.3 +1.2 +1.1 +S1.0 +F0.9 +0.8 +0.7 +0.6 +0.5 +1 +5 +10 +15 +20 +25 +30 +0 +Time (s)Actual Speed +. Set Speed +Vehicle Speed (km/h) +Time (s)Actual Path +:- Desired Pathateral deviation +Time (s)Steering wheel angle (degree) +Time (s)Time (s)35 +30 +25 +Vehicle Speed (km/h) +20 +15 +10 +5 +Actual Speed +Desired Speed +10 +15 +20 +25 +5 +30 +Time (s)500 +400 +degree) +300 +200 +100 +0 +-100 +200 +-300 +S +-400 +-500 +0 +5 +10 +15 +20 +25 +30 +Time (s)2 +0 +-2 +-4 +-8 +-10 + - Actual Path +-12 +Desired Path +-14 +20 +40 +0 +60 +80 +100 +120 +140 +160 +X(m)0.5 +0.4 +0.3 +(m) +0.2 +Lateral deviation +0.1 +V +0.0 +-0.2 +-0.3 +-0.4 +-0.5 +0 +5 +10 +15 +20 +25 +30 +Time (s)0.6 +0.4 +IS-2 +I acceleration (g/m +0.2 +0.0 + Lateral +-0.2 +-0.4 +-0.6 +5 +10 +15 +20 +25 +30 +0 +Time (s)0.04 +0.02 +(degree/s) +0.00 +Yaw rate ( +-0.02 +-0.04 +-0.06 +5 +0 +10 +15 +20 +25 +30 +Time (s)11 +(a) +(b) +(c) +(d) +(e) +(f) +Fig. 9. (a) Tp change curve; (b) Speed tracking results; (c) Path tracking results; (d) Lateral deviation results; (e) Corresponding steering wheel angle results; +(f) Lateral acceleration results. The setting with MPC-based method under 50 km/h. +(2) Vehicle speed is 50 km/h +The simulation results are depicted in Fig. 9 following +an increase in the vehicle speed. The vehicle is capable of +maintaining the desired speed. The path control deviation in +this complicated environment can be controlled to within about +0.5 m, and it can be seen from the lateral acceleration and +yaw rate that an increase in vehicle speed does not result in +a decrease in the performance of the vehicle’s stability. Simi- +larly, except for the abrupt bends with the greatest curvature, +the lateral acceleration will reach its maximum to give an +adequate amount of lateral force, but the longitudinal speed +will decrease. Because the combined effect is required to +help the vehicle safely in a more complicated environment, in +which decoupling the longitudinal and lateral motion is tough. +MPC tracking controller can produce satisfactory tracking +results as well, when the speeds are relatively modest, there +is no significant difference in the tracking precision provided +by the two methods. However, in a complicated environment, +when the vehicle speed rises, the curvature is still rather +significant in some locations. MPC with the vehicle dynamics +model can adjust the vehicle’s state and predict the vehicle’s +output for a future period by continuous feedback optimization +correction. Consequently, it also has an improved tracking +capacity at faster speeds. +VI. CONCLUSION +Based on the comprehensive control strategy of FDWEV +chassis, different tracking controllers are designed to realize +the trajectory tracking control of the vehicle in low and +medium-speed working conditions with considerable curva- +tures to verify the presented algorithms. First, the repulsive +force function and the direction are optimized to solve the +unreachability of the target point in the standard APF ap- +proach. Then the corresponding speed planning is designed +for the kinematic and dynamic constraints of the vehicle. +Along with the kinematic and lateral dynamics models, two +tracking controllers based on curvature calculation and MPC +are designed in a complicated unstructured environment to +improve tracking accuracy and stability. Furthermore, co- +simulation in Simulink/Carsim/Amesim was conducted to val- +idate the presented algorithms. Overall, based on the relatively +significant curvature of the generated path, the two tracking +controllers have strong tracking capabilities under low-speed +working conditions with about 0.3 m deviation; when speed +increases, the tracking capabilities can reduce, and the MPC- +based controller with 0.5 m deviations can improve the fol- +lowing ability while maintaining better control compared to +the CC-based controller with 0.6 m deviation. Hence, MPC +can handle the optimization problem with constraints better in +more dynamic unstructured environments. +Future studies will be focused on the real-world comparison +of the two methods and the integration of collision avoidance +control with dynamic environment information by radar and +vision sensing systems. + +55 +50 +45 +1 (km/h) +40 +35 +Vehicle Speed +30 +25 +20 +15 +10 +5 + Actual Speed +- - Desired Speed +0 +10 +15 +20 +25 +0 +Time (s)400 +300 + angle (degree) +200 +100 +0 +neel +-100 +wh +g +-200 +Steering +-300 +400 +-500 +5 +10 +15 +20 +25 +0 +Time (s)2 +0 +-2 +-4 +Y-6 +-8 +-10 +- Actual Path +-12 +Desired Path +-14 +0 +20 +40 +60 +80 +100 +120 +140 +160 +X(m)1.0 +0.8 +0.6 +0.4 +I deviation +0.2 +0.0 +_ateral +-0.2 +-0.4 +-0.6 +-0.8 +-1.0 +1 +0 +5 +10 +15 +20 +25 +Time (s)0.6 +0.4 +27 +I acceleration (g/ms*2 +0.2 +-0.2 +Latera +-0.4 +-0.6 +5 +10 +15 +0 +20 +25 +Time (s)0.04 +0.03 +0.02 +S +0.01 + (degree +0.00 +Yaw rate ( +-0.01 +-0.02 +入 +-0.03 +-0.04 +-0.05 +10 +15 +0 +5 +20 +25 +Time (s)12 +REFERENCES +[1] J. 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Hassanien, “Bezier curve based path +planning in a dynamic field using modified genetic algorithm,” Journal +of Computational Science, vol. 25, pp. 339–350, 2018. + diff --git a/edE0T4oBgHgl3EQf5gLh/content/tmp_files/load_file.txt b/edE0T4oBgHgl3EQf5gLh/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..007792d3fc58cf683e02597473db66618d9d25ad --- /dev/null +++ b/edE0T4oBgHgl3EQf5gLh/content/tmp_files/load_file.txt @@ -0,0 +1,965 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf,len=964 +page_content='1 Planning and Tracking Control of Full Drive-by-Wire Electric Vehicles in Unstructured Scenario Guoying Chen, Min Hua, Wei Liu, Jinhai Wang, Shunhui Song, Changsheng Liu Abstract—Full drive-by-wire electric vehicles (FDWEV) with X-by-wire technology can achieve independent driving, braking, and steering of each wheel, providing a good application platform for autonomous driving technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Path planning and tracking control, in particular, are critical components of autonomous driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' However, It is challenging to comprehensively design an robust control algorithm by integrating vehicle path planning in a complicated unstructured scenario for FDWEV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' To address the above issue, this paper first proposes the artificial potential field (APF) method for path planning in the prescribed park with different static obstacles to generate the reference path information, where speed planning is incorporated considering kinematics and dynamic constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Second, two tracking control methods, curvature calculation (CC-based) and model predictive control (MPC-based) methods with the lateral dynamics model, are proposed to track the desired path under different driving conditions, in which a forward-looking behavior model of the driver with variable preview distance is designed based on fuzzy control theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' CarSim-AMESim-Simulink co-simulation is conducted with the existence of obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' The simulation results show that the proposed two control approaches are effective for many driving scenarios and the MPC-based path-tracking controller enhances dynamic tracking performance and ensures good maneuverability under high-dynamic driving conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Index Terms—Full drive-by-wire electric vehicles, path plan- ning, tracking control, curvature calculation, model predictive control I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' INTRODUCTION V EHICLE driving safety has always been an important research field for vehicle manufacturing [1]–[4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' With the increasingly intelligent, electrified, wire-controlled, and net- worked vehicles [5], the intelligent full drive-by-wire electric vehicle possesses more advantages than traditional vehicles, such as independent driving, braking, and steering control [6], [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' In addition, each wheel in the steering system can achieve positive and negative 180-degree rotation around the z-axis Guoying Chen is with the State Key Laboratory of Automotive Simulation and Control, Jilin University, Changchun, 130022, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' (e-mail: cgy- 011@163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='com), Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Min Hua is with the School of Engineering, University of Birmingham, Birmingham, B15 2TT, UK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' (e-mail: mxh623@student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='bham.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='uk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Wei Liu is with the School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907, USA (e-mail: liu3044@purdue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='edu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Jinhai Wang is with the School of Automotive Engineering, Wuhan University of Technology, Wuhan, 430070, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' China (e-mail: wangjin- hai@whut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Shunhui Song is with the School of Automotive Studies, Tongji University, Shanghai, 218074, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' China (e-mail: songshunhui@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='com).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Changsheng Liu is with the College of Computer Science and Technol- ogy, Zhejiang University, Hangzhou, 310027, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' (e-mail: chang- shengliu23@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='com).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' of the tire for various steering modes, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=', crab, traverse, diagonal travel, and steer in place [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Controllable degrees of freedom can improve the mobility and flexibility of the vehicle at low speeds to achieve automatic parking and other functions in low-speed and narrow spaces, as well as the stability and safety of the vehicle when driving at high speeds [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Furthermore, due to the application of X-by-wire technology, four-wheel motor torque, and other parameters can be obtained in real time, which will provide accurate information for the advanced dynamics control system and the intelligent chassis integration control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' With the introduction of the additional sensing signals and position information, the lane change, overtaking and car-following, and a series of autonomous operations will be achieved [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Autonomous driving(AD) integrates the information from the surroundings and the vehicle’s states via advanced sensing systems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=', LIDAR, cameras, GPS, and IMU) [11]–[15], and communication networks [16] to replace or assist the driver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' In the last decade, with the development and advances in sensors and computer technologies, AD-related research have become hot topics in both path planning and autonomous tracking control, and have been conducted to ensure vehicle safety, comfort, and energy efficiency [17], [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Many studies have been dedicated to solving the collision- free path planning problem to meet the constraints of col- lision avoidance, steering speed, and road curvature under continuous conditions [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' State lattice is a graph theory- based approach that has emerged in recent years, where kinematic and dynamical constraints can be regarded through state space rules and repeatable sampling in autonomous driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Furthermore, unstructured road environments give more possibilities for employing this method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Lattice edges are computed offline to achieve real-time online planning using look-up tables to reduce the computational burden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' However, several problems exist for urban areas with highly variable structured environments, such as the discretization of heading angles that may lead to oscillations between two oriented samples [20], [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' A spline-based search tree utilizes a search tree to generate a path that is tangent to all objects and assumes that the optimal path is oriented straight ahead or touches at least one object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Each path segment from one object to the next is defined by a constant acceleration, which is effective in certain scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' However, transient lateral acceleration and sudden changes make this approach infeasible for vehicle robust control [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Alternatively, model predictive control (MPC) [23] has been widely applied in the field arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='02753v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='SY] 7 Jan 2023 2 of automotive, where the optimization goal is to find the optimal value of a series of curvature changes to minimize the steering effort defined by the road curvature in a collision scenario, along a prediction domain from a fixed sampling time [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' A potential field-based path planner is presented to provide obstacle avoidance with an adaptive multi-speed scheduler using a fuzzy system [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' In [26], a time-horizon based MPC method is proposed to reduce the calculation load of vehicle speed predictive control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' And the performance is verified through real road simulation experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Nowadays, the vehicle models adopted for autonomous tracking control are divided into three types: geometric, kinematic, and dynamic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' The mainstream control algorithms include PID control, sliding mode control (SMC), neural network control, and MPC [27] to obtain the corresponding control parameters: the steering wheel angle, throttle opening, and braking strength [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' There is a certain delay in the actual vehicle control when executing the control command immediately, which is the biggest problem of PID algorithm in autonomous control [29];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' in addition, advanced control algorithms, such as SMC and neural networks, are also widely employed [30], [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' However, these algorithms greatly depend on parameter sensitivities and environment information, and different vehicle constraints need to be considered when the vehicle model is embedded;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' there are great limitations for these optimization algorithms with constraints, although MPC is capable of solving the constraints problem with high com- putation cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Therefore, various tracking control algorithms need to be traded off in all aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' The main challenge with current tracking control algorithms is a reasonable simplifica- tion of the vehicle dynamics modeling to improve the real- time performance of the algorithm while ensuring tracking accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Unfortunately, most algorithms mainly attempt to find collision-free paths with no guarantee of feasibility in the real world due to poor quality and small turns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' To mitigate the above limitations in the harsh condition, in this paper, a com- plicated path with considerable curvatures will be created, and the precise and independent control of four-wheel angles and motor torque can be fully utilized to achieve obstacle-avoiding tracking control based on the multi-degree-of-freedom control platform of the FDWEV [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' The autonomous tracking con- trol of the FDWEV can provide a theoretical and practical basis for developing AD technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' The main contribution of this paper is fourfold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' 1) Artificial potential field (APF) for path planning through comprehensive comparison is optimized to address the poten- tial unreachable and local optimal problems with the improved repulsive force function and the direction of repulsive force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' 2) For the high maneuverability of the FDWEV, the relevant kinematics and dynamic constraints are considered to be inte- grated into the vehicle speed planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Then interpolation and piecewise fitting are conducted to provide achievable position information for tracking control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' 3) Based on fuzzy control theory, a forward-looking driver behavior model with variable preview distance is designed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' The curvature calculation is proposed based on the vehicle lateral dynamics and kinematic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' For the shortcoming of the tracking control algorithm based on curvature calculation, the MPC algorithm is designed by considering the vehicle constraints, real-time prediction, and feedback optimization of the vehicle states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' 4) Based on the path planning and vehicle speed planning in the prescribed park, the autonomous tracking controls are ver- ified based on the co-simulation of Simulink/Carsim/Amesim platform under different conditions with speeds of 30 km/h and 50 km/h in a harsh environment with considerable curvatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' The rest of this paper is organized as follows: Section 2 formulates the lateral dynamics model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' In section 3, path planning with APF method is proposed with the generated reachable points of vehicle tracking control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' In Section 4, tracking control based on the curvature method and MPC algorithm are proposed according to the kinematic and dy- namics model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Section 5 conducts simulation experiments and analyzes the results for validation and evaluation, followed by the conclusions in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' LATERAL DYNAMICS MODEL The lateral dynamics model is established firstly to design the tracking controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' The global coordinate system XY and the vehicle coordinate system xy are defined as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Schematic diagram for lateral dynamics model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' In detail, x is along the vehicle’s longitudinal direction, and y is along the lateral direction, a fixed coordinate system x′y′ is defined, where x′ is defined in the direction of the tangential velocity V and y′ is pointing at the rotation center O of the vehicle from the position of the mass center c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='g of the vehicle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' the yaw angle of the x-direction of the vehicle coordinate system with respect to the global X-axis direction is denoted by the ϕ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' V is the tangential velocity of the vehicle, and Vx (called longitudinal velocity) is set as the x-axis component of V;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' δf is the front wheel steering angle, ϕdes is the desired yaw angle of the vehicle, and ∆ϕ is the yaw angle deviation with respect to the desired path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' In addition, the lateral position error from the vehicle center of mass along the y-direction to R Y 20 Ay12 R 12 sin( y V eff x V des X3 the corresponding desired path point is ∆y, ∆y, also known as the current deviation [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Assuming that the desired path has a constant curvature and the vehicle has a constant speed in the longitudinal direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Based on the two-degree-of-freedom model derived above,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' the lateral motion control at a constant speed ¨y = ˙Vy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' ˙y = Vy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' are expressed as follows: ¨y = −(Cf + Cr Vxm ) ˙y + (−Vx − Cflf − Crlr Vxm ) ˙ϕ + Cf m δf (1) ¨ϕ = −(Cflf − Crlr IzVx ) ˙y − ( Cfl2 f + Crl2 r IzVx ) ˙ϕ + Cflf Iz δf (2) ∆ϕ = ϕdes − ϕ (3) ˙∆ϕ = ˙ϕdes − ˙ϕ (4) ˙∆y = − ˙y + Vx∆ϕ (5) The preview deviation yL2 is the lateral position deviation of the vehicle at the effective preview distance Leff from the vehicle center of mass,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' The preview deviation yL2 can be considered as the sum of three different distances, which is expressed as follows: yL2 = ∆yL2 + ∆y + Leff sin(∆ϕ) (6) where ∆y is the current lateral deviation, ∆ϕ is the heading angle error, and ϕ is the yaw rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' And the only variable yL2 depends on the change of the future path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' The current lateral position deviation ∆y depends on the current lateral position of the vehicle, and the third term of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' (6) depends on the heading angle of the path and the current heading angle of the vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' For small angles ∆ϕ, the above equation can be approximated as: yL2 ≈ ∆yL2 + ∆y + Leff sin(∆ϕ) (7) To calculate the preview deviation precisely, it is necessary to find the relationship between the variable yL2 and the other variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' If the vehicle travels with a constant tangential velocity V on a circular path with radius R, the relationship with the ideal yaw rate ˙ϕdes is expressed as: ˙ϕdes = V R (8) It can also be seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' 1 that the relationship between the radius of R and the effective preview distance Leff is: sin(2θ) = Leff R (9) Where θ is the angle between the current driving direction of the vehicle and the point (Leff, ∆y′ L2) defined in the coordinate system x′y′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' When combining Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' (8) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' (9) and θ = α tan( ∆y′ L2 Leff ) (as can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' 1), the following relationship is derived as follows: ˙ϕdes = V sin(2θ) Leff = V sin(2 · α tan(∆y′ L2/Leff)) Leff ≈ 2V ∆y′ L2 L2 eff (10) Then ∆y′ L2 ≈ L2 eff ˙ϕdes 2V (11) where the distance ∆y′ L2 is not equal to ∆yL2, since there is the lateral angle α between the driving direction of the vehicle body and the movement direction of the actual wheels due to the side slip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Therefore, it can be approximated as follows: ∆y′ L2 ≈ ∆yL2 − Leff tan(α) (12) As for the small side slip, the lateral angle α can be expressed as: α ≈ tan(α) = dy dx = dy dt · dt dx = ˙y Vx (13) In addition, the preview deviation yL2 needs to be compen- sated for the side slip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' The compensated preview deviation is described as: ∆yL2Slipcomp ≈ ∆yL2 − Leff · ˙y Vx ≈ ∆yL2 − ˙y δfVx Leff · δf ≈ ∆yL2 − β · Leff · δf (14) β = Gδf ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' ˙y(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Vx) = ˙y δfVx = CfVx(−lfmV 2 x + CrL2 r + Crlflr) −CfmV 2 x lf + CrmV 2 x lr + CfCrl2 f + 2CfCrlflr + CfCrl2r (15) When the vehicle is in steady state ¨y = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' ¨ϕ = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' the gain of the above equation can be derived based on the two-freedom model and the expression for the preview deviation considering the vehicle side slip can be obtained as: yL2Slipcomp = ∆yL2Slipcomp + ∆y + Leff∆ϕ = L2 eff ˙ϕdes 2V − β · Leff · δf + ∆y + Leff∆ϕ (16) The current lateral deviation ∆y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' the lateral velocity y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' the yaw angle deviation ∆ϕ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' and the yaw rate ϕ are set as the state variables x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' the input variables u are the front wheel angle δf and the desired yaw rate ϕdes, and then the output variables are current lateral deviation ∆y, the yaw angle deviation ∆ϕ, and the preview deviation yL2slipcomp considering vehicle side slip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Assuming V ≈ Vx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' the states equations can be ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='described as follows: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='dt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='∆y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='˙y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='∆ϕ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='˙ϕ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='Vx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='− Cf +Cr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='Vxm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='−Vx − Cf lf −Crlr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='Vxm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='− Cf lf −Crlr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='IzVx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='Cf l2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='f +Crl2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='IzVx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='∆y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='˙y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='∆ϕ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='˙ϕ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='Cf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='Cf lf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='Iz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='δf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='˙ϕdes ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='˙ϕdes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='(18) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='The lateral dynamics model is established and used to ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='design the tracking controllers,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' followed by path planning to provide the tracking input information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' PATH PLANNING This section focuses on the path planning of the FDWEV in the presence of static obstacles to provide path point infor- mation for tracking control by avoiding obstacles reasonably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Because of the varying sizes and shapes of the various barriers, it is necessary to project the obstacles onto the ground to create a two-dimensional environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Then, an external circle is utilized to simplify the obstacles so that they fit within a circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Within the confines of a predetermined park, the vehicle is initially simplified to drive in the form of a circle, with the origin being the mass center of the vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' The corresponding diagram is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Diagram of the simplified obstacle avoidance in a complicated unstructured scenario A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Path generator The artificial potential field (APF) method is widely used for the path generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' However, it currently suffers from the local optimal solution problem and target unreachability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Based on it, some scholars have proposed some improved algorithms [34], [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' In this paper, modifying the repulsive force function and repulsive direction have been presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' (1) Repulsive force function optimization When the moving vehicle, the target point, and the obstacle are all in the same line and the target point is close to the obstacle, the attractive force of the target point to the moving vehicle decreases while the repulsive force of the obstacle from the moving vehicle increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' As a result, the moving vehicle cannot reach the target point since the repulsive force does not vary as the driving vehicle gets closer to the target point in this scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Therefore, modifying the repulsive force function decreases the repulsive force as the moving vehicle gets closer to the target point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' In this way, the repulsive and the attractive force are equal to zero at the target point, and the vehicle can maintain the target point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' The improved repulsive field function is described as follows: Urep,j = � � � 1 2ηj( 1 dj(x) − 1 Q∗ j )2d2(x, xgoal) dj(x) ≤ Q∗ j 0 dj(x) > Q∗ j (19) where Urep,j denotes the improved repulsive force function, ηj is the coefficient constant of the repulsive force function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' dj(x) is the shortest distance between the moving vehicle and the jth obstacle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Q∗ j is the maximum range of the influence for jth obstacle on the moving vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' When the shortest distance between the moving vehicle and the obstacle is less than Q∗ j, the repulsive force function will affect the moving vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' d(x, xgoal) denotes the shortest distance between the moving vehicle and the target point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' The corresponding repulsive force function is: Frep = −grad(Urep,j) = � Frep1N1 + Frep2N2 dj(x) ≤ Q∗ j 0 dj(x) > Q∗ j (20) where: Frep1 = ηj( 1 dj(x) − 1 Q∗ j )d2(x, xgoal) d2 j(x) (21) Frep2 = ηj( 1 dj(x) − 1 Q∗ j )2d(x, xgoal) (22) Frep1 and Frep2 are the two splitting forces of Frep, where the direction of Frep1 is from the center of the obstacle pointing to the moving vehicle, the direction of Frep2 and the attractive force are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' This improved method incorporates an adjustment factor d(x, xgoal) so that the repulsive force decreases while the moving vehicle is approaching the target point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' The repulsive force is zero, and the attractive force is also zero when reaching the target point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' (2) Repulsive force direction reformation Altering the repulsive force described above is insufficient to solve either of the difficulties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' As a result, based on the repulsive force function, modifying the direction of repulsive force Frep1 has been conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Frep1 and Frep2 are the two splitting forces of the repulsive force, and the direction of Frep1 is from the obstacle to the moving vehicle, and the direction of Frep2 is from the moving vehicle to the target point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' When the angle between Frep2 and the attractive force is larger than 90◦, this may nevertheless result in a locally optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Therefore, modifying the direction of the repulsive force is to define Frep1 as a tangent direction along the influencing range of the obstacle and the angle between it and the attractive force, which is less than or equal to 90◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Speed planning (1) Vehicle dynamics constraints 40 30 Influence range of obstacle Simplified obstacle 20 Irregular obstacle 10 Start point Target point 10 Circle-Simplified Vehicle 20 0 20 40 60 80 100 120 140 160 180 2005 According to the vehicle dynamics theory analysis, the under-steering gain can be defined as K = m (lf +lr)2 ( lf Cαr − lr Cαf ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Then, steering gain is the ratio of the yaw rate to the front wheel angle, which is ϕ δf = Vx/(lf +lr) 1+KV 2 x , ensuring the vehicle’s stability and safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' � � � ˙ϕ δ = Vx lf + lr Vx = ˙ϕR (23) In the vehicle coordinate system, the vehicle lateral accel- eration can be expressed as: αy = lim ∆t→0 ∆Vy + Vx · ∆θ ∆t (24) To avoid obstacles, the vehicle is steering in such a way as to satisfy steady-state steering and to make certain that the lateral side slip angle of the center of mass is as small as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Assuming the velocity of the vehicle along the longitudinal axis remains constant, the lateral acceleration can be represented as follows: αy = V 2 x lf + lr δ (25) To make the tires in the linear area, the lateral acceleration should be no more than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='4 g, and the front wheel needs to be satisfied as follows: δ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='4g · lf + lr V 2 x (26) Therefore, the vehicle’s steering would be simultaneously impacted by the velocity and steering radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' And the front wheel steering angle should meet smoothly and securely: δ = min(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='4g · lf + lr V 2 x , lf + lr Rmin ) (27) During steering, the longitudinal speed is far too high to have any impact on the safety of the vehicle, and the vehicle speed at any given curve can be satisfied: Vlim = � gµ/ρd (28) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' 28 the limited speed under ideal conditions, it is required to adjust the vehicle speed by inserting an adjustment factor λd by considering the real instantaneous scenario: Vlim = λd � gµ/ρd (29) It can be seen that the only factors that influence the limited speed are the radius of the road, denoted by λd, and the maximum adhesion coefficient, denoted by µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Then the longitudinal reference speed should also consider the maximum legal speed restriction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Therefore, the following can be obtained [37] : Vref = min(Vset, Vlim) (30) where Vset is the desired target speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' (2) Vehicle kinematic constraints The front wheels need to be in accordance with the ideal Ackermann geometry steering, and the vehicle kinematic model can be described as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' � � � � � � � � � ˙X = V sin θ ˙Y = V cos θ θ = V tan δf lf + lr (31) The incomplete constraint can be expressed as: � ˙X = ˙X sin θ − ˙Y cos θ ˙Y = ˙X sin(δf + θ) − ˙Y cos(δf + θ) − (lf + lr)θ cos δf (32) where θ is the angle between the vehicle and the X-axis direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Based on the above constraints, it is clear that the kinematic constraints include a speed limit Vlim without side slip, with different curvatures: Vlim = � µg � (1 + (lf + lr)2ρ2 d) · � R2 − (lf + lr)2 ρd = 1 R (33) The reference speed Vref will be selected as the smallest speed value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' TRACKING CONTROL The tracking control algorithms are designed to accurately follow the path and the corresponding vehicle speed to avoid static obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Firstly, the fuzzy control is used to obtain the preview time, combined with speed planning to get the variable preview distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Furthermore, the vehicle kinematics and dynamics models are considered respectively with vehicle constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Thus, a tracking control model based on curvature calculation (CC-based) and model prediction control (MPC- based) methods are established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Design of tracking controller based on curvature calcula- tion In the lateral dynamics model, the defined preview deviation yL2 contains three parts, ∆yL2 indicating the future path change, ∆y representing the current lateral position deviation, and ∆ϕ indicating the current heading angle deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' (1) Curvature calculation based on current lateral deviation ∆y To calculate and correct the deviation between the current path and the reference path, PID controllers, as the most common feedback controllers used in industrial fields, were first employed due to no representation of the internal system and the straightforward application to determine the ideal curvature κ∆y based on the present deviance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' The following is an equation representing the PID controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' κ∆y(k) = Kp∆y(k)+Ki k � j=0 ∆y(j)+Kd[∆y(k)−∆y(k−1)] (34) The Kp is the proportional coefficient, the Ki is the integral coefficient, and the Kd is the differential coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' The I 6 controller provides the most effective results by modifying the three parameters in the repeated simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' As a result, the I controller is used to determine the ideal curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Then, the sensor measurement has a certain delay in the real-world case, which results in a delay in the acquired current lateral deviation ∆y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' So a delayed module esTDelay needs to be introduced to provide a more accurate control in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Block diagram of the current deviation control loop (2) Curvature calculation based on preview deviation yL2 For the control loop of the yL2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' the data delay is first considered to get the delayed preview deviation yL2delayed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' and then the derivation from the lateral dynamics model is given: ˙ϕdes = V sin (2θ) Leff ≈ 2V ∆y′ L2 L2eff (35) The preview deviation distance can therefore be converted into the ideal curvature as shown below: κyL2 = ˙ϕdes Vx ≈ 2∆yL2 L2eff (36) The conversion of the preview deviation into the corre- sponding curvature value by gain 2/L2 eff is conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Then a parameter Gout can be added between the two distances, the range of which is set to 0 < Gout < 1 to make the steering less aggressive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Since the vehicle is subject to side slip, a compensation amount is considered to calculate the preview distance deviation to obtain yL2slipcomp, which is: yL2slipcomp = yL2 − dy dxLeff (37) where dy dx = dy dt dt dx = ˙y δwheel 1 Vx δf = Gδwheel, ˙y(0, Vx) Vx δsteeringwheel SteerRatio (38) The compensated preview distance deviation is brought into the curvature calculation module as the actual preview deviation, which is yL2slipcomp = yL2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' The control block diagram can be obtained as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Block diagram of the preview deviation control loop Finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' the ideal curvature κ∆y based on the current de- viation and the ideal curvature κyL2 based on the preview distance are transformed into the corresponding steering angle in different cases,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' the conversion between the steering angle and the ideal curvature is derived as follows: Gcurvature,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='δf (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Vx) = δf κ = δfVx ˙ϕ = G−1 δf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' ˙Ψ(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Vx) · Vx = −CfmV 2 x lf + CrmV 2 x lr + CfCrl2 f + 2CfCrlflr + CfCrl2 r CfCrVx(lf + lr) Vx = −CfmV 2 x lf + CrmV 2 x lr + CfCrl2 f + 2CfCrlflr + CfCrl2 r CfCr(lf + lr) (39) The effective preview time can be obtained from the fuzzy controller and the speed limit obtained from the vehicle dynamics and kinematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' An effective preview distance can be obtained by multiplying the speed with the effective pre- view time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' The CC-based tracking controller framework can therefore be obtained as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Gout is an adaptive parameter, and LocalOffsetLimit is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='2m, and the logic pseudo-code for the judgment is: Algorithm 1 The logic pseudo-code for the judgment if ALatLimit ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='4g then Gout = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='8 else Gout = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='65 end if if |∆y| > LocalOffsetLimit then DisableLocalOffsetIntegrator = True else DisableLocalOffsetIntegrator = False end if B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Design of tracking controller based on model predictive control (1) Optimization objectives The aim is not only to require the vehicle to produce the smallest lateral deviation but also to achieve the smallest steering wheel angle variation based on the consideration of the stability and safety [38], [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Therefore, the optimization objective function is defined as follows: V (k) = Hp � i=1 ∥z(k + i|k) − r(k + i)∥2 Q(i) + Hp � i=1 ∥u(k + i|k)∥2 Ru(i) + Hp � i=1 ∥∆u(k + i|k)∥2 Q(i) (40) The reference path information is defined as: R(k) = [r(k + 1|k) · · · r(k + Hp|k)]T (41) Normalization is required due to different magnitudes of three terms for the objective function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' then the objective function can be expressed as: V (k) = ∥Z(k) − R(k)∥2 Q + ∥U(k)∥2 Ru + ∥∆U(k)∥2 R (42) The vehicle lateral tracking deviation is defined as: ε(k) = R(k) − Hx(k) − Pu(k − 1) (43) Current position coordinates (X,Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='9) Desired position Ay Ayaike sTpeloy KNY coordinates e KT/ (X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='0)Current position coordinates (x,Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='0) Desired position coordinates Yi2 sTDeday V12delared (x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='p) e JL2 Effectivepreview distance Ls7 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Block diagram of CC-based control strategy design ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='Bring the lateral tracking deviation into the objective func- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='tion as follows: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='V (k) = ∥S∆U(k) − ε(k)∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='Q + ∥U(k)∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='Ru + ∥∆U(k)∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='∆U(k)T ST − ε(k)T � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='Q [S∆U(k) − ε(k)] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='u(k − 1)T ΓT + ∆u(k)T ΛT � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='Ru [Γu(k − 1) + Λ∆u(k)] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='+ ∆U(k)T R∆U(K) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='(44) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='The above equation is simplified as follows: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='V (k) = C − ∆U(k)T M + ∆U(k)T N∆U(K) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='M = 2ST Qε(k) − 2ΛT RuΓu(k − 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='N = ΛT RuΛ + R + ST QS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='(45) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='It is easy to see that M and N are irrelevant to ∆U(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Then the dimensions of the input, output, and state variables in the system are assumed to be l, m, and n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' therefore, the dimensions of the matrix can be obtained as shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' TABLE I TABLE OF MATRIX DIMENSION Matrix Dimension Γ lHu × l Λ lHu × lHu Q mHp × mHp Ru lHu × lHu R lHu × lHu H mHp × n P mHp × l S mHp × lHu ε m × 1 M lHu × 1 N lHu × lHu (2) Constraints To ensure the safety and stability of the vehicle, limitations on the steering wheel angle, steering wheel angle rate, lateral acceleration, lateral deviation, and actuator capacity should be considered when designing the control algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Firstly, the constraint of the front wheel angle as the model input is set as −25◦ − 25◦, which is the extreme value of the steering angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Thus, the constraints on the front wheel angle and the corresponding variation are designed as follows: − 25◦ ≤ δf ≤ 25◦ (46) − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='47◦ ≤ ∆δf ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='47◦ (47) Then the lateral acceleration is αy ≤ µg in the theoretical case, and this paper sets the limit as αy ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='85µg for safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' The yaw rate can also be obtained as follows: ψ · Vx ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='85µg ˙ψ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='85µg Vx (48) Finally, the side slip angle cannot be exceeded to 5◦, within which there is a linear relationship between the lateral force and side slip angle according to the tyre characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Therefore, the constraint of the side slip angle of the front wheel is: − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='5◦ ≤ α ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='5◦ (49) (3) Optimization with constraints The above constraints can first be expressed as linear inequalities as: E � ∆U(k) 1 � ≤ 0 (50) F � U(k) 1 � ≤ 0 (51) G � Z(k) 1 � ≤ 0 (52) where the matrix F is denoted as f = [F1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' F2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' FHu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' f],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Fi is the matrix of q × m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' f is the matrix of q × 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' it can be described as: Hu � i=1 Fiu(k + i − 1|k) + f ≤ 0 (53) Because u(k + i − 1|k) = u(k − 1) + i−1 � j=1 ∆u(k + j|k) (54) PID controller Data preprocessing Current position coordinates Current Desire path information (X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='0) deviation Desired curvature Desiredposition Preview Comprehensives Steering Vehiclemodel Gain model coordinates deviation control module angle in Carsim udge between curvature of intelligentfull (x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=') and s teering angle G (0,V) drive-by-wire Speed Fuzzy electricvehicle controller Vehicle current state Efficientpreview Desired curvature time and distance Vehicle kinematic Data model preprocessing8 Bring this equation into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' (53): Hu � j=1 Fj∆u(k|k) + Hu � j=2 Fj∆u(k + 2|k) + · · · + FHu∆u(k + Hu − 1|k) + Hu � j=1 Fju(k − 1) + f ≤ 0 (55) where Fi = �Hu j=1 Fj F∆U(k) ≤ −F1u(k − 1) − f (56) Thus Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' (51) can be transformed into a linear constraint on ∆U(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Similarly, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' (52) can be transformed into a constraint on ∆U(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' G � Hx(k|k) + Pu(k − 1) + S∆U(K) 1 � ≤ 0 (57) where G = [T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' g],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' g is the last column of G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' since it is obtained: T [Hx(k|k) + Pu(k − 1)] + TS∆U(k) + g ≤ 0 (58) Furthermore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' it can be described as: TS∆U(k) ≤ −T [Hx(k|k) + Pu(k − 1)] − g (59) Finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' it can be summarized as: W∆U(k) ≤ w (60) Combining with three equations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' it can be described as: � � E TS W � � ∆U(k) ≤ � � −E1u(k − 1) − e −T [Hx(k|k) + Pu(k − 1)] − g w � � (61) Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' the optimization problem can be expressed as follows: min − ∆U(k)T M + ∆U(k)T N∆U(k) sub λ∆U(k) ≤ κ (62) The optimization solution is obtained by solving the gradi- ent function for the objective function and making it equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' ∇∆U(k)V = −M + 2N∆U(k) = 0 (63) So the optimal solution is obtained as follows: ∆U(k)opt = 1 2N −1M (64) V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' SIMULATION AND RESULTS The Simulink/Carsim/Amesim co-simulation is developed under the vehicle speed of 30 km/h and 50 km/h, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Based on the path planning of the tracking control module in the static environment, the target steering angle and vehicle speed are obtained, which will be utilized as the inputs for full drive-by-wire electric vehicles chassis control [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Following the output of the hierarchical chassis control, the ideal torque of four wheels and four angles will be produced, which are utilized as the inputs of Carsim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' This paper will verify the path-tracking control strategies under various working situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Results of planning and tracking control algorithm based on curvature calculation A practicable path within a specific range is designed for the FDWEV, with various obstacles, the start point, and a target point to offer correct position information for the tracking control module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' After obtaining the path point information, the generated points will be interpolated with a three-order spline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Then the points are processed using a five-order polynomial fitting for every 10 path points to make them meet the vehicle driving requirements and generate a reasonable path [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Then simulation settings with vehicle speeds of 30 km/h and 50 km/h are created in a specific scenario with considerable curvatures to validate the effectiveness of the tracking control algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' (1) Vehicle speed is 30 km/h According to the various deviations and curvature inputs, the output corresponding to the preview time Tp can be generated using the fuzzy controller, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' 6 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Based on the speed limitations for the curvature of the road, in this paper, we use the safety factor, which is defined as the adjustment factor λd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='65, and the adhesion coefficient µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' As a result, the reference speed curve with the restriction can be obtained, and the speed tracking results show good performance by the chassis integrated control strategy, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' 6 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' 6 (c) depicts the path tracking results at a speed of 30 km/h working condition, and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' 6 (d) depicts the corresponding lateral deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' To determine the present curvature, a delay is introduced with TDelay = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='065s, and the PID controller computes a curvature value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' By repeated simulations, Ki = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='5 is eventually determined, implying that the I controller is employed to regulate the deviation directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' A delay with TDelay = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='065s is also added to determine the preview deviation, the preview time determined by the fuzzy controller can then be multiplied by the reference speed to obtain the variable preview distance and the gain is set as Gout = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='8, which would produce a curvature value with LocalOffsetLimit = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' As a result, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' 6 (e), the steering wheel angle can be calculated, where the transmission ratio is chosen as 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Further, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' 6 (f) with the lateral acceleration, it shows that the vehicle is mostly in the stable area and the steering curvature is significant, and near to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='1, the lateral acceleration reaches its maximum, increasing the lateral force and allowing it to negotiate the curve successfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' (2) Vehicle speed is 50 km/h Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' 7 (a) depicts the obtained preview time Tp when the vehicle speed increases to 50 km/h, where the road curvature is large, the preview time and distance are both reduced, and the majority of the preview time is one second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Because the maximum adhesion given by the tires is regarded as the major speed restriction and the vehicle’s limit in speed planning, when the speed is raised, however, the vehicle speed cannot be instantly lowered if the curvature is larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Therefore, it is necessary to slow down the vehicle earlier by establishing a lower speed threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' The vehicle’s speed planning and tracking results are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' 7 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' The tracking 9 (a) (b) (c) (d) (e) (f) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' (a) Tp change curve;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' (b) Speed tracking results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' (c) Path tracking results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' (d) Lateral deviation results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' (e) Corresponding steering wheel angle results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' (f) Lateral acceleration results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' The setting with CC-based method under 30 km/h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' path and accompanying lateral deviation are determined by lowering the vehicle speed immediately before approaching the curve and allowing the vehicle to finish the steering operation for varied curvatures, as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' 7 (c) and 7 (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' It creates a greater deviation than 30 km/h since increasing the speed makes the vehicle more prone to side slip during the steering, and even if the vehicle is in a stable zone in advance, by reducing the speed, the actuators will have a delay in taking action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' As a result, although the curvature calculation (CC- based) method is straightforward to be calculated, it does not take into account different restrictions and real-time feedback correction according to the vehicle’s condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Therefore, it will generate larger lateral deviations at higher vehicle speeds when the road curvature is rather considerable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Comparing the tracking results under different speeds, the CC-based tracking controller performs better when the vehicle speed is relatively lower in such a complicated environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' However, when the vehicle speed increases, it is not enough to reduce the speed to a relatively low level in advance where the curvature is significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' This is mostly because this method does not have real-time feedback correction of the vehicle’s state and considers different limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' As a result, it is not ideally suitable for an environment with greater operational difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' On the other hand, this approach is straightforward and does not require significant computation, so it is more applicable under typical working conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Results of tracking control algorithm based on MPC The simulation settings of vehicle speed 30 km/h and 50 km/h have been specified separately to validate the effec- tiveness of the tracking controller based on the MPC, with the control goal and the process of continuous feedback optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' (1) Vehicle speed is 30 km/h Based on the established vehicle model with both lateral and yaw directions in MPC, the road surface adhesion coefficient is µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='85, and the MPC controller is written using the S-function in Simulink with parameters: sample period T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='05s, prediction step Np = 25, control step Nc = 10, and the objective function weights are set as follows: Q = � ��� 2000 0 0 0 0 1000 0 0 0 0 1000 0 0 0 0 1000 � ��� , R = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='5 × 105 (65) The slack factor is taken as 1000, the output deviation constraint is [-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='5], and the front wheel steering angle con- straint is [−20◦, 20◦].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Therefore, the results can be obtained and presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' 8 (a), the vehicle can maintain the longitudinal speed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' the tracking control precision is within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='3 m, which has a strong tracking capability in extremely complicated environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' From the lateral acceleration, the vehicle is in a rather stable area;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' only when the curvature is at its greatest extent will the lateral acceleration reach its greatest value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' This is because there will be a greater requirement to increase the lateral force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' 400 300 I angle (degree) 200 100 wheel 100 200 Steering 300 400 500 0 5 10 15 20 25 30 Time (s)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='2 acceleration 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='2 Lateral 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='6 1 5 10 15 20 25 30 0 Time (s)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='0 P0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='5 5 10 15 20 25 0 30 Time (s)35 : Actual Speed - Set Speed 30 Vehicle Speed (km/h) 25 20 15 10 5 5 10 15 20 25 30 0 Time (s)- -Actual Path :-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Desired Path0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='4 Lateral deviation (m) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='8 0 5 10 15 20 25 30 Time (s)10 (a) (b) (c) (d) (e) (f) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' (a) Tp change curve;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' (b) Speed tracking results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' (c) Path tracking results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' (d) Lateral deviation results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' (e) Corresponding steering wheel angle results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' (f) Lateral acceleration results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' The setting with CC-based method under 50 km/h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' (a) (b) (c) (d) (e) (f) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' (a) Tp change curve;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' (b) Speed tracking results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' (c) Path tracking results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' (d) Lateral deviation results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' (e) Corresponding steering wheel angle results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' (f) Lateral acceleration results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' The setting with MPC-based method under 30 km/h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='1 S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='0 F0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='5 1 5 10 15 20 25 30 0 Time (s)Actual Speed .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Set Speed Vehicle Speed (km/h) Time (s)Actual Path :- Desired Pathateral deviation Time (s)Steering wheel angle (degree) Time (s)Time (s)35 30 25 Vehicle Speed (km/h) 20 15 10 5 Actual Speed Desired Speed 10 15 20 25 5 30 Time (s)500 400 degree) 300 200 100 0 100 200 300 S 400 500 0 5 10 15 20 25 30 Time (s)2 0 2 4 8 10 Actual Path 12 Desired Path 14 20 40 0 60 80 100 120 140 160 X(m)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='3 (m) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='2 Lateral deviation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='1 V 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='5 0 5 10 15 20 25 30 Time (s)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='4 IS-2 I acceleration (g/m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='0 Lateral 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='6 5 10 15 20 25 30 0 Time (s)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='02 (degree/s) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='00 Yaw rate ( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='06 5 0 10 15 20 25 30 Time (s)11 (a) (b) (c) (d) (e) (f) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' (a) Tp change curve;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' (b) Speed tracking results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' (c) Path tracking results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' (d) Lateral deviation results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' (e) Corresponding steering wheel angle results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' (f) Lateral acceleration results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' The setting with MPC-based method under 50 km/h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' (2) Vehicle speed is 50 km/h The simulation results are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' 9 following an increase in the vehicle speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' The vehicle is capable of maintaining the desired speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' The path control deviation in this complicated environment can be controlled to within about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='5 m, and it can be seen from the lateral acceleration and yaw rate that an increase in vehicle speed does not result in a decrease in the performance of the vehicle’s stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Simi- larly, except for the abrupt bends with the greatest curvature, the lateral acceleration will reach its maximum to give an adequate amount of lateral force, but the longitudinal speed will decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Because the combined effect is required to help the vehicle safely in a more complicated environment, in which decoupling the longitudinal and lateral motion is tough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' MPC tracking controller can produce satisfactory tracking results as well, when the speeds are relatively modest, there is no significant difference in the tracking precision provided by the two methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' However, in a complicated environment, when the vehicle speed rises, the curvature is still rather significant in some locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' MPC with the vehicle dynamics model can adjust the vehicle’s state and predict the vehicle’s output for a future period by continuous feedback optimization correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Consequently, it also has an improved tracking capacity at faster speeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' CONCLUSION Based on the comprehensive control strategy of FDWEV chassis, different tracking controllers are designed to realize the trajectory tracking control of the vehicle in low and medium-speed working conditions with considerable curva- tures to verify the presented algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' First, the repulsive force function and the direction are optimized to solve the unreachability of the target point in the standard APF ap- proach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Then the corresponding speed planning is designed for the kinematic and dynamic constraints of the vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Along with the kinematic and lateral dynamics models, two tracking controllers based on curvature calculation and MPC are designed in a complicated unstructured environment to improve tracking accuracy and stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Furthermore, co- simulation in Simulink/Carsim/Amesim was conducted to val- idate the presented algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Overall, based on the relatively significant curvature of the generated path, the two tracking controllers have strong tracking capabilities under low-speed working conditions with about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='3 m deviation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' when speed increases, the tracking capabilities can reduce, and the MPC- based controller with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='5 m deviations can improve the fol- lowing ability while maintaining better control compared to the CC-based controller with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='6 m deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Hence, MPC can handle the optimization problem with constraints better in more dynamic unstructured environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' Future studies will be focused on the real-world comparison of the two methods and the integration of collision avoidance control with dynamic environment information by radar and vision sensing systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content=' 55 50 45 1 (km/h) 40 35 Vehicle Speed 30 25 20 15 10 5 Actual Speed - Desired Speed 0 10 15 20 25 0 Time (s)400 300 angle (degree) 200 100 0 neel 100 wh g 200 Steering 300 400 500 5 10 15 20 25 0 Time (s)2 0 2 4 Y-6 8 10 Actual Path 12 Desired Path 14 0 20 40 60 80 100 120 140 160 X(m)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='4 I deviation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='0 _ateral 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='0 1 0 5 10 15 20 25 Time (s)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='4 27 I acceleration (g/ms*2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='2 Latera 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='6 5 10 15 0 20 25 Time (s)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='02 S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='01 (degree 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='00 Yaw rate ( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='02 入 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE0T4oBgHgl3EQf5gLh/content/2301.02753v1.pdf'} +page_content='04 0.' metadata={'source': 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0000000000000000000000000000000000000000..aae15537ab02498a2caafddcce6b298ba44b2878 --- /dev/null +++ b/f9FJT4oBgHgl3EQfUSzI/content/tmp_files/2301.11508v1.pdf.txt @@ -0,0 +1,1398 @@ +Theme-driven Keyphrase Extraction from Social Media on Opioid Recovery +William Romano*1, Omar Sharif*1, Madhusudan Basak1,2, Joseph Gatto1, Sarah Preum1 +1Department of Computer Science, Dartmouth College, USA +2Department of Computer Science and Engineering, BUET, Dhaka, Bangladesh +{william.j.romano.gr,omar.sharif.gr,madhusudan.basak.gr,joseph.m.gatto.gr,sarah.masud.preum}@dartmouth.edu +Abstract +An emerging trend on social media platforms is their use as +safe spaces for peer support. Particularly in healthcare, where +many medical conditions contain harsh stigmas, social me- +dia has become a stigma-free way to engage in dialogues +regarding symptoms, treatments, and personal experiences. +Many existing works have employed NLP algorithms to fa- +cilitate quantitative analysis of health trends. Notably absent +from existing works are keyphrase extraction (KE) models +for social health posts—a task crucial to discovering emerg- +ing public health trends. This paper presents a novel, theme- +driven KE dataset, SuboxoPhrase, and a qualitative annota- +tion scheme with an overarching goal of extracting targeted +clinically-relevant keyphrases. To the best of our knowledge, +this is the first study to design a KE schema for social media +healthcare texts. To demonstrate the value of this approach, +this study analyzes Reddit posts regarding medications for +opioid use disorder, a paramount health concern worldwide. +Additionally, we benchmark ten off-the-shelf KE models on +our new dataset, demonstrating the unique extraction chal- +lenges in modeling user-generated health texts. The proposed +theme-driven KE approach lays the foundation of future work +on efficient, large-scale analysis of social health texts, allow- +ing researchers to surface useful public health trends, pat- +terns, and knowledge gaps. +Introduction +Social media platforms like Twitter, Facebook, and Red- +dit gather spontaneous, self-reported lived experiences from +thousands of individuals with a diverse array of medical and +socio-demographic conditions. People seeking or undergo- +ing treatment often resort to social media for informational +and emotional support (Chen, Wang, and others 2021). Find- +ings in public health research reports increased reliance on +social media and other online health platforms for address- +ing health information needs (Neely, Eldredge, and Sanders +2021; Bhandari, Shi, and Jung 2014). Thus social media +has become an exceptional source of clinically relevant data +to understand population-level concerns, knowledge gaps, +treatment perceptions, and barriers (Chen, Wang, and others +2021). In addition, social media platforms like Reddit sup- +port anonymity and thus encourage rich engagement among +*Both authors contributed equally to the paper. +Copyright © 2023, Association for the Advancement of Artificial +Intelligence (www.aaai.org). All rights reserved. +peers on stigmatized topics like mental health and substance +use recovery (Naslund et al. 2016). Qualitative analysis of +social media data has been used to understand various health +issues, including COVID-19 (Sleigh et al. 2021), cancer +(Levonian et al. 2020), depression, and other mental health +conditions (Lachmar et al. 2017). +Contemporary Natural Language Processing (NLP) algo- +rithms have enabled large-scale, quantitative analysis of so- +cial media platforms. For example, advancements in text +classification have produced works in suicide risk detection +(Mathur, Sawhney, and Shah 2020), detection of adverse +drug reactions (Aroyehun and Gelbukh 2019), and misin- +formation classification (Dharawat et al. 2022). Advance- +ments in information extraction have led to models which +can automatically extract medical entities from social me- +dia texts (ˇS´cepanovi´c et al. 2021; Santosh et al. 2020). Other +works have conducted a data-driven analysis of social me- +dia health (SMH) texts in the context of Topic Modeling +(El-Bassel et al. 2021) and Medical Named Entity Recog- +nition (MedNER) (Batbaatar and Ryu 2019). Notably ab- +sent from the literature on social media for health analysis is +keyphrase extraction (KE), the task of extracting the most +salient terms from a given text. We note it is essential to dis- +tinguish KE from topic modeling and MedNER, as the latter +two do not focus on extracting key terms. To illustrate, con- +sider a social media user who posts: +Am I the only one taking subs that feels nervus all of +the time? I know anxiety is a symptom of other opioid +recovery drugs like methadone but I don’t take those. +The keyphrases in this post are subs1, nervus, and anxiety. A +Topic Model, defined over a collection of documents and not +designed for analysis of an individual text, would likely ex- +tract none of these keywords if they were not topics of other +texts in the corpus. A MedNER model may extract all of the +drug/symptom mentions, but it would be unable to distin- +guish that methadone is not a keyword in this case, rather +is only mentioned as context. Finally, both Topic Models +and MedNER models may struggle with misspellings and +shorthand, such as nervus and subs, respectively. By using +KE for SMH texts, we employ a method that operates at the +1shorthand for Suboxone, a prescription medication used to +treat opioid use disorder (OUD) +arXiv:2301.11508v1 [cs.CL] 27 Jan 2023 + +post-level, can dynamically extract different lexical varia- +tions of a concept (e.g., misspellings, slang, or other collo- +quial forms), and ensure the information extracted is mean- +ingful and relevant to the target user. Applying KE to SMH +texts poses a unique challenge as the notion of “keyphrase” +is historically difficult to define and subject to annotator bi- +ases and end-goal/downstream applications (i.e., how to uti- +lize the extracted keyphrases). Thus existing works on KE +are not well-suited for modeling long texts comprised of col- +loquial, user-generated health narratives with a wide variety +in length, ambiguity, content, and style. +To address this problem, this study defines a novel theme- +driven KE annotation strategy for SMH texts for identi- +fying keyphrases relevant for clinical researchers to sur- +face population-level knowledge gaps, treatment percep- +tions, and experience. KE for SMH analysis is vital since +standard information extraction methods are limited by rely- +ing on narrowly defined entity types. Conversely, keyphrases +can cover a wide variety of entities given that they are key or +significant to the post. This is of value to researchers who +wish to explore social media discourse 1) with the knowl- +edge that the returned terms are meaningful to the original +poster, and 2) with the ability to discover valuable public +health trends and patterns unconstrained to pre-defined en- +tity classes. To our knowledge, no existing models enable a +broad-scale, theme-driven keyphrase analysis of SMH texts. +To illustrate the value of KE for SMH, we analyze Red- +dit posts regarding Medications for Opioid Use Disorder +(MOUD). MOUD analysis is of great importance as the opi- +oid crisis has become one of the most pressing health con- +cerns in the United States. In 2020 alone, 68,630 people +died due to an opioid-related overdose (CDC 2022a). Ac- +cording to the 2021 PEW survey on substance use preven- +tion and treatment, opioid overdose, misuse, and dependence +result in 35 billion dollars in healthcare costs, 14.8 billion +dollars in criminal justice costs, and 92 billion dollars in +lost productivity annually (Florence, Luo, and Rice 2021; +Spencer, Mini˜no, and Warner 2022). Medications for opioid +use disorder (MOUD), such as buprenorphine, methadone, +and naltrexone, are the most effective, evidence-based treat- +ment for OUD (Wakeman et al. 2020; Yarborough et al. +2016). Given the prevalence of opioids, patients suffering +from OUD often experience multi-level stigma, including +inter-and intra-personal, social, provider-level, and policy- +level stigma (Cheetham et al. 2022). Thus, MOUD forums +are often utilized by those in recovery. By analyzing online +MOUD self-reports, researchers can gather valuable, clini- +cally relevant insights from large-scale self-disclosed patient +data on their lived experiences, psychophysical effects, and +treatment concerns, identifying knowledge gaps. +In this work, we study the subreddit r/Suboxone, the most +populous Reddit forum among those dedicated to discussing +buprenorphine-based drugs. Reddit is well suited for such +analysis as it is anonymous, publicly available data with +theme-specific forums ready for exploration. To facilitate +KE on SMH, this work provides the following contributions: +1. We define a systematic qualitative KE annotation/coding +scheme for SMH texts relevant to opioid recovery with +guidance from our multi-disciplinary domain experts on +substance use research and clinical practice. We develop a +dataset, SuboxoPhrase, with human-annotated KE sam- +ples from the r/Suboxone subreddit. To the best of our +knowledge, this is the first KE dataset of SMH texts and +will be made publicly available upon the acceptance of +the paper. +2. We demonstrate the ability of keyphrases to draw clinical +insights on MOUD-based recovery from a relevant sub- +reddit through extensive quantitative and qualitative anal- +ysis of keyphrases. Specifically, our research questions +explore (i) drawing clinical insights from keyphrase fre- +quency and keyphrase co-occurrences in the labeled data, +(ii) analyzing the effect of the COVID-19 pandemic on the +keyphrases, and (iii) the association of keyphrases with +peer engagement—identifying terms which stimulate on- +line discussions among peers. +3. We benchmark ten existing Unsupervised KE (UKE) +models to demonstrate the limitations of exiting UKEs +on SMH data. Our results serve as references for future +works on KE for SMH and motivate the need for SMH- +specific approaches to KE. +Related Work +Keyword and Keyphrase Extraction +KE is a widely explored research problem (Nomoto 2022; +Hasan and Ng 2014), where the goal is to extract salient +phrases that best summarize the document. The extracted +set of keyphrases can vary significantly based on the task +one is focusing on (Firoozeh et al. 2020). Existing KE ap- +proaches generally work in two steps. 1) Select candidate +keyphrases by heuristic rules or manual annotation (Wang, +Zhao, and Huang 2016). 2) Apply supervised (Lopez and +Romary 2010) or unsupervised (Gu et al. 2021) approaches +to rank keyphrases as per their relationship to the document +and return the top-k keyphrases. Popular KE techniques can +be characterized as either statistical (Campos et al. 2020), +graph-based (Mihalcea and Tarau 2004), or embedding- +based (Bennani-Smires et al. 2018) methods. However, with +the emergence of pre-trained language models, recent KE +works have seen a paradigm shift, as contextual embedding- +based approaches have become the primary building block +of recent KE models (Zhang et al. 2022). +Most previous studies have performed KE on scientific lit- +erature, where author-assigned keyphrases are leveraged for +ground truth (Augenstein et al. 2017). In contrast, this work +focuses on extracting keyphrases from social media data +which, compared to scientific literature, is challenging to in- +terpret, contains domain-specific vocabulary, shorthand, and +slang, and is often not amenable to existing NLP resources. +We note some existing works on KE for social media, such +as the work from Zhang et al. (2016), where keyphrase labels +are inferred from Twitter hashtags. In contrast, this study fo- +cuses on SMH texts from Reddit posts where all samples are +human-annotated. + +Social Media Discourse Analysis for Substance Use +and Opioid Use Disorder (OUD) +Recently, social media has emerged as a place for people +to share their experiences, provide support, and engage in +discussions with others who have a similar interest in sub- +stance use, addiction, and recovery (Chancellor, Mitra, and +De Choudhury 2016). Social media discourse on substance +use refers to posts or discussions on drugs and other re- +lated topics such as addiction, harm reduction, treatment +options, and recovery in social media platforms (Lavertu, +Hamamsy, and Altman 2021). Example works on this topic +include Chen, Johnny, and Conway (2022), who analyzed +Reddit discussions about cannabis, alcohol, and opioids to +examine the nature of stigma related to these substances. +MacLean et al. (2015) demonstrated a positive correlation +between forum use and recovery by analyzing online discus- +sions. Chancellor et al. (2019) attempted to uncover poten- +tial alternative treatment options for OUD by investigating +posts from opioid recovery subreddits. Unlike Chancellor et +al., our study differs in the NLP task, domain (we focus on +a subreddit specific to a MOUD, namely Suboxone), as well +as scope (we explore a broader range of categories than treat- +ment options, including psychophysical effects, medical his- +tory, and substance dependency & recovery). +Peer Engagement Modeling +Quantifying peer engagement on social media often depends +on the platforms, and researchers use various indicators to +assess engagement (Sharma et al. 2020). Low et al. (2021) +analyzed the effect of engagement in the r/SuicideWatch +subreddit based on the number of comments received. They +argued that receiving more comments increases a user’s +likelihood of posting and engaging in the future. Previous +works also studied other engagement indicators, including +but not limited to the number of posts, the number of com- +ments/responses, the time between responses, and the num- +ber of links shared (De Choudhury, Counts, and Horvitz +2013). In this work, we utilize the number of comments and +the number of upvotes a post received to measure engage- +ment as these (i) are readily available through the Reddit +API and (ii) capture peer engagement in a tangible way. +Dataset +Data Source +We collected data from Reddit, an internet forum with over +2.8 million communities catering to various topics, interests, +and social groups called “subreddits”. Users may join these +subreddits to engage with other users in anonymous discus- +sions. Since Reddit is an anonymous platform, the only pub- +lic information available for a given user is their username, +join date, and activity (i.e., post and comment history). The +subreddit from which we collected the data used in this study +is r/Suboxone, a community with over 25 thousand users as +of December 2022. Created in 2011, r/Suboxone is a com- +munity forum for dialogue between users who share their +relationship with the opioid recovery medication Suboxone. +This subreddit is strictly moderated, and any posts about +buying/selling illicit drugs, exposing other users’ personal +information (also known as doxxing), or bullying/abuse are +removed. Analyzing activity on r/Suboxone makes it possi- +ble to achieve a unique and authentic perspective on a user’s +opioid recovery journey. While this subreddit explicitly fo- +cuses on the treatment of OUD using Suboxone, users fre- +quently discuss their experiences and concerns related to +other treatment options. This is because individuals with +OUD may try many treatment options to support their re- +covery. Discussions extend to various related topics and co- +occurring substance and medication usage. +Data Collection +All posts between the 2nd of January, 2018, and the 6th of +August, 2022 on the r/Suboxone subreddit were scraped us- +ing the PRAW and PushShift APIs (Boe 2022; Baumgartner +2022). We select this timeline as it allows us to (i) focus +on more recent user data, and (ii) collect a dataset with a +comparable amount of posts before and after the onset of +the COVID-19 pandemic. Also, we observed very limited +interaction in this subreddit before 2018. After filtering the +corpus for irrelevant posts (e.g., those containing polls, links +with no texts, or posts which had been deleted), we devel- +oped a corpus of 15,253 posts. Sample posts are presented +in Table 2. +Sample Selection +We chose 1,000 sample posts for keyphrase annotation, a +set of comparable size to other notable works in KE (Au- +genstein et al. 2017). Given that one of the goals of this +work is to explore the effect of the COVID-19 pandemic +on opioid recovery discussions (i.e., RQ1), the sub-sample +of 1,000 posts was chosen to equally represents activity +before and after the COVID-19 pandemic (with respect to +the date COVID-19 was declared a pandemic by the World +Health Organization (Domenico and Vanelli 2020)), consist- +ing of 500 posts from each half. The title and body of each +post were combined and annotated to capture the post’s full +context. This is important because many posts contain the +most insightful details in their titles. Larger social media +datasets for KE also exist (Zhang et al. 2016), but those +datasets leverage automated approaches like using Twitter +hashtags as keyphrases. However, our theme-driven defini- +tion of keyphrases requires rigorous manual annotation and +can lead to data-driven knowledge discovery. Thus we limit +our sample size for manual annotation to 1,000 posts. +Schema & Guidelines Design Process +Determining a phrase to be key to a given text is subjec- +tive, given annotator biases and target downstream applica- +tions. In this study, we aim to extract keyphrases relevant +to Suboxone-assisted OUD recovery. We collaborate with +a team of five domain experts to define the themes of key- +ness for medication-based OUD recovery. All of our collab- +orators are well-versed in substance use disorder. Together, +they cover a wide variety of expertise, including psychi- +atry, biomedical data science, digital technology for sub- +stance use and mental health, epidemiology, public health +policy, addiction medicine, and addiction psychiatry. Two + +of our collaborators are clinicians and administer MOUD +treatment. All of them reviewed our samples and helped us +to contextualize different aspects of opioid recovery and the +shared lived experiences of the affected individuals. Based +on guidance from our collaborators, we identify four main +themes: Treatment Options, Substance Dependency & Re- +covery, Medical History, and Psychophysical Effects. The +definitions of these four themes are presented below, along +with an extra category Others intended to capture additional +keyphrases that do not belong to any of these categories but +are still key to the original post. +• Treatment Options: This category covers keyphrases re- +lated to different treatment options used for recovery. +Such as medications used to treat OUD (e.g., Buprenor- +phine, Methadone, or their formulations), psycho-therapy, +behavioral counseling, or other medications used to cope +with withdrawal or other psychophysical effects (e.g., us- +ing melatonin to help with insomnia while in recovery). +We consider prescribed medications, over-the-counter +medications, herbal supplements, and other therapeutic +options as potential candidates for keyphrases. +• Substance Dependency & Recovery: This category cov- +ers keyphrases related to the history of substance use (e.g., +fentanyl), co-occurring substance use (e.g., tobacco, al- +cohol), and critical factors in recovery (e.g., recovery, re- +lapse). We consider both prescribed and self-administered +substances. +• Medical History: Keyphrases concerning medical his- +tory, including diagnosis and self-diagnosis of any phys- +ical and mental health conditions, relevant medical pro- +cedures (e.g., major surgery), or critical family medical +history. This category covers medical history beyond sub- +stance dependency/recovery. +• Psychophysical Effects: Keyphrases regarding any phys- +ical or psychological effects and symptoms associated +with OUD recovery, e.g., psychological effects relevant +to withdrawal, precipitated withdrawal, and side effects +of medications. +• Others: All keyphrases that do not belong to any of the +above four categories are assigned to this category. +Category +Fre- +quency +Example Keyphrases +Treatment Options +182 +Adderall, antidepressant, +naloxone +Substance Dependency +& Recovery +77 +cocaine, fentanyl, heroin +Medical History +35 +COVID, osteoarthritis, +PTSD +Psychophysical Effects +331 +aches, constipation, panic +attack +Others +256 +scam, travel, boyfriend +Table 1: Examples from each keyphrase category. Frequency +denotes the number of occurrences in the category in the +whole SuboxoPhrase dataset. +Example keyphrases of each category from SuboxoPhrase +are exhibited in Table 1. To meet our goal of extracting +keyphrases relevant to opioid recovery, we thoroughly it- +erated over our definition of a keyphrase in SuboxoPhrase +via multiple rounds of trial annotations and discussions be- +tween annotators. Keyphrases were annotated to satisfy the +general criteria for “keyness” inspired by Firoozeh et al. +(2020), primarily “conformity”, “homogeneity”, and “uni- +vocity”. Conformity, for our purposes here, was reflected +by capturing domain-specific terminology. Homogeneity ap- +plies to normalizing the diverse vocabulary, slang, and mis- +spellings associated with informal discussions online. Uni- +vocity and generalizability are two crucial and opposing cri- +teria we set for our annotation guidelines. Univocity refers to +keyphrases that are specific and non-ambiguous. This prop- +erty is achievable by increasing the number of terms in a +keyphrase—e.g., a bigram is less ambiguous than a unigram. +This principle is directly opposed to generalizability in that +the more specific a keyphrase is, the fewer text documents +it will apply to. Annotators were encouraged to balance the +trade-offs between these two constraints to produce high- +quality keyphrases. Note that our process is distinguished +from named entity recognition and traditional KE in that our +goal was to extract keyphrases aligned with specific themes +grounded in substance use research while maintaining the +representativeness and generalizability of the keyphrases. +Data Annotation and Quality Check +The complete data annotation was manually carried out by +four graduate students who are both regular social media +users and active in NLP research. They studied/used opioid +recovery subreddits, so they possessed better domain knowl- +edge than mTurk annotators. Moreover, they were trained on +the annotation task and provided background on MOUD and +suboxone through multiple sessions led by experts. We used +LightTag, an online platform, to label the keyphrases (Light- +Tag 2023). Two annotators annotated each sample to gener- +ate high-quality keyphrases. Experts resolved any confusion +during annotation through discussion. +Since the annotation involved multiple annotators, we cal- +culated the inter-annotator agreement score to ensure the +dataset’s quality. We used two lists of normalized keyphrases +for each sample from the annotators. We use the Jaccard +index to measure the agreement/similarity between annota- +tions (Sarwar, Noor, and Miah 2022). Jaccard index is de- +fined as: +JI = +len(A ∩ B) +len(A) + len(B) − len(A ∪ B) +(1) +Let A and B be the respective set of keyphrases from an- +notators 1 and 2 for a given sample i in SuboxoPhrase. JI +computes the Jaccard index for sample i and represents the +annotator agreement for that sample. While calculating the +intersection and union of the two sets, we considered the ex- +act string match between the elements of the sets as used in +Schopf, Klimek, and Matthes (2022). We used Avg.(JI) to +capture the average Jaccard index for the whole dataset of n +samples. The average Jaccard similarity index obtained us- +ing the exact string match approach was 61.36%. The score + +indicates a substantial similarity between keyphrases ex- +tracted by multiple annotators. Sample posts with extracted +keyphrases and JI-score are presented in Table 2. +Normalization +Reddit users commonly use various unique phrasings of the +same word, e.g., shorthand, slang, and misspellings. For ex- +ample, the opiate heroin may also be referred to as h, dope, +smack, and speedball. For this study to draw general con- +clusions about MOUD forum discussions, we must be able +to normalize all keyphrases to a common parent term. We +manually map all keyphrase variations to their most mean- +ingful representative parent phrases to solve this problem. +In our analysis, we thus ensure that different slang terms for +drugs like suboxone (sub, zub, bupe) or oxycodone (contin, +oxy, perc) are mapped to a common term. Additionally, we +normalize terms from all non-treatment categories with ex- +amples such as muscle pain (tore a muscle, muscle aches, +muscle tension), sobriety (sober), heart rate (high heart +rate, low heart rate), and memory issues (memory problems, +memory loss, short term memory problems). The steps and +guidelines we followed to normalize keyphrases are illus- +trated in this section. +Identification of “Other” keyphrases +Some keyphrases +may be important to the post but were not in line with our +research questions. Singleton verbs, adjectives, or modifiers +all fall into this category. Examples include, adjust, expect, +acute, accidentally, expensive, etc. We used manual inspec- +tion to identify such keyphrases. These keyphrases were +added to the “Other” category. +Mapping unigram keyphrases with common stems +We +additionally perform stemming of keyphrases. All unigrams +in our dataset that shared a common stem pattern were +mapped to that single variation. For example, craving and +cravings were generated from the stem crave, whereas detox +is the stem for both detoxing and detoxification. +Mapping semantically similar keyphrases +An essential +normalization step was to map all keyphrases with the same +meaning. The shorthand of keyphrases (pwd for precipi- +tated withdrawal, dr for doctor) and misspelled keyphrases +(depressssssed instead of depressed) fall into this category. +Since slang and misspellings can arise in numerous varia- +tions, we perform the semantic mapping manually. Included +in this category are keyphrases consisting of more than one +word. Some constituent words can dominate this type of +keyphrase (e.g., online clinic is dominated by clinic). Also, +they may possess a different meaning than each of their con- +stituent words (e.g., precipitated withdrawal has a differ- +ent meaning than precipitated and withdrawal). Thus proper +mapping of such keyphrases was identified after careful +manual inspection. Finally, keyphrases with similar mean- +ings (e.g., vomiting and puking) were also mapped in this +step using human judgment. It is worth mentioning that we +tried to select the generic keyphrases as the representative +ones while mapping. For instance, the specific drug clon- +azepam was assigned to the generic group benzodiazepines +whereas the specialist sub-category of doctor endocrinolo- +gist was mapped to the more general group doctor. How- +ever, we did not map some of the specific keyphrases (e.g., +suboxone, sublocade) to their generic group due to their fre- +quent occurrences in the posts. Multiple annotators reviewed +all the manual tasks to avoid the potential subjective bias +of manual inspections. Each conflicting case was resolved +based on majority voting. Finally, an expert reviewed the +whole normalization output list and confirmed the validity of +the process. Each normalized keyphrase is assigned to pre- +defined categories after a manual review of the keyphrase’s +context in a sample of posts. +Methods +RQ1: What clinical insights can we draw from +theme-driven opioid recovery-related keyphrases? +This question aims to uncover potential clinical insights +from self-reports on r/Suboxone. Specifically, we explore +two questions in our annotated data: +1. What insights can we draw from keyphrase occurrences +and co-occurrences? +We aim to provide a quantitative analysis of keyphrases +in the SuboxoPhrase dataset. First, we report the most +frequently occurring keyphrases in opioid recovery dis- +cussions, providing analysis on why specific keyphrases +are more prevalent. Then, we discuss the most frequent +keyphrase categories in SuboxoPhrase. Category analysis +allows us to draw high-level conclusions on discussion +themes in r/Suboxone. +We also examine the frequency with which specific +keyphrases co-occur. Notably, we are interested in draw- +ing clinical insights from co-occurrences between treat- +ment options and psychophysical effects (e.g., suboxone, +nausea) to probe the discussion issues about various treat- +ment options at scale. This analysis could help uncover +unknown side effects and reveal new treatment options +unbeknownst to opioid recovery researchers. +2. How did the onset of the COVID-19 pandemic effect the +discussion on opioid recovery on Reddit? +To analyze this question, we investigate how keyphrase +frequency changes over time—specifically concerning the +onset of the pandemic, using the time-stamped data. This +information is useful to substance use clinicians and re- +searchers who want to assess the effects of the pandemic +on MOUD treatment. +RQ2: How does peer engagement associate with +keyphrases in opioid recovery-related Reddit +discourse? +In recent years, Reddit has become a place for sharing re- +covery experiences from various drug dependency issues +and providing support to peers (Chancellor et al. 2019; +MacLean et al. 2015). The success of any peer discussion +group heavily depends on the engagement between its mem- +bers (Sharma et al. 2020). Unfortunately, very few works +in NLP explore how users interact or seek support in opi- +oid recovery discourse on Reddit. This work aims to fill this +knowledge gap by characterizing a user’s experience with + +Title +Post +Annotation-1 +Annotation-2 +JI +Getting on +suboxone +I just came from Florida and have been clean from dope for 9 +months but the cravings are setting in. Can anyone suggest me the +best way to go about getting on suboxone?? +craving, heroin, +clean, suboxone +suboxone, +craving, heroin +0.75 +Tapering +from 24mg +Will I go through any withdrawals tapering off 24mg of suboxone +if I taper down to 22mg? Been on 24mg for 4 months. +suboxone, taper, +withdrawal +suboxone, taper, +withdrawal +1.0 +Table 2: Some sample excerpts with titles, posts, and annotations. Annotated keyphrases are normalized (e.g., dope is normal- +ized to heroin). The Jaccard Index (JI) is calculated over the normalized keyphrase list. +opioid dependency and recovery through keyphrase analy- +sis. We explore the following research questions to draw in- +sights from user engagement on r/Suboxone. +1. Which keyphrase-containing posts provoke the most peer +engagement? +Reddit facilitates discussion in a thread-like setting where +a user starts the conversation by posting, and others in- +teract via comments, upvotes, and downvotes. We mod- +eled the engagement of a post by counting comments and +upvotes. The engagement score for a given post is calcu- +lated using the average number of comments or upvotes +received. If a post has n keyphrases, each will receive a +similar engagement score. This allows us to inspect what +thematic topics attract more users to a discussion. +2. How does peer engagement vary across posts with the +most frequently used keyphrases in our data? +We obtain two sets of keyphrases from our annotation +efforts—the union and intersection of each annotator’s +keyphrases. The union set comprises all the keyphrases +marked by both annotators, while the intersection set only +contains those keyphrases that exactly match both an- +notators. We chose the intersection set for analysis as +both annotators agreed on these keyphrases. We select the +top-most frequent keyphrases and empirically analyze the +associated post to get data-driven insights. Specifically, +we explore the association between frequent keyphrases +and engagement scores to determine whether frequent +keyphrases result in more peer engagement. +For both RQ1 and RQ2, our domain experts guided us to +contextualize the results and conduct qualitative analysis. +RQ3: How effective are Off-The-Shelf Keyphrase +Extraction models on theme-driven opioid +recovery-related keyphrases? +In this section, we aim to evaluate the performance of 10 +off-the-shelf KE models on the SuboxoPhrase dataset. +Statistical Methods: +We evaluate the performance of two +statistical methods, TfIdf (Manning and Prabhakar 2010) +and YAKE (Campos et al. 2020). +Graph-Based Methods: +We explore four graph-based ap- +proaches to keyphrase extraction, namely TextRank (Mihal- +cea and Tarau 2004), TopicRank (Bougouin, Boudin, and +Daille 2013), PositionRank (Florescu and Caragea 2017), +and MultipartiteRank (Boudin 2018). These classic mod- +els are commonly used as baselines in modern UKE studies +(Liang et al. 2021; Zhang et al. 2022). +Embedding Methods: +All of our embedding methods use +the following pipeline: 1) Extract candidate keyphrases 2) +Embed each candidate and the full Reddit post 3) Return the +top-k candidate keyphrases with the highest cosine similar- +ity to the embedding of the entire Reddit post. We evaluate +the performance of DistilBERT (Sanh et al. 2019), BERT +(Devlin et al. 2018), DeBERTa-v3 (He, Gao, and Chen +2021), and SBERT (Reimers and Gurevych 2019), as they +are all widely-used transformer models with proven ability +to produce meaningful contextual word embeddings. +Experimental Setup: +All models evaluated in this study +are unsupervised, as the dataset is too small for supervised +learning. For the statistical and graph-based methods, we use +the PKE library (Boudin 2016) to perform keyphrase extrac- +tion. For the embedding methods, we use KeyBERT (Groo- +tendorst 2020) to extract and rank keyphrases. All mod- +els use the same candidate keyphrases. Specifically, we use +PKE’s grammar selection tool to extract only noun phrases +as candidates for each post. We report the Recall@k for +each model, where k represents the number of keyphrases +returned by the model. +Results +RQ1: What clinical insights can we draw from +theme-driven opioid recovery-related keyphrases? +What insights can we draw from keyphrase occur- +rences? After normalization, we obtained a total of 881 +unique keyphrases. The distribution of the keyphrases +among different groups can be found in Table 1. We ob- +served that the highest frequency keyphrase category is psy- +chophysical effects with 331 in total. The prevalence of psy- +chophysical effects can be attributed to the fact that users +discuss different types of effects they had faced due to the +concurrent treatment or substance use, thus seeking sugges- +tions. Withdrawal, sleep, and precipitated withdrawals were +the top mentioned psychophysical effects. In contrast, we +found the lowest number (35) of keyphrases in the medical +history category. The possible reason is that users focused on +the ongoing problems or effects and did not mention medical +history, or the reported medical histories were not often the +keyphrase in a post. Examples of medical history include +but are not limited to pregnancy, adhd, and bipolar disor- +der. Many users posted about treatment options (182), indi- +cating that they were concerned about their treatments and +thus tried different treatment options. Commonly used treat- +ment options were suboxone, doctor, buprenorphine, etc. We +also found reasonable evidence (77) of keyphrases related + +to substance dependency & recovery. Frequently occurring +keyphrases in this category are taper, heroin, oxycodone, +and fentanyl. The 256 keyphrases that did not belong to any +of the four main categories were categorized as others, e.g., +clean, dissolve, water, etc. +Our thematic analysis of frequent keyphrases is valuable +for clinicians and researchers focusing on MOUD treat- +ment. For example, it is advantageous for researchers to +understand which other treatment options patients consider +while undergoing suboxone-based treatment for opioid re- +covery. This information can uncover potentially harmful +trends in self-prescribed or new treatment options criti- +cal to a patient’s recovery experience. Similarly, quantify- +ing the prevalence of various psychophysical effects can +guide researchers toward emerging, potentially significant +side effects which require attention from public health offi- +cials. Identifying medical/substance use histories can inform +about clinically relevant subpopulations seeking MOUD +treatment who frequently engage on Reddit, e.g., individ- +uals with fentanyl or heroin dependency, pregnant women +seeking MOUD treatment, or individuals interested in ta- +pering their medications. These findings can inform tailored +intervention design for MOUD treatment, i.e., who can be +reached through Reddit and when. +What +insights +can +we +draw +from +keyphrase +co- +occurrences? +Our analysis found that at least one instance +of the keyphrase suboxone co-occurs with each of the known +adult side-effects listed on the U.S. Substance Abuse and +Mental Health Service Administration webpage (SAMHSA +2022). This relationship to standard substance abuse guide- +lines validates the quality of our annotated data. Addition- +ally, such annotations allow one to gather many self-reports +based on suboxone and a corresponding psychophysical ef- +fect. This facilitates analysis of the context and severity of +various opioid-related issues. For example, one user posted: +Is anyone getting heart or chest pain, back pain while +breathing in sometimes and pain in right upper ab- +domen randomly while you’re detoxing yourself with +subs?...day 5 [and] still finding quite a few of these +scary symptoms ... 1st time I was kicking tranq dope +so idk if it’s just that? or the subs somehow? +Here we find a user experiencing a physical side-effect +of starting suboxone (i.e., lingering withdrawal symptoms) +being perceived as a general suboxone side-effect. MOUD +researchers are interested in such cases as misperceptions +in MOUD treatment can negatively impact treatment induc- +tion, adherence, and retention. +Additionally, we find many side effects co-occurring with +suboxone which are not officially listed as known psy- +chophysical responses to suboxone. For example, almost +2% of user posts in SuboxoPhrase expressed issues with +their sex drive. Other emerging co-occurring psychophysi- +cal effects of suboxone include depression, depersonaliza- +tion, anger, skin crawling, mood swings, hunger, and mem- +ory issues. Researchers interested in monitoring rare adverse +drug reactions or unknown drug effects may benefit from KE +co-occurrence analysis. Accurate extraction of keyphrases +in recovery-related social media discourse can help iden- +tify similar discussions and inform the design of qualitative, +prospective studies to identify population-level perceptions +and misperceptions related to MOUD treatment. +In addition to co-occurrence patterns with the keyphrase +suboxone, there are intriguing patterns with tapering. Co- +occurrence patterns—such as between tapering and with- +drawal are unsurprising given that dose reduction usually +induces withdrawal symptoms. More interesting is the co- +occurrence pattern that tapering has with both sleep and anx- +iety. Such significant associations present in SuboxoPhrase +but not explored by the literature are primary candidates for +exploration in future work. +How did the onset of the COVID-19 pandemic affect the +discussion of opioid recovery treatments? +Crucial to our +analysis of SuboxoPhrase is the temporal component of the +data. Opioid use, abuse, and treatment are subject to and in- +fluenced by global events. One such global event that has +had far-reaching impacts in many domains worldwide was +the COVID-19 pandemic. Illustrating this point, Currie et +al. (2021) found that, despite no notable changes in existing +OUD treatment plans, admission of new patients to OUD +treatments lagged behind predicted levels before rebound- +ing in August 2020. +To uncover potential patterns around the pandemic, we +divided the SuboxoPhrase dataset along the start date of the +COVID-19 pandemic as declared by the WHO—the 11th +of March, 2020 (Domenico and Vanelli 2020). This allows +us to make qualitative comparisons between the frequencies +of keyphrases before and after the onset of the pandemic. +Figure 1: The relative percent increase or decrease in +keyphrase frequencies after the onset of COVID-19. We can +see a large increase in certain keyphrases, e.g., fentanyl, and +a notable decrease in certain keyphrases, e.g., loperamide. + +fentanyl +prescription +gabapentin +sublocade +high +pill +opioid +oxycodone +SICk +precipitated withdrawals +pain +film +tired +methadone +surgery +constipation +sweat +Keyphrase +headache +rehab +withdrawal +xanax +drug test +insurance +insomnia +clonidine +sleep +stomach +psychiatrist +benzodiazepines +back,pain +symptoms +painmedicine +restless leg +klonopin +urine +addiction +volumetric dose +sobriety +testosterone +loperamide +100%-50% +0% +50% +100% +150% +200% +250% +Percent changeWe aim to analyze keyphrases with the highest relative per- +centage increase or decrease, as such samples demonstrate +the most considerable shift in the discussion. Figure 1 con- +tains the top 20 keyphrases (each with a minimum of 5 oc- +currences in the dataset) displaying the highest relative in- +crease or decrease as shown in blue and red, respectively. +After the start of the pandemic, the terms with the highest +keyness for their posts shifted to the terms fentanyl, prescrip- +tion, gabapentin, sublocade, and others. Keyphrases that oc- +cur less frequently following the COVID-19 pandemic in- +clude loperamide, testosterone, addiction, and sobriety. The +reason for the reduced frequency of such terms may be ex- +plained by the logistical challenges created by the pandemic. +For example, testing testosterone levels require in-person +visits and might explain a decision to delay such testing due +to the COVID-19 restrictions. +Increases of terms such as prescription suggest an acceler- +ation in new treatment plans and administration of MOUDs +in the years after the pandemic (Currie et al. 2021). How- +ever, the most striking change by far is increased fentanyl +mentions. Not only is there substantial evidence that fen- +tanyl use has been growing over the last decade, but the pan- +demic is also associated with an increase in fentanyl over- +dose beyond projected levels (CDC 2023; Morin et al. 2021). +According to the CDC, there was a less extreme increase +in 2021 compared to 2020 (CDC 2022b). The rise in pre- +scription discussions likely stems from new access barriers +around treatment, pharmacies, and users’ frustrations. These +changes in crucial discussion subjects demonstrate the ex- +tractive capabilities of a theme-driven KE approach. Future +models and larger datasets could guide policies and outreach +efforts through real-time monitoring compared to surveys’ +cost, time use, and lack of scalability. +Figure 2: Top 20 engaging keyphrases in terms of the num- +ber of comments (#comments). Keyphrases related to psy- +chophysical effects like mood swings, mental health, and +melatonin (to cope with sleep trouble) spark more comment- +based engagement. +RQ2: How does peer engagement associate with +keyphrases in opioid recovery-related discourse? +Figure 3: Engagement score in terms of comments and up- +votes of the top 20 most frequent keyphrases in the Subox- +oPhrase dataset. The frequency of a keyphrase is shown in +parentheses, e.g., heroin appears 52 times as a keyphrase. +Which keyphrase-containing posts provoke the most +peer engagement? Figure 2 shows the top 20 engaging +keyphrases regarding the number of comments. By empir- +ically observing the keyphrases and associated posts, we +found that more than 60% of engaging keyphrases are re- +lated to psychophysical effects. People tend to involve in +conversations that discuss potential side effects of with- +drawals, such as mental health, body aches, etc. Other en- +gaging posts include posts mentioning different treatment +options to cope with withdrawal symptoms (e.g., melatonin +for helping with sleep, bupropion, or cognitive behavioral +therapy (CBT)). Due to space constraints, we omitted the +graph corresponding to upvote-based engagement, which +yielded results similar to comment-based engagement. +How does peer engagement vary across posts with the +most frequently used keyphrases in our data? +Figure +3 shows the annotated dataset’s twenty most frequently +mentioned keyphrases and their peer engagement score. +Keyphrases related to treatment options (e.g., suboxone, +kratom) are more frequently used than psychophysical ef- +fects. The reason behind the frequent use of treatment- +related keyphrases is that they generally co-occur with the +mention of psychophysical effects. The mention of previous +substance usage (e.g., heroin, oxycodone) as a keyphrase is +also significant. People mention their substance usage ex- +perience to explain their situation while seeking recovery- +related information. +From Figure 3, it is evident that engagement does not di- +rectly correlate with the frequency of the keyphrases. If a +keyphrase is frequent, it does not guarantee it will have a +higher engagement score. For example, frequent keyphrases +such as buprenorphine, fentanyl, opiate have low engage- + +0 +50 +100 +mood swings +128 +128 +mental health +114 +melatonin +numb +68 +Top 20 engaging keyphrases +63 +opioid addiction +bupropion +62 +irritation +58 +dizzy +54 +52 +pregnancy +cbt +47 +overdose +47 +foggy head +47 +body aches +45 +leg cramps +41 +40 +gas pain +energy +40 +introversion +38 +38 +dark thoughts +37 +grapefruit juice +robitussin +37 +Engagement score (#comments)30 +avg comments +avg_upvotes +25 +Engagement-score +20 +15 +10 +5 +suboxone (516) +withdrawal (120) +(52) +(26) +naloxone (18) +depression (13) +《(59) +I (33) +(33) +(30) +(29) +(15) +xanax (13) +(63) +(63) +(45) +(44) +(15) +7 +oxycodone (（ +sleep ( +sublocade ( +buprenorphine +subutex +kratom +fentanyl +heroin +opiate +J withdrawals +gabapentin +benzodiazepines +clonidine +taper +methadone +anxiety +precipitated +Top 2o freguent keyphrases in the Intersection setModel +R@5 +R@10 +R@15 +Statistical Methods +TfIdf +0.38 +0.50 +0.55 +YAKE +0.35 +0.48 +0.51 +Graph-Based Methods +TextRank +0.16 +0.33 +0.43 +TopicRank +0.30 +0.44 +0.49 +PositionRank +0.28 +0.42 +0.50 +MultipartiteRank +0.32 +0.46 +0.52 +Embedding Methods +DistilBERT +0.30 +0.45 +0.52 +BERT +0.31 +0.46 +0.52 +DeBERTa-v3 +0.26 +0.39 +0.47 +SBERT +0.40 +0.51 +0.56 +Table 3: Recall@K for each model. Note: On average +only 72% of annotated keyphrases appear in the candidate +keyphrases. Thus, the maximum recall is moderately low. +ment. While less frequent keyphrases can be more engag- +ing (e.g., methadone, anxiety). The engagement trend of fre- +quent keyphrases seems quite similar for comments and up- +votes. However, for specific keyphrases (xanax), the trend +is the opposite. In summary, the qualitative analysis uncov- +ers that posts having keyphrases related to psychophysical +effects/treatment options engage more users in direct inter- +actions, i.e., comments/upvotes. +RQ3: How effective are Off-The-Shelf Keyphrase +Extraction models on theme-driven opioid +recovery-related keyphrases? +Our experimental results find that the SBERT embeddings +produce the highest recall across all experiments. This is ex- +pected as 1) Embedding approaches are unique in produc- +ing contextual representations of the full post, which explic- +itly encode textual semantics—a useful feature in UKE 2) +SBERT is the only embedding method employed which was +trained to output embeddings for longer sequences of texts +(other methods are only explicitly trained to produce only +high-quality token embeddings). Graph-based methods per- +formed consistently lower than SBERT at all values of k. +The method which achieved the second-best performance, +TfIdf, displays the viability of a low computational cost +method for use in resource-constrained settings. +Unfortunately, +none +of +the +standard +UKE +models +achieved recall above 0.56. This limits the use of off-the- +shelf models to draw insights from silver-labeled samples +across all of r/suboxone. Large-scale data collection us- +ing our described annotation guidelines could enable su- +pervised training, which may alleviate this problem. Addi- +tionally, social media-specific UKE models may better han- +dle long self-reports filled with informal user-generated lan- +guage. Improving the performance of UKE models on Sub- +oxoPhrase will be the focus of future works. +Ethical Considerations +This research is conducted under Institutional Review Board +(IRB) approval at the authors’ institution. Since Reddit users +can make an account using only an email address (not dis- +closed by Reddit), the site consists of anonymous users who +can only be identified by data they voluntarily disclose in +their posts. Even if a user shares some self-identifying in- +formation (e.g., location, age, gender identity), it is still +nearly impossible to identify the true identity of a Reddit +user without further information. Additionally, the Reddit +user agreement requires users to consent to share their pub- +lic posts/comments with the Reddit API. +If the user deletes a post, we do not include it in our +dataset as it is the user’s right to remove information from +the Reddit platform. Additionally, posts included as exam- +ples in the paper were paraphrased to prevent the reader from +directly identifying the Reddit user account that posted each +example. This is a standard precautionary measure. +Broader Impact +Implications for KE on Social Media: +Existing works on +keyphrase extraction in social media have two limitations: +(1) Posts are derived from Twitter, where low character lim- +its preclude capturing long-range dependency and rich infor- +mation from long, complex colloquial text. Analysis of Sub- +oxoPhrase requires understanding texts of variable length, +some of which reach even 10,000 characters. Thus Subox- +oPhrase will promote the creation of better HealthNLP mod- +els capable of modelling keyphrases in more extended so- +cial media discourse on health. (2) Large Twitter datasets +utilize hashtags as surrogate keywords—a strategy based on +the error-prone assumption that hashtags are always indica- +tors of keyness. This is the first Reddit dataset with keywords +extracted by human annotators. This work thus provides re- +liable annotation for clinically relevant keyphrase extraction +from Reddit on MOUD-based treatment for opioid recovery. +Implications for MOUD Research: SuboxoPhrase can in- +form the development of clinician-facing tools that facilitate +the discovery of tangible insights that can inform MOUD +research and practice. For example, a SuboxoPhrase-based +keyphrase extraction tool can help facilitate the discovery +of the perceived effectiveness of different MOUD treatment +options, strategies to cope with side effects, rare/new adverse +drug reactions, and uncover patterns in the patient-reported +experience with different MOUDs. Such findings may guide +future research in opioid recovery by clinicians and public +health researchers. Also, results from such theme-driven KE +may guide the development of tailored patient communica- +tion tools and programs, e.g., when and how to taper or po- +tentially severe side effects of suboxone. +Limitations and Future Work +A limitation of this work is the insufficient amount of gold- +labeled data, constraining supervised approaches. As higher- +performance UKEs are applied to the SuboxoPhrase dataset, +it will be possible to extract reasonably accurate keyphrases +from a much larger number of Reddit posts. 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MDERank: A masked document embedding +rank approach for unsupervised keyphrase extraction. In Findings +of the Association for Computational Linguistics: ACL 2022, 396– +409. Dublin, Ireland: Association for Computational Linguistics. +ˇS´cepanovi´c, S.; Aiello, L. M.; Zhou, K.; Joglekar, S.; and Quercia, +D. 2021. The healthy states of america: Creating a health tax- +onomy with social media. Proceedings of the International AAAI +Conference on Web and Social Media 15(1):621–632. + diff --git a/f9FJT4oBgHgl3EQfUSzI/content/tmp_files/load_file.txt b/f9FJT4oBgHgl3EQfUSzI/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..82d789b9fd506d3989e302a16b041105f6059e02 --- /dev/null +++ b/f9FJT4oBgHgl3EQfUSzI/content/tmp_files/load_file.txt @@ -0,0 +1,1259 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf,len=1258 +page_content='Theme-driven Keyphrase Extraction from Social Media on Opioid Recovery William Romano*1, Omar Sharif*1, Madhusudan Basak1,2, Joseph Gatto1, Sarah Preum1 1Department of Computer Science, Dartmouth College, USA 2Department of Computer Science and Engineering, BUET, Dhaka, Bangladesh {william.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='romano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='gr,omar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='sharif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='gr,madhusudan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='basak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='gr,joseph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='gatto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='gr,sarah.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='masud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='preum}@dartmouth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='edu Abstract An emerging trend on social media platforms is their use as safe spaces for peer support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Particularly in healthcare, where many medical conditions contain harsh stigmas, social me- dia has become a stigma-free way to engage in dialogues regarding symptoms, treatments, and personal experiences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Many existing works have employed NLP algorithms to fa- cilitate quantitative analysis of health trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Notably absent from existing works are keyphrase extraction (KE) models for social health posts—a task crucial to discovering emerg- ing public health trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' This paper presents a novel, theme- driven KE dataset, SuboxoPhrase, and a qualitative annota- tion scheme with an overarching goal of extracting targeted clinically-relevant keyphrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' To the best of our knowledge, this is the first study to design a KE schema for social media healthcare texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' To demonstrate the value of this approach, this study analyzes Reddit posts regarding medications for opioid use disorder, a paramount health concern worldwide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Additionally, we benchmark ten off-the-shelf KE models on our new dataset, demonstrating the unique extraction chal- lenges in modeling user-generated health texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' The proposed theme-driven KE approach lays the foundation of future work on efficient, large-scale analysis of social health texts, allow- ing researchers to surface useful public health trends, pat- terns, and knowledge gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Introduction Social media platforms like Twitter, Facebook, and Red- dit gather spontaneous, self-reported lived experiences from thousands of individuals with a diverse array of medical and socio-demographic conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' People seeking or undergo- ing treatment often resort to social media for informational and emotional support (Chen, Wang, and others 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Find- ings in public health research reports increased reliance on social media and other online health platforms for address- ing health information needs (Neely, Eldredge, and Sanders 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Bhandari, Shi, and Jung 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Thus social media has become an exceptional source of clinically relevant data to understand population-level concerns, knowledge gaps, treatment perceptions, and barriers (Chen, Wang, and others 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' In addition, social media platforms like Reddit sup- port anonymity and thus encourage rich engagement among Both authors contributed equally to the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Copyright © 2023, Association for the Advancement of Artificial Intelligence (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='aaai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='org).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' All rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' peers on stigmatized topics like mental health and substance use recovery (Naslund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Qualitative analysis of social media data has been used to understand various health issues, including COVID-19 (Sleigh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' 2021), cancer (Levonian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' 2020), depression, and other mental health conditions (Lachmar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Contemporary Natural Language Processing (NLP) algo- rithms have enabled large-scale, quantitative analysis of so- cial media platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' For example, advancements in text classification have produced works in suicide risk detection (Mathur, Sawhney, and Shah 2020), detection of adverse drug reactions (Aroyehun and Gelbukh 2019), and misin- formation classification (Dharawat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Advance- ments in information extraction have led to models which can automatically extract medical entities from social me- dia texts (ˇS´cepanovi´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Santosh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Other works have conducted a data-driven analysis of social me- dia health (SMH) texts in the context of Topic Modeling (El-Bassel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' 2021) and Medical Named Entity Recog- nition (MedNER) (Batbaatar and Ryu 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Notably ab- sent from the literature on social media for health analysis is keyphrase extraction (KE), the task of extracting the most salient terms from a given text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' We note it is essential to dis- tinguish KE from topic modeling and MedNER, as the latter two do not focus on extracting key terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' To illustrate, con- sider a social media user who posts: Am I the only one taking subs that feels nervus all of the time?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' I know anxiety is a symptom of other opioid recovery drugs like methadone but I don’t take those.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' The keyphrases in this post are subs1, nervus, and anxiety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' A Topic Model, defined over a collection of documents and not designed for analysis of an individual text, would likely ex- tract none of these keywords if they were not topics of other texts in the corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' A MedNER model may extract all of the drug/symptom mentions, but it would be unable to distin- guish that methadone is not a keyword in this case, rather is only mentioned as context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Finally, both Topic Models and MedNER models may struggle with misspellings and shorthand, such as nervus and subs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' By using KE for SMH texts, we employ a method that operates at the 1shorthand for Suboxone, a prescription medication used to treat opioid use disorder (OUD) arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='11508v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='CL] 27 Jan 2023 post-level, can dynamically extract different lexical varia- tions of a concept (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=', misspellings, slang, or other collo- quial forms), and ensure the information extracted is mean- ingful and relevant to the target user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Applying KE to SMH texts poses a unique challenge as the notion of “keyphrase” is historically difficult to define and subject to annotator bi- ases and end-goal/downstream applications (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=', how to uti- lize the extracted keyphrases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Thus existing works on KE are not well-suited for modeling long texts comprised of col- loquial, user-generated health narratives with a wide variety in length, ambiguity, content, and style.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' To address this problem, this study defines a novel theme- driven KE annotation strategy for SMH texts for identi- fying keyphrases relevant for clinical researchers to sur- face population-level knowledge gaps, treatment percep- tions, and experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' KE for SMH analysis is vital since standard information extraction methods are limited by rely- ing on narrowly defined entity types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Conversely, keyphrases can cover a wide variety of entities given that they are key or significant to the post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' This is of value to researchers who wish to explore social media discourse 1) with the knowl- edge that the returned terms are meaningful to the original poster, and 2) with the ability to discover valuable public health trends and patterns unconstrained to pre-defined en- tity classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' To our knowledge, no existing models enable a broad-scale, theme-driven keyphrase analysis of SMH texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' To illustrate the value of KE for SMH, we analyze Red- dit posts regarding Medications for Opioid Use Disorder (MOUD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' MOUD analysis is of great importance as the opi- oid crisis has become one of the most pressing health con- cerns in the United States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' In 2020 alone, 68,630 people died due to an opioid-related overdose (CDC 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Ac- cording to the 2021 PEW survey on substance use preven- tion and treatment, opioid overdose, misuse, and dependence result in 35 billion dollars in healthcare costs, 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='8 billion dollars in criminal justice costs, and 92 billion dollars in lost productivity annually (Florence, Luo, and Rice 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Spencer, Mini˜no, and Warner 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Medications for opioid use disorder (MOUD), such as buprenorphine, methadone, and naltrexone, are the most effective, evidence-based treat- ment for OUD (Wakeman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Yarborough et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Given the prevalence of opioids, patients suffering from OUD often experience multi-level stigma, including inter-and intra-personal, social, provider-level, and policy- level stigma (Cheetham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Thus, MOUD forums are often utilized by those in recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' By analyzing online MOUD self-reports, researchers can gather valuable, clini- cally relevant insights from large-scale self-disclosed patient data on their lived experiences, psychophysical effects, and treatment concerns, identifying knowledge gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' In this work, we study the subreddit r/Suboxone, the most populous Reddit forum among those dedicated to discussing buprenorphine-based drugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Reddit is well suited for such analysis as it is anonymous, publicly available data with theme-specific forums ready for exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' To facilitate KE on SMH, this work provides the following contributions: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' We define a systematic qualitative KE annotation/coding scheme for SMH texts relevant to opioid recovery with guidance from our multi-disciplinary domain experts on substance use research and clinical practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' We develop a dataset, SuboxoPhrase, with human-annotated KE sam- ples from the r/Suboxone subreddit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' To the best of our knowledge, this is the first KE dataset of SMH texts and will be made publicly available upon the acceptance of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' We demonstrate the ability of keyphrases to draw clinical insights on MOUD-based recovery from a relevant sub- reddit through extensive quantitative and qualitative anal- ysis of keyphrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Specifically, our research questions explore (i) drawing clinical insights from keyphrase fre- quency and keyphrase co-occurrences in the labeled data, (ii) analyzing the effect of the COVID-19 pandemic on the keyphrases, and (iii) the association of keyphrases with peer engagement—identifying terms which stimulate on- line discussions among peers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' We benchmark ten existing Unsupervised KE (UKE) models to demonstrate the limitations of exiting UKEs on SMH data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Our results serve as references for future works on KE for SMH and motivate the need for SMH- specific approaches to KE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Related Work Keyword and Keyphrase Extraction KE is a widely explored research problem (Nomoto 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Hasan and Ng 2014), where the goal is to extract salient phrases that best summarize the document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' The extracted set of keyphrases can vary significantly based on the task one is focusing on (Firoozeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Existing KE ap- proaches generally work in two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' 1) Select candidate keyphrases by heuristic rules or manual annotation (Wang, Zhao, and Huang 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' 2) Apply supervised (Lopez and Romary 2010) or unsupervised (Gu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' 2021) approaches to rank keyphrases as per their relationship to the document and return the top-k keyphrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Popular KE techniques can be characterized as either statistical (Campos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' 2020), graph-based (Mihalcea and Tarau 2004), or embedding- based (Bennani-Smires et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' 2018) methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' However, with the emergence of pre-trained language models, recent KE works have seen a paradigm shift, as contextual embedding- based approaches have become the primary building block of recent KE models (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Most previous studies have performed KE on scientific lit- erature, where author-assigned keyphrases are leveraged for ground truth (Augenstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' In contrast, this work focuses on extracting keyphrases from social media data which, compared to scientific literature, is challenging to in- terpret, contains domain-specific vocabulary, shorthand, and slang, and is often not amenable to existing NLP resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' We note some existing works on KE for social media, such as the work from Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' (2016), where keyphrase labels are inferred from Twitter hashtags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' In contrast, this study fo- cuses on SMH texts from Reddit posts where all samples are human-annotated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Social Media Discourse Analysis for Substance Use and Opioid Use Disorder (OUD) Recently, social media has emerged as a place for people to share their experiences, provide support, and engage in discussions with others who have a similar interest in sub- stance use, addiction, and recovery (Chancellor, Mitra, and De Choudhury 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Social media discourse on substance use refers to posts or discussions on drugs and other re- lated topics such as addiction, harm reduction, treatment options, and recovery in social media platforms (Lavertu, Hamamsy, and Altman 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Example works on this topic include Chen, Johnny, and Conway (2022), who analyzed Reddit discussions about cannabis, alcohol, and opioids to examine the nature of stigma related to these substances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' MacLean et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' (2015) demonstrated a positive correlation between forum use and recovery by analyzing online discus- sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Chancellor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' (2019) attempted to uncover poten- tial alternative treatment options for OUD by investigating posts from opioid recovery subreddits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Unlike Chancellor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=', our study differs in the NLP task, domain (we focus on a subreddit specific to a MOUD, namely Suboxone), as well as scope (we explore a broader range of categories than treat- ment options, including psychophysical effects, medical his- tory, and substance dependency & recovery).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Peer Engagement Modeling Quantifying peer engagement on social media often depends on the platforms, and researchers use various indicators to assess engagement (Sharma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Low et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' (2021) analyzed the effect of engagement in the r/SuicideWatch subreddit based on the number of comments received.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' They argued that receiving more comments increases a user’s likelihood of posting and engaging in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Previous works also studied other engagement indicators, including but not limited to the number of posts, the number of com- ments/responses, the time between responses, and the num- ber of links shared (De Choudhury, Counts, and Horvitz 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' In this work, we utilize the number of comments and the number of upvotes a post received to measure engage- ment as these (i) are readily available through the Reddit API and (ii) capture peer engagement in a tangible way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Dataset Data Source We collected data from Reddit, an internet forum with over 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='8 million communities catering to various topics, interests, and social groups called “subreddits”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Users may join these subreddits to engage with other users in anonymous discus- sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Since Reddit is an anonymous platform, the only pub- lic information available for a given user is their username, join date, and activity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=', post and comment history).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' The subreddit from which we collected the data used in this study is r/Suboxone, a community with over 25 thousand users as of December 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Created in 2011, r/Suboxone is a com- munity forum for dialogue between users who share their relationship with the opioid recovery medication Suboxone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' This subreddit is strictly moderated, and any posts about buying/selling illicit drugs, exposing other users’ personal information (also known as doxxing), or bullying/abuse are removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Analyzing activity on r/Suboxone makes it possi- ble to achieve a unique and authentic perspective on a user’s opioid recovery journey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' While this subreddit explicitly fo- cuses on the treatment of OUD using Suboxone, users fre- quently discuss their experiences and concerns related to other treatment options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' This is because individuals with OUD may try many treatment options to support their re- covery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Discussions extend to various related topics and co- occurring substance and medication usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Data Collection All posts between the 2nd of January, 2018, and the 6th of August, 2022 on the r/Suboxone subreddit were scraped us- ing the PRAW and PushShift APIs (Boe 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Baumgartner 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' We select this timeline as it allows us to (i) focus on more recent user data, and (ii) collect a dataset with a comparable amount of posts before and after the onset of the COVID-19 pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Also, we observed very limited interaction in this subreddit before 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' After filtering the corpus for irrelevant posts (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=', those containing polls, links with no texts, or posts which had been deleted), we devel- oped a corpus of 15,253 posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Sample posts are presented in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Sample Selection We chose 1,000 sample posts for keyphrase annotation, a set of comparable size to other notable works in KE (Au- genstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Given that one of the goals of this work is to explore the effect of the COVID-19 pandemic on opioid recovery discussions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=', RQ1), the sub-sample of 1,000 posts was chosen to equally represents activity before and after the COVID-19 pandemic (with respect to the date COVID-19 was declared a pandemic by the World Health Organization (Domenico and Vanelli 2020)), consist- ing of 500 posts from each half.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' The title and body of each post were combined and annotated to capture the post’s full context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' This is important because many posts contain the most insightful details in their titles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Larger social media datasets for KE also exist (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' 2016), but those datasets leverage automated approaches like using Twitter hashtags as keyphrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' However, our theme-driven defini- tion of keyphrases requires rigorous manual annotation and can lead to data-driven knowledge discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Thus we limit our sample size for manual annotation to 1,000 posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Schema & Guidelines Design Process Determining a phrase to be key to a given text is subjec- tive, given annotator biases and target downstream applica- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' In this study, we aim to extract keyphrases relevant to Suboxone-assisted OUD recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' We collaborate with a team of five domain experts to define the themes of key- ness for medication-based OUD recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' All of our collab- orators are well-versed in substance use disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Together, they cover a wide variety of expertise, including psychi- atry, biomedical data science, digital technology for sub- stance use and mental health, epidemiology, public health policy, addiction medicine, and addiction psychiatry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Two of our collaborators are clinicians and administer MOUD treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' All of them reviewed our samples and helped us to contextualize different aspects of opioid recovery and the shared lived experiences of the affected individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Based on guidance from our collaborators, we identify four main themes: Treatment Options, Substance Dependency & Re- covery, Medical History, and Psychophysical Effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' The definitions of these four themes are presented below, along with an extra category Others intended to capture additional keyphrases that do not belong to any of these categories but are still key to the original post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Treatment Options: This category covers keyphrases re- lated to different treatment options used for recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Such as medications used to treat OUD (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=', Buprenor- phine, Methadone, or their formulations), psycho-therapy, behavioral counseling, or other medications used to cope with withdrawal or other psychophysical effects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=', us- ing melatonin to help with insomnia while in recovery).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' We consider prescribed medications, over-the-counter medications, herbal supplements, and other therapeutic options as potential candidates for keyphrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Substance Dependency & Recovery: This category cov- ers keyphrases related to the history of substance use (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=', fentanyl), co-occurring substance use (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=', tobacco, al- cohol), and critical factors in recovery (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=', recovery, re- lapse).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' We consider both prescribed and self-administered substances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Medical History: Keyphrases concerning medical his- tory, including diagnosis and self-diagnosis of any phys- ical and mental health conditions, relevant medical pro- cedures (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=', major surgery), or critical family medical history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' This category covers medical history beyond sub- stance dependency/recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Psychophysical Effects: Keyphrases regarding any phys- ical or psychological effects and symptoms associated with OUD recovery, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=', psychological effects relevant to withdrawal, precipitated withdrawal, and side effects of medications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Others: All keyphrases that do not belong to any of the above four categories are assigned to this category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Category Fre- quency Example Keyphrases Treatment Options 182 Adderall, antidepressant, naloxone Substance Dependency & Recovery 77 cocaine, fentanyl, heroin Medical History 35 COVID, osteoarthritis, PTSD Psychophysical Effects 331 aches, constipation, panic attack Others 256 scam, travel, boyfriend Table 1: Examples from each keyphrase category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Frequency denotes the number of occurrences in the category in the whole SuboxoPhrase dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Example keyphrases of each category from SuboxoPhrase are exhibited in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' To meet our goal of extracting keyphrases relevant to opioid recovery, we thoroughly it- erated over our definition of a keyphrase in SuboxoPhrase via multiple rounds of trial annotations and discussions be- tween annotators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Keyphrases were annotated to satisfy the general criteria for “keyness” inspired by Firoozeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' (2020), primarily “conformity”, “homogeneity”, and “uni- vocity”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Conformity, for our purposes here, was reflected by capturing domain-specific terminology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Homogeneity ap- plies to normalizing the diverse vocabulary, slang, and mis- spellings associated with informal discussions online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Uni- vocity and generalizability are two crucial and opposing cri- teria we set for our annotation guidelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Univocity refers to keyphrases that are specific and non-ambiguous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' This prop- erty is achievable by increasing the number of terms in a keyphrase—e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=', a bigram is less ambiguous than a unigram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' This principle is directly opposed to generalizability in that the more specific a keyphrase is, the fewer text documents it will apply to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Annotators were encouraged to balance the trade-offs between these two constraints to produce high- quality keyphrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Note that our process is distinguished from named entity recognition and traditional KE in that our goal was to extract keyphrases aligned with specific themes grounded in substance use research while maintaining the representativeness and generalizability of the keyphrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Data Annotation and Quality Check The complete data annotation was manually carried out by four graduate students who are both regular social media users and active in NLP research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' They studied/used opioid recovery subreddits, so they possessed better domain knowl- edge than mTurk annotators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Moreover, they were trained on the annotation task and provided background on MOUD and suboxone through multiple sessions led by experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' We used LightTag, an online platform, to label the keyphrases (Light- Tag 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Two annotators annotated each sample to gener- ate high-quality keyphrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Experts resolved any confusion during annotation through discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Since the annotation involved multiple annotators, we cal- culated the inter-annotator agreement score to ensure the dataset’s quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' We used two lists of normalized keyphrases for each sample from the annotators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' We use the Jaccard index to measure the agreement/similarity between annota- tions (Sarwar, Noor, and Miah 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Jaccard index is de- fined as: JI = len(A ∩ B) len(A) + len(B) − len(A ∪ B) (1) Let A and B be the respective set of keyphrases from an- notators 1 and 2 for a given sample i in SuboxoPhrase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' JI computes the Jaccard index for sample i and represents the annotator agreement for that sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' While calculating the intersection and union of the two sets, we considered the ex- act string match between the elements of the sets as used in Schopf, Klimek, and Matthes (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' We used Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' (JI) to capture the average Jaccard index for the whole dataset of n samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' The average Jaccard similarity index obtained us- ing the exact string match approach was 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='36%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' The score indicates a substantial similarity between keyphrases ex- tracted by multiple annotators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Sample posts with extracted keyphrases and JI-score are presented in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Normalization Reddit users commonly use various unique phrasings of the same word, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=', shorthand, slang, and misspellings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' For ex- ample, the opiate heroin may also be referred to as h, dope, smack, and speedball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' For this study to draw general con- clusions about MOUD forum discussions, we must be able to normalize all keyphrases to a common parent term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' We manually map all keyphrase variations to their most mean- ingful representative parent phrases to solve this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' In our analysis, we thus ensure that different slang terms for drugs like suboxone (sub, zub, bupe) or oxycodone (contin, oxy, perc) are mapped to a common term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Additionally, we normalize terms from all non-treatment categories with ex- amples such as muscle pain (tore a muscle, muscle aches, muscle tension), sobriety (sober), heart rate (high heart rate, low heart rate), and memory issues (memory problems, memory loss, short term memory problems).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' The steps and guidelines we followed to normalize keyphrases are illus- trated in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Identification of “Other” keyphrases Some keyphrases may be important to the post but were not in line with our research questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Singleton verbs, adjectives, or modifiers all fall into this category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Examples include, adjust, expect, acute, accidentally, expensive, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' We used manual inspec- tion to identify such keyphrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' These keyphrases were added to the “Other” category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Mapping unigram keyphrases with common stems We additionally perform stemming of keyphrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' All unigrams in our dataset that shared a common stem pattern were mapped to that single variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' For example, craving and cravings were generated from the stem crave, whereas detox is the stem for both detoxing and detoxification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Mapping semantically similar keyphrases An essential normalization step was to map all keyphrases with the same meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' The shorthand of keyphrases (pwd for precipi- tated withdrawal, dr for doctor) and misspelled keyphrases (depressssssed instead of depressed) fall into this category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Since slang and misspellings can arise in numerous varia- tions, we perform the semantic mapping manually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Included in this category are keyphrases consisting of more than one word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Some constituent words can dominate this type of keyphrase (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=', online clinic is dominated by clinic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Also, they may possess a different meaning than each of their con- stituent words (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=', precipitated withdrawal has a differ- ent meaning than precipitated and withdrawal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Thus proper mapping of such keyphrases was identified after careful manual inspection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Finally, keyphrases with similar mean- ings (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=', vomiting and puking) were also mapped in this step using human judgment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' It is worth mentioning that we tried to select the generic keyphrases as the representative ones while mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' For instance, the specific drug clon- azepam was assigned to the generic group benzodiazepines whereas the specialist sub-category of doctor endocrinolo- gist was mapped to the more general group doctor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' How- ever, we did not map some of the specific keyphrases (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=', suboxone, sublocade) to their generic group due to their fre- quent occurrences in the posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Multiple annotators reviewed all the manual tasks to avoid the potential subjective bias of manual inspections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Each conflicting case was resolved based on majority voting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Finally, an expert reviewed the whole normalization output list and confirmed the validity of the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Each normalized keyphrase is assigned to pre- defined categories after a manual review of the keyphrase’s context in a sample of posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Methods RQ1: What clinical insights can we draw from theme-driven opioid recovery-related keyphrases?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' This question aims to uncover potential clinical insights from self-reports on r/Suboxone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Specifically, we explore two questions in our annotated data: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' What insights can we draw from keyphrase occurrences and co-occurrences?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' We aim to provide a quantitative analysis of keyphrases in the SuboxoPhrase dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' First, we report the most frequently occurring keyphrases in opioid recovery dis- cussions, providing analysis on why specific keyphrases are more prevalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Then, we discuss the most frequent keyphrase categories in SuboxoPhrase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Category analysis allows us to draw high-level conclusions on discussion themes in r/Suboxone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' We also examine the frequency with which specific keyphrases co-occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Notably, we are interested in draw- ing clinical insights from co-occurrences between treat- ment options and psychophysical effects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=', suboxone, nausea) to probe the discussion issues about various treat- ment options at scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' This analysis could help uncover unknown side effects and reveal new treatment options unbeknownst to opioid recovery researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' How did the onset of the COVID-19 pandemic effect the discussion on opioid recovery on Reddit?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' To analyze this question, we investigate how keyphrase frequency changes over time—specifically concerning the onset of the pandemic, using the time-stamped data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' This information is useful to substance use clinicians and re- searchers who want to assess the effects of the pandemic on MOUD treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' RQ2: How does peer engagement associate with keyphrases in opioid recovery-related Reddit discourse?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' In recent years, Reddit has become a place for sharing re- covery experiences from various drug dependency issues and providing support to peers (Chancellor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' MacLean et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' The success of any peer discussion group heavily depends on the engagement between its mem- bers (Sharma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Unfortunately, very few works in NLP explore how users interact or seek support in opi- oid recovery discourse on Reddit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' This work aims to fill this knowledge gap by characterizing a user’s experience with Title Post Annotation-1 Annotation-2 JI Getting on suboxone I just came from Florida and have been clean from dope for 9 months but the cravings are setting in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Can anyone suggest me the best way to go about getting on suboxone?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' craving, heroin, clean, suboxone suboxone, craving, heroin 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='75 Tapering from 24mg Will I go through any withdrawals tapering off 24mg of suboxone if I taper down to 22mg?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Been on 24mg for 4 months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' suboxone, taper, withdrawal suboxone, taper, withdrawal 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='0 Table 2: Some sample excerpts with titles, posts, and annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Annotated keyphrases are normalized (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=', dope is normal- ized to heroin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' The Jaccard Index (JI) is calculated over the normalized keyphrase list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' opioid dependency and recovery through keyphrase analy- sis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' We explore the following research questions to draw in- sights from user engagement on r/Suboxone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Which keyphrase-containing posts provoke the most peer engagement?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Reddit facilitates discussion in a thread-like setting where a user starts the conversation by posting, and others in- teract via comments, upvotes, and downvotes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' We mod- eled the engagement of a post by counting comments and upvotes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' The engagement score for a given post is calcu- lated using the average number of comments or upvotes received.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' If a post has n keyphrases, each will receive a similar engagement score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' This allows us to inspect what thematic topics attract more users to a discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' How does peer engagement vary across posts with the most frequently used keyphrases in our data?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' We obtain two sets of keyphrases from our annotation efforts—the union and intersection of each annotator’s keyphrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' The union set comprises all the keyphrases marked by both annotators, while the intersection set only contains those keyphrases that exactly match both an- notators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' We chose the intersection set for analysis as both annotators agreed on these keyphrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' We select the top-most frequent keyphrases and empirically analyze the associated post to get data-driven insights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Specifically, we explore the association between frequent keyphrases and engagement scores to determine whether frequent keyphrases result in more peer engagement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' For both RQ1 and RQ2, our domain experts guided us to contextualize the results and conduct qualitative analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' RQ3: How effective are Off-The-Shelf Keyphrase Extraction models on theme-driven opioid recovery-related keyphrases?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' In this section, we aim to evaluate the performance of 10 off-the-shelf KE models on the SuboxoPhrase dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Statistical Methods: We evaluate the performance of two statistical methods, TfIdf (Manning and Prabhakar 2010) and YAKE (Campos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Graph-Based Methods: We explore four graph-based ap- proaches to keyphrase extraction, namely TextRank (Mihal- cea and Tarau 2004), TopicRank (Bougouin, Boudin, and Daille 2013), PositionRank (Florescu and Caragea 2017), and MultipartiteRank (Boudin 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' These classic mod- els are commonly used as baselines in modern UKE studies (Liang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Embedding Methods: All of our embedding methods use the following pipeline: 1) Extract candidate keyphrases 2) Embed each candidate and the full Reddit post 3) Return the top-k candidate keyphrases with the highest cosine similar- ity to the embedding of the entire Reddit post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' We evaluate the performance of DistilBERT (Sanh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' 2019), BERT (Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' 2018), DeBERTa-v3 (He, Gao, and Chen 2021), and SBERT (Reimers and Gurevych 2019), as they are all widely-used transformer models with proven ability to produce meaningful contextual word embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Experimental Setup: All models evaluated in this study are unsupervised, as the dataset is too small for supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' For the statistical and graph-based methods, we use the PKE library (Boudin 2016) to perform keyphrase extrac- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' For the embedding methods, we use KeyBERT (Groo- tendorst 2020) to extract and rank keyphrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' All mod- els use the same candidate keyphrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Specifically, we use PKE’s grammar selection tool to extract only noun phrases as candidates for each post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' We report the Recall@k for each model, where k represents the number of keyphrases returned by the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Results RQ1: What clinical insights can we draw from theme-driven opioid recovery-related keyphrases?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' What insights can we draw from keyphrase occur- rences?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' After normalization, we obtained a total of 881 unique keyphrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' The distribution of the keyphrases among different groups can be found in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' We ob- served that the highest frequency keyphrase category is psy- chophysical effects with 331 in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' The prevalence of psy- chophysical effects can be attributed to the fact that users discuss different types of effects they had faced due to the concurrent treatment or substance use, thus seeking sugges- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Withdrawal, sleep, and precipitated withdrawals were the top mentioned psychophysical effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' In contrast, we found the lowest number (35) of keyphrases in the medical history category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' The possible reason is that users focused on the ongoing problems or effects and did not mention medical history, or the reported medical histories were not often the keyphrase in a post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Examples of medical history include but are not limited to pregnancy, adhd, and bipolar disor- der.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Many users posted about treatment options (182), indi- cating that they were concerned about their treatments and thus tried different treatment options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Commonly used treat- ment options were suboxone, doctor, buprenorphine, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' We also found reasonable evidence (77) of keyphrases related to substance dependency & recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Frequently occurring keyphrases in this category are taper, heroin, oxycodone, and fentanyl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' The 256 keyphrases that did not belong to any of the four main categories were categorized as others, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=', clean, dissolve, water, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Our thematic analysis of frequent keyphrases is valuable for clinicians and researchers focusing on MOUD treat- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' For example, it is advantageous for researchers to understand which other treatment options patients consider while undergoing suboxone-based treatment for opioid re- covery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' This information can uncover potentially harmful trends in self-prescribed or new treatment options criti- cal to a patient’s recovery experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Similarly, quantify- ing the prevalence of various psychophysical effects can guide researchers toward emerging, potentially significant side effects which require attention from public health offi- cials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Identifying medical/substance use histories can inform about clinically relevant subpopulations seeking MOUD treatment who frequently engage on Reddit, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=', individ- uals with fentanyl or heroin dependency, pregnant women seeking MOUD treatment, or individuals interested in ta- pering their medications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' These findings can inform tailored intervention design for MOUD treatment, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=', who can be reached through Reddit and when.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' What insights can we draw from keyphrase co- occurrences?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Our analysis found that at least one instance of the keyphrase suboxone co-occurs with each of the known adult side-effects listed on the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Substance Abuse and Mental Health Service Administration webpage (SAMHSA 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' This relationship to standard substance abuse guide- lines validates the quality of our annotated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Addition- ally, such annotations allow one to gather many self-reports based on suboxone and a corresponding psychophysical ef- fect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' This facilitates analysis of the context and severity of various opioid-related issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' For example, one user posted: Is anyone getting heart or chest pain, back pain while breathing in sometimes and pain in right upper ab- domen randomly while you’re detoxing yourself with subs?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='.day 5 [and] still finding quite a few of these scary symptoms .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' 1st time I was kicking tranq dope so idk if it’s just that?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' or the subs somehow?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Here we find a user experiencing a physical side-effect of starting suboxone (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=', lingering withdrawal symptoms) being perceived as a general suboxone side-effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' MOUD researchers are interested in such cases as misperceptions in MOUD treatment can negatively impact treatment induc- tion, adherence, and retention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Additionally, we find many side effects co-occurring with suboxone which are not officially listed as known psy- chophysical responses to suboxone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' For example, almost 2% of user posts in SuboxoPhrase expressed issues with their sex drive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Other emerging co-occurring psychophysi- cal effects of suboxone include depression, depersonaliza- tion, anger, skin crawling, mood swings, hunger, and mem- ory issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Researchers interested in monitoring rare adverse drug reactions or unknown drug effects may benefit from KE co-occurrence analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Accurate extraction of keyphrases in recovery-related social media discourse can help iden- tify similar discussions and inform the design of qualitative, prospective studies to identify population-level perceptions and misperceptions related to MOUD treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' In addition to co-occurrence patterns with the keyphrase suboxone, there are intriguing patterns with tapering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Co- occurrence patterns—such as between tapering and with- drawal are unsurprising given that dose reduction usually induces withdrawal symptoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' More interesting is the co- occurrence pattern that tapering has with both sleep and anx- iety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Such significant associations present in SuboxoPhrase but not explored by the literature are primary candidates for exploration in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' How did the onset of the COVID-19 pandemic affect the discussion of opioid recovery treatments?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Crucial to our analysis of SuboxoPhrase is the temporal component of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Opioid use, abuse, and treatment are subject to and in- fluenced by global events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' One such global event that has had far-reaching impacts in many domains worldwide was the COVID-19 pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Illustrating this point, Currie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' (2021) found that, despite no notable changes in existing OUD treatment plans, admission of new patients to OUD treatments lagged behind predicted levels before rebound- ing in August 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' To uncover potential patterns around the pandemic, we divided the SuboxoPhrase dataset along the start date of the COVID-19 pandemic as declared by the WHO—the 11th of March, 2020 (Domenico and Vanelli 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' This allows us to make qualitative comparisons between the frequencies of keyphrases before and after the onset of the pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Figure 1: The relative percent increase or decrease in keyphrase frequencies after the onset of COVID-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' We can see a large increase in certain keyphrases, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=', fentanyl, and a notable decrease in certain keyphrases, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=', loperamide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' fentanyl prescription gabapentin sublocade high pill opioid oxycodone SICk precipitated withdrawals pain film tired methadone surgery constipation sweat Keyphrase headache rehab withdrawal xanax drug test insurance insomnia clonidine sleep stomach psychiatrist benzodiazepines back,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='pain symptoms painmedicine restless leg klonopin urine addiction volumetric dose sobriety testosterone loperamide 100%-50% 0% 50% 100% 150% 200% 250% Percent changeWe aim to analyze keyphrases with the highest relative per- centage increase or decrease,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' as such samples demonstrate the most considerable shift in the discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Figure 1 con- tains the top 20 keyphrases (each with a minimum of 5 oc- currences in the dataset) displaying the highest relative in- crease or decrease as shown in blue and red, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' After the start of the pandemic, the terms with the highest keyness for their posts shifted to the terms fentanyl, prescrip- tion, gabapentin, sublocade, and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Keyphrases that oc- cur less frequently following the COVID-19 pandemic in- clude loperamide, testosterone, addiction, and sobriety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' The reason for the reduced frequency of such terms may be ex- plained by the logistical challenges created by the pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' For example, testing testosterone levels require in-person visits and might explain a decision to delay such testing due to the COVID-19 restrictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Increases of terms such as prescription suggest an acceler- ation in new treatment plans and administration of MOUDs in the years after the pandemic (Currie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' How- ever, the most striking change by far is increased fentanyl mentions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Not only is there substantial evidence that fen- tanyl use has been growing over the last decade, but the pan- demic is also associated with an increase in fentanyl over- dose beyond projected levels (CDC 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Morin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' According to the CDC, there was a less extreme increase in 2021 compared to 2020 (CDC 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' The rise in pre- scription discussions likely stems from new access barriers around treatment, pharmacies, and users’ frustrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' These changes in crucial discussion subjects demonstrate the ex- tractive capabilities of a theme-driven KE approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Future models and larger datasets could guide policies and outreach efforts through real-time monitoring compared to surveys’ cost, time use, and lack of scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Figure 2: Top 20 engaging keyphrases in terms of the num- ber of comments (#comments).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Keyphrases related to psy- chophysical effects like mood swings, mental health, and melatonin (to cope with sleep trouble) spark more comment- based engagement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' RQ2: How does peer engagement associate with keyphrases in opioid recovery-related discourse?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Figure 3: Engagement score in terms of comments and up- votes of the top 20 most frequent keyphrases in the Subox- oPhrase dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' The frequency of a keyphrase is shown in parentheses, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=', heroin appears 52 times as a keyphrase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Which keyphrase-containing posts provoke the most peer engagement?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Figure 2 shows the top 20 engaging keyphrases regarding the number of comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' By empir- ically observing the keyphrases and associated posts, we found that more than 60% of engaging keyphrases are re- lated to psychophysical effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' People tend to involve in conversations that discuss potential side effects of with- drawals, such as mental health, body aches, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Other en- gaging posts include posts mentioning different treatment options to cope with withdrawal symptoms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=', melatonin for helping with sleep, bupropion, or cognitive behavioral therapy (CBT)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Due to space constraints, we omitted the graph corresponding to upvote-based engagement, which yielded results similar to comment-based engagement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' How does peer engagement vary across posts with the most frequently used keyphrases in our data?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Figure 3 shows the annotated dataset’s twenty most frequently mentioned keyphrases and their peer engagement score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Keyphrases related to treatment options (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=', suboxone, kratom) are more frequently used than psychophysical ef- fects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' The reason behind the frequent use of treatment- related keyphrases is that they generally co-occur with the mention of psychophysical effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' The mention of previous substance usage (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=', heroin, oxycodone) as a keyphrase is also significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' People mention their substance usage ex- perience to explain their situation while seeking recovery- related information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' From Figure 3, it is evident that engagement does not di- rectly correlate with the frequency of the keyphrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' If a keyphrase is frequent, it does not guarantee it will have a higher engagement score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' For example,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' frequent keyphrases such as buprenorphine,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' fentanyl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' opiate have low engage- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='mood swings ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='128 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='128 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='mental health ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='114 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='melatonin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='numb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='68 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='Top 20 engaging keyphrases ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='63 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='opioid addiction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='bupropion ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='62 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='irritation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='58 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='dizzy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='54 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='52 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='pregnancy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='cbt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='47 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='overdose ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='47 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='foggy head ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='47 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='body aches ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='45 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='leg cramps ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='41 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='gas pain ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='energy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='introversion ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='38 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='38 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='dark thoughts ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='37 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='grapefruit juice ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='robitussin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='37 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='Engagement score (#comments)30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='avg comments ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='avg_upvotes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='Engagement-score ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='suboxone (516) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='withdrawal (120) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='(52) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='(26) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='naloxone (18) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='depression (13) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='《(59) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='I (33) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='(33) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='(30) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='(29) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='(15) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='xanax (13) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='(63) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='(63) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='(45) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='(44) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='(15) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='7 ' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='heroin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='opiate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='J withdrawals ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='gabapentin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='benzodiazepines ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='clonidine ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} 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0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='52 Embedding Methods DistilBERT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='52 BERT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='52 DeBERTa-v3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='47 SBERT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='56 Table 3: Recall@K for each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Note: On average only 72% of annotated keyphrases appear in the candidate keyphrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Thus, the maximum recall is moderately low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' While less frequent keyphrases can be more engag- ing (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=', methadone, anxiety).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' The engagement trend of fre- quent keyphrases seems quite similar for comments and up- votes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' However, for specific keyphrases (xanax), the trend is the opposite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' In summary, the qualitative analysis uncov- ers that posts having keyphrases related to psychophysical effects/treatment options engage more users in direct inter- actions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=', comments/upvotes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' RQ3: How effective are Off-The-Shelf Keyphrase Extraction models on theme-driven opioid recovery-related keyphrases?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Our experimental results find that the SBERT embeddings produce the highest recall across all experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' This is ex- pected as 1) Embedding approaches are unique in produc- ing contextual representations of the full post, which explic- itly encode textual semantics—a useful feature in UKE 2) SBERT is the only embedding method employed which was trained to output embeddings for longer sequences of texts (other methods are only explicitly trained to produce only high-quality token embeddings).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Graph-based methods per- formed consistently lower than SBERT at all values of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' The method which achieved the second-best performance, TfIdf, displays the viability of a low computational cost method for use in resource-constrained settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Unfortunately, none of the standard UKE models achieved recall above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' This limits the use of off-the- shelf models to draw insights from silver-labeled samples across all of r/suboxone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Large-scale data collection us- ing our described annotation guidelines could enable su- pervised training, which may alleviate this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Addi- tionally, social media-specific UKE models may better han- dle long self-reports filled with informal user-generated lan- guage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Improving the performance of UKE models on Sub- oxoPhrase will be the focus of future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Ethical Considerations This research is conducted under Institutional Review Board (IRB) approval at the authors’ institution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Since Reddit users can make an account using only an email address (not dis- closed by Reddit), the site consists of anonymous users who can only be identified by data they voluntarily disclose in their posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Even if a user shares some self-identifying in- formation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=', location, age, gender identity), it is still nearly impossible to identify the true identity of a Reddit user without further information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Additionally, the Reddit user agreement requires users to consent to share their pub- lic posts/comments with the Reddit API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' If the user deletes a post, we do not include it in our dataset as it is the user’s right to remove information from the Reddit platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Additionally, posts included as exam- ples in the paper were paraphrased to prevent the reader from directly identifying the Reddit user account that posted each example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' This is a standard precautionary measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Broader Impact Implications for KE on Social Media: Existing works on keyphrase extraction in social media have two limitations: (1) Posts are derived from Twitter, where low character lim- its preclude capturing long-range dependency and rich infor- mation from long, complex colloquial text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Analysis of Sub- oxoPhrase requires understanding texts of variable length, some of which reach even 10,000 characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Thus Subox- oPhrase will promote the creation of better HealthNLP mod- els capable of modelling keyphrases in more extended so- cial media discourse on health.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' (2) Large Twitter datasets utilize hashtags as surrogate keywords—a strategy based on the error-prone assumption that hashtags are always indica- tors of keyness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' This is the first Reddit dataset with keywords extracted by human annotators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' This work thus provides re- liable annotation for clinically relevant keyphrase extraction from Reddit on MOUD-based treatment for opioid recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Implications for MOUD Research: SuboxoPhrase can in- form the development of clinician-facing tools that facilitate the discovery of tangible insights that can inform MOUD research and practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' For example, a SuboxoPhrase-based keyphrase extraction tool can help facilitate the discovery of the perceived effectiveness of different MOUD treatment options, strategies to cope with side effects, rare/new adverse drug reactions, and uncover patterns in the patient-reported experience with different MOUDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Such findings may guide future research in opioid recovery by clinicians and public health researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Also, results from such theme-driven KE may guide the development of tailored patient communica- tion tools and programs, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=', when and how to taper or po- tentially severe side effects of suboxone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Limitations and Future Work A limitation of this work is the insufficient amount of gold- labeled data, constraining supervised approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' As higher- performance UKEs are applied to the SuboxoPhrase dataset, it will be possible to extract reasonably accurate keyphrases from a much larger number of Reddit posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' This study demonstrates that standard off-the-shelf baseline models are inadequate for such analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Future works can focus on KEs better suited to the idiosyncrasies of Reddit posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Finally, we note that our four main keyphrase categories: Treatment Options, Substance Dependency & Recovery, Medical History, and Psychophysical Effects, are not ex- haustive and may be expanded in future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' We stress that this mixed methods study demonstrates the feasibility of our novel approach, and future works will benefit from a large volume of gold-labeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' We plan to investigate KEs on r/Suboxone to facilitate opioid research via an inter- active interface for researchers looking to uncover valuable opioid recovery patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' We also plan to scale up the analy- sis to other relevant subreddits, social media platforms, and other options for MOUD to extend this strategically impor- tant area of 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Creating a health tax- onomy with social media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} +page_content=' Proceedings of the International AAAI Conference on Web and Social Media 15(1):621–632.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FJT4oBgHgl3EQfUSzI/content/2301.11508v1.pdf'} diff --git a/fNE3T4oBgHgl3EQf3QsN/content/tmp_files/2301.04761v1.pdf.txt b/fNE3T4oBgHgl3EQf3QsN/content/tmp_files/2301.04761v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..7b130ecd4fcadae0de070ac1456b3798ca21dd00 --- /dev/null +++ b/fNE3T4oBgHgl3EQf3QsN/content/tmp_files/2301.04761v1.pdf.txt @@ -0,0 +1,673 @@ +NarrowBERT: Accelerating Masked Language Model +Pretraining and Inference +Haoxin Li1 and Phillip Keung2 and Daniel Cheng1 and Jungo Kasai1 and Noah A. Smith1 +Paul G. Allen School of Computer Science & Engineering, University of Washington, USA +1{lihaoxin,d0,jkasai,nasmith}@cs.washington.edu +2pkeung@uw.edu +Abstract +Large-scale language model pretraining is a +very successful form of self-supervised learn- +ing in natural language processing, but it is +increasingly expensive to perform as the mod- +els and pretraining corpora have become larger +over time. We propose NarrowBERT, a mod- +ified transformer encoder that increases the +throughput for masked language model pre- +training by more than 2×. NarrowBERT spar- +sifies the transformer model such that the self- +attention queries and feedforward layers only +operate on the masked tokens of each sentence +during pretraining, rather than all of the tokens +as with the usual transformer encoder. We also +show that NarrowBERT increases the through- +put at inference time by as much as 3.5× with +minimal (or no) performance degradation on +sentence encoding tasks like MNLI. Finally, +we examine the performance of NarrowBERT +on the IMDB and Amazon reviews classifica- +tion and CoNLL NER tasks and show that it +is also comparable to standard BERT perfor- +mance. +1 +Introduction +Pretrained masked language models, such as BERT +(Devlin et al., 2019), RoBERTa (Liu et al., 2019), +and DeBERTa (He et al., 2021), have pushed the +state-of-the-art in a wide range of downstream tasks +in natural language processing. At their core is the +transformer architecture (Vaswani et al., 2017) that +consists of interleaved self-attention and feedfor- +ward sublayers. Since the former sublayer implies +quadratic time complexity in the input sequence +length (Vaswani et al., 2017), many have proposed +methods to make the self-attention computation +more efficient (Katharopoulos et al., 2020; Choro- +manski et al., 2021; Wang et al., 2020; Peng et al., +2021, 2022, inter alia). +In this work, we explore an orthogonal approach +to efficiency: can we make masked language mod- +els efficient by reducing the length of the input se- +quence that each layer needs to process? In particu- +lar, pretraining by masked language modeling only +involves prediction of masked tokens (typically, +only 15% of the input tokens; Devlin et al., 2019; +Liu et al., 2019). Despite this sparse pretraining +objective, each transformer layer computes a repre- +sentation for every token. In addition to pretraining, +many downstream applications only use a single +vector representation (i.e., only the [CLS] token) +for prediction purposes, which is much smaller than +the number of input tokens (e.g., sequence classifi- +cation tasks as in GLUE/SuperGLUE; Wang et al., +2018, 2019). By narrowing the input sequence for +transformer layers, we can accelerate both pretrain- +ing and inference. +We present NarrowBERT, a new architecture +that takes advantage of the sparsity in the training +objective. We present two NarrowBERT meth- +ods in the sections that follow (Figure 1). We +provide the code to reproduce our experiments at +redacted-during-review. The first method re- +duces the input sequence for the feedforward sub- +layers by reordering the interleaved self-attention +and feedforward sublayers in the standard trans- +former architecture (Press et al., 2020): +after +two standard, interleaved transformer layers, self- +attention sublayers are first applied, followed only +by feedforward sublayers. This way, the feedfor- +ward sublayer computations are only performed +for masked tokens, resulting in a 1.3× speedup in +pretraining (§3). The second approach reduces the +input length to the attention sublayers: queries are +only computed for masked tokens in the attention +mechanism (Bahdanau et al., 2015), while the keys +and values are not re-computed for non-masked +tokens, which leads to a greater than 2× speedup +in pretraining. +We extensively evaluate our efficient pretrained +models on well-established downstream tasks (e.g., +Wang et al., 2018; Tjong Kim Sang and De Meul- +der, 2003.) We find that our modifications result +arXiv:2301.04761v1 [cs.CL] 11 Jan 2023 + +(a) {6,sf} model: standard BERT with the transformer encoder, trained on MLM loss. +(b) sf{5,s}:{5,f} ContextFirst model: Transformer encoder with re-ordered layers. Attentional contextualization is performed +all-at-once near the beginning of the model. +(c) sf:{5,sf} SparseQueries model: Transformer encoder with sparsified queries. Contextualization is focused on [MASK] +tokens only. (See Fig. 2.) +Figure 1: Examples of standard BERT and NarrowBERT variations. NarrowBERT takes advantage of the spar- +sity in the masking (i.e., only 15% of tokens need to be predicted) to reduce the amount of computation in the +transformer encoder. +in almost no drop in downstream performance, +while providing substantial pretraining and infer- +ence speedups (§3). While efficient attention vari- +ants are promising research directions, this work +presents a different and simple approach to mak- +ing transformers efficient, with minimal changes in +architecture. +2 +NarrowBERT +In Figures 1b and 1c, we illustrate two variations +of NarrowBERT. We define some notation to de- +scribe the configuration of our models. s refers to +a single self-attention layer and f refers to a sin- +gle feedforward layer. The colon : refers to the +‘narrowing’ operation, which gathers the masked +positions from the output of the previous layer. +The first variation (‘ContextFirst’ in Fig. 1b) +uses attention to contextualize all-at-once at the +beginning of the model. In short, the transformer +layers have been rearranged to frontload the atten- +tion components. The example given in the fig- +ure specifies the model as sf{5,s}:{5,f}, which +means that the input sentence is encoded by a self- +attention layer, a feedforward layer, and 5 consecu- +tive self-attention layers. At that point, the masked +positions from the encoded sentence are gathered +into a tensor and passed through 5 feedforward lay- +ers, thereby avoiding further computations for +all non-masked tokens. Finally, the masked posi- +tions are unmasked and the MLM loss is computed. +The second variation (‘SparseQueries’ in Fig. 1c) +does not reorder the layers at all. Instead, the +sf:{5,sf} model contextualizes the input sen- +tence in a more limited way. As shown in Figure +2, the input sentence is first contextualized by a s +and a f layer, but the non-masked tokens are never +contextualized again afterwards. Only the masked +tokens are contextualized by the remaining {5,sf} +layers. +Since the masked tokens are only about 15% +of the total sentence length, the potential speedup +is ~6.6× for every feedforward or attention layer +downstream of a narrowing : operation. The mem- +ory usage can also decrease by ~6.6× for those lay- +ers since the sequence length has decreased, which +allows us to use larger batch sizes during training. +For GLUE, Amazon, and IMDB text classifica- +tion tasks, only the [CLS] token is used for predic- +tion. When we finetune or predict with ContextFirst +on a GLUE/Amazon/IMDB task, the feedforward +layers only need to operate on the [CLS] token. +When we finetune or predict with SparseQueries, + +W1 +. +W6 +. +[MASK] +Ws +W4 +s +S +S +S +S +W3 +. +[MASK] +W2 +W +.W7 +W5 +W6 +f +f +W2 +[MASK] +W4 +S +S +S +S +S +S +W3 +[MASK] +W1W7 +W5 +W6 +f +s +f +s +f +s +f +S +s +W2 +[MASK] +W4 +s +W3 +[MASK] +W1Figure 2: Sparse queries in the attention layers. Only the masked positions are contextualized as query vectors in +subsequent s layers. The inputs are contextualized once by the first s layer and f layer, and reused as the keys and +values in all subsequent attention layers. +Pretrain +Finetune +Inference +GLUE +Speedup +Speedup +Speedup +MNLI +QNLI +SST2 +STS-B +QQP +WNLI +Baseline BERT ({12,sf}) +1× +1× +1× +0.83 +0.91 +0.93 +0.89 +0.87 +0.56 +Funnel Transformer (B4-4-4) +0.88× +0.86× +0.78× +0.78 +0.87 +0.88 +0.86 +0.86 +0.56 +ContextFirst (sfsf{10,s}:{10,f}) +1.33× +1.24× +1.64× +0.82 +0.90 +0.91 +0.89 +0.87 +0.56 +SparseQueries: +{1,sf}:{11,sf} +2.47× +4.73× +4.64× +0.77 +0.87 +0.89 +0.84 +0.80 +0.56 +{2,sf}:{10,sf} +2.34× +2.82× +3.49× +0.81 +0.88 +0.91 +0.88 +0.87 +0.59 +{3,sf}:{9,sf} +2.15× +2.43× +2.79× +0.81 +0.89 +0.91 +0.86 +0.87 +0.56 +{4,sf}:{8,sf} +1.63× +2.13× +2.33× +0.82 +0.88 +0.91 +0.89 +0.87 +0.57 +Table 1: Test scores on various GLUE tasks. (‘MNLI’ scores refer to the MNLI matched dev set.) Finetuning and +inference speedups refer to speeds on the MNLI task. +only the [CLS] token is used in the queries of the +attention layers. Everything after the narrowing : +operation only operates on the [CLS] token, which +dramatically speeds up the NarrowBERT variants. +3 +Experiments +We focus on 2 models in our experiments: +ContextFirst (sfsf{10,s}:{10,f}) and Sparse- +Queries ({1,sf}:{11,sf}, · · · , {4,sf}:{8,sf}). +Our NarrowBERT models all contain 12 self- +attention and 12 feedforward layers in total, with +the narrowing operation used at different points +in the model. We compare NarrowBERT with +the baseline BERT model and the Funnel Trans- +former model (Dai et al., 2020), which is a pre- +trained encoder-decoder transformer model where +the encoder goes through a sequence of length bot- +tlenecks. +In our experiments, we use 15% masking in +masked language model (MLM) training. +Fol- +lowing Liu et al. (2019), we do not use next sen- +tence prediction as a pretraining task. We use large +batch sizes and high learning rates to fully utilize +GPU memory, as suggested in Izsak et al. (2021). +Batches are sized to be the largest that fit in GPU +memory. We use a learning rate of 0.0005. Models +are trained for 70k steps, where each step contains +1728 sequences of 512 tokens, and gradient accu- +mulation is used to accumulate the minibatches +needed per step. Models were trained on hosts +with 8 Nvidia A100 GPUs. We used the Hugging +Face implementations of the baseline BERT and +Funnel Transformer models. We pretrained the +baseline BERT, Funnel Transformer, and Narrow- +BERT models using the same Wikipedia and Books +corpora and total number of steps. +In Figure 3, we see the evolution of the develop- +ment MLM loss over the course of model training. +The BERT and NarrowBERT models all converge +to similar values, with the NarrowBERT models +reaching a slightly higher MLM loss near the end +of training. +We report the accuracy for MNLI (Williams +et al., 2018), QNLI (Rajpurkar et al., 2016), SST2 +(Socher et al., 2013), WNLI (Levesque et al., +2012), IMDB (Maas et al., 2011), and English +Amazon reviews (Keung et al., 2020), F1 for +QQP (Sharma et al., 2019) and CoNLL-2003 NER +(Tjong Kim Sang and De Meulder, 2003), and +Spearman correlation for STS-B (Cer et al., 2017). + +[MASK] tokens only +hs +queries +queries +h +f +S +S +h2 +h2 +h2 +↑ keys, values +↑ keys, values +h1 +h1 +hi +W7 +ha +ha +ha +W6 +hg +hs +hs +[MASK] +W4 +f +h4 +h4 +h4 +W3 +h3 +h3 +hg +[MASK] +h2 +h2 +h2 +W1 +h1 +h1 +h1(a) All training steps. +(b) Near the end of training. +Figure 3: Development MLM loss over the course of pretraining. At the end of training, the BERT, ContextFirst, +and SparseQueries ({2,sf}:{10,sf}) dev MLM losses are 1.41, 1.43, and 1.47 respectively. +CoNLL NER +IMDB +Amazon2 +Amazon5 +Baseline BERT ({12,sf}) +0.90 +0.93 +0.96 +0.66 +Funnel Transformer +0.87 +0.92 +0.95 +0.65 +ContextFirst (sfsf{10,s}:{10,f}) +0.89 +0.93 +0.95 +0.65 +SparseQueries: +{1,sf}:{11,sf} +0.87 +0.91 +0.94 +0.65 +{2,sf}:{10,sf} +0.89 +0.91 +0.95 +0.65 +{3,sf}:{9,sf} +0.89 +0.92 +0.95 +0.65 +{4,sf}:{8,sf} +0.89 +0.93 +0.95 +0.65 +Table 2: Test scores on CoNLL NER, IMDB, binarized Amazon reviews, and 5-star Amazon reviews tasks. +For the Amazon reviews corpus, we consider both +the usual 5-star prediction task and the binarized +(i.e., 1-2 stars versus 4-5 stars) task. +In Table 1, we present the results for our extrinsic +evaluation on various GLUE tasks. The reduction +in performance is small or non-existent, and on +WNLI, the NarrowBERT variations perform better +than the baseline. For SparseQueries, it is clear that +using more layers prior to the narrowing operation +improves performance, though the training and in- +ference speedups become smaller. We note that the +Funnel Transformer implementation in Pytorch is +slower than the baseline BERT model; this may be +due to the fact that the original implementation was +written in Tensorflow and optimized for Google +TPUs.1 +In Table 2, we provide results on the IMDB +and Amazon reviews classification tasks and the +CoNLL NER task. Generally, NarrowBERT is +close to the baseline in performance, and the +SparseQueries performance improves as more lay- +ers are used before the narrowing operation. +It is well known that the variability in the per- +formance of BERT on certain GLUE tasks is ex- +treme (Mosbach et al., 2020; Dodge et al., 2020; +Lee et al., 2019), where the differences in perfor- +1See https://github.com/laiguokun/Funnel-Transformer. In +their paper, the Funnel Transformer authors claim to have a +finetuning FLOPs that is 0.58× of the BERT baseline’s. +mance between finetuning runs can exceed 20% +(absolute). We have also observed this extreme +variability in the course of our own GLUE fine- +tuning experiments. While many techniques have +been proposed to address this issue, it is not the +goal of this work to apply finetuning stabilization +methods to maximize BERT’s performance. For +this reason, we have excluded the RTE, MRPC, and +COLA tasks (which are high-variance tasks studied +in the aforementioned papers) from our evaluation. +4 +Discussion and Conclusion +We have explored two straightforward ways of ex- +ploiting the sparsity in the masked language model +loss: rearranging the layers of the transformer +encoder to allow the feedforward components to +avoid computations on the non-masked positions, +and sparsifying the queries in the attention mech- +anism to only contextualize the masked positions. +The NarrowBERT variants can speed up training +by a factor of ~2× and inference by a factor of +~3×, while maintaining very similar performance +on GLUE, IMDB, Amazon, and CoNLL NER tasks. +Based on the favorable trade-off between speed +and performance seen in Section 3, we recommend +that practitioners consider using the SparseQueries +NarrowBERT model with 2 or 3 layers before nar- +rowing. + +Oss +BERT +0. +3 +ContextFirst +SparseQueries +> +0 +2 +M +0. +0 +10000 +20000 +30000 +40000 +50000 +60000 +70000 +StepsOss +8 +BERT +ContextFirst +9 +SparseQueries +4. +ML +2 +40000 +45000 +50000 +55000 +60000 +65000 +70000 +StepsLimitations +Due to our budget constraint, we only performed +pretraining and downstream experiments with base- +sized transformer models. We also only applied the +masked language modeling objective, but there are +other effective pretraining objectives (e.g., Clark +et al., 2020). Nonetheless, since we introduced +minimal changes in architecture, we hope that sub- +sequent work will benefit from our narrowing oper- +ations and conduct a wider range of pretraining and +downstream experiments. While pretrained models +can be applied to even more downstream tasks, we +designed a reasonable task suite in this work, con- +sisting of both GLUE sentence classification and +the CoNLL NER sequential classification tasks. +References +Dzmitry Bahdanau, Kyunghyun Cho, and Yoshua Ben- +gio. 2015. +Neural machine translation by jointly +learning to align and translate. 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In Proc. +of NAACL. + diff --git a/fNE3T4oBgHgl3EQf3QsN/content/tmp_files/load_file.txt b/fNE3T4oBgHgl3EQf3QsN/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f1776c7cbec6be583f5ff073b8f94fb20dee2903 --- /dev/null +++ b/fNE3T4oBgHgl3EQf3QsN/content/tmp_files/load_file.txt @@ -0,0 +1,414 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf,len=413 +page_content='NarrowBERT: Accelerating Masked Language Model Pretraining and Inference Haoxin Li1 and Phillip Keung2 and Daniel Cheng1 and Jungo Kasai1 and Noah A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' Smith1 Paul G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' Allen School of Computer Science & Engineering, University of Washington, USA 1{lihaoxin,d0,jkasai,nasmith}@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='washington.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='edu 2pkeung@uw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='edu Abstract Large-scale language model pretraining is a very successful form of self-supervised learn- ing in natural language processing, but it is increasingly expensive to perform as the mod- els and pretraining corpora have become larger over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' We propose NarrowBERT, a mod- ified transformer encoder that increases the throughput for masked language model pre- training by more than 2×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' NarrowBERT spar- sifies the transformer model such that the self- attention queries and feedforward layers only operate on the masked tokens of each sentence during pretraining, rather than all of the tokens as with the usual transformer encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' We also show that NarrowBERT increases the through- put at inference time by as much as 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='5× with minimal (or no) performance degradation on sentence encoding tasks like MNLI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' Finally, we examine the performance of NarrowBERT on the IMDB and Amazon reviews classifica- tion and CoNLL NER tasks and show that it is also comparable to standard BERT perfor- mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' 1 Introduction Pretrained masked language models, such as BERT (Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=', 2019), RoBERTa (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=', 2019), and DeBERTa (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=', 2021), have pushed the state-of-the-art in a wide range of downstream tasks in natural language processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' At their core is the transformer architecture (Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=', 2017) that consists of interleaved self-attention and feedfor- ward sublayers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' Since the former sublayer implies quadratic time complexity in the input sequence length (Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=', 2017), many have proposed methods to make the self-attention computation more efficient (Katharopoulos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' Choro- manski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=', 2021, 2022, inter alia).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' In this work, we explore an orthogonal approach to efficiency: can we make masked language mod- els efficient by reducing the length of the input se- quence that each layer needs to process?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' In particu- lar, pretraining by masked language modeling only involves prediction of masked tokens (typically, only 15% of the input tokens;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' Despite this sparse pretraining objective, each transformer layer computes a repre- sentation for every token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' In addition to pretraining, many downstream applications only use a single vector representation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=', only the [CLS] token) for prediction purposes, which is much smaller than the number of input tokens (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=', sequence classifi- cation tasks as in GLUE/SuperGLUE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=', 2018, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' By narrowing the input sequence for transformer layers, we can accelerate both pretrain- ing and inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' We present NarrowBERT, a new architecture that takes advantage of the sparsity in the training objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' We present two NarrowBERT meth- ods in the sections that follow (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' We provide the code to reproduce our experiments at redacted-during-review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' The first method re- duces the input sequence for the feedforward sub- layers by reordering the interleaved self-attention and feedforward sublayers in the standard trans- former architecture (Press et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=', 2020): after two standard, interleaved transformer layers, self- attention sublayers are first applied, followed only by feedforward sublayers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' This way, the feedfor- ward sublayer computations are only performed for masked tokens, resulting in a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='3× speedup in pretraining (§3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' The second approach reduces the input length to the attention sublayers: queries are only computed for masked tokens in the attention mechanism (Bahdanau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=', 2015), while the keys and values are not re-computed for non-masked tokens, which leads to a greater than 2× speedup in pretraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' We extensively evaluate our efficient pretrained models on well-established downstream tasks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=', Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' Tjong Kim Sang and De Meul- der, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=') We find that our modifications result arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='04761v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='CL] 11 Jan 2023 (a) {6,sf} model: standard BERT with the transformer encoder, trained on MLM loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' (b) sf{5,s}:{5,f} ContextFirst model: Transformer encoder with re-ordered layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' Attentional contextualization is performed all-at-once near the beginning of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' (c) sf:{5,sf} SparseQueries model: Transformer encoder with sparsified queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' Contextualization is focused on [MASK] tokens only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' (See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=') Figure 1: Examples of standard BERT and NarrowBERT variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' NarrowBERT takes advantage of the spar- sity in the masking (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=', only 15% of tokens need to be predicted) to reduce the amount of computation in the transformer encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' in almost no drop in downstream performance, while providing substantial pretraining and infer- ence speedups (§3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' While efficient attention vari- ants are promising research directions, this work presents a different and simple approach to mak- ing transformers efficient, with minimal changes in architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' 2 NarrowBERT In Figures 1b and 1c, we illustrate two variations of NarrowBERT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' We define some notation to de- scribe the configuration of our models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' s refers to a single self-attention layer and f refers to a sin- gle feedforward layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' The colon : refers to the ‘narrowing’ operation, which gathers the masked positions from the output of the previous layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' The first variation (‘ContextFirst’ in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' 1b) uses attention to contextualize all-at-once at the beginning of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' In short, the transformer layers have been rearranged to frontload the atten- tion components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' The example given in the fig- ure specifies the model as sf{5,s}:{5,f}, which means that the input sentence is encoded by a self- attention layer, a feedforward layer, and 5 consecu- tive self-attention layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' At that point, the masked positions from the encoded sentence are gathered into a tensor and passed through 5 feedforward lay- ers, thereby avoiding further computations for all non-masked tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' Finally, the masked posi- tions are unmasked and the MLM loss is computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' The second variation (‘SparseQueries’ in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' 1c) does not reorder the layers at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' Instead, the sf:{5,sf} model contextualizes the input sen- tence in a more limited way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' As shown in Figure 2, the input sentence is first contextualized by a s and a f layer, but the non-masked tokens are never contextualized again afterwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' Only the masked tokens are contextualized by the remaining {5,sf} layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' Since the masked tokens are only about 15% of the total sentence length, the potential speedup is ~6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='6× for every feedforward or attention layer downstream of a narrowing : operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' The mem- ory usage can also decrease by ~6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='6× for those lay- ers since the sequence length has decreased, which allows us to use larger batch sizes during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' For GLUE, Amazon, and IMDB text classifica- tion tasks, only the [CLS] token is used for predic- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' When we finetune or predict with ContextFirst on a GLUE/Amazon/IMDB task, the feedforward layers only need to operate on the [CLS] token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' When we finetune or predict with SparseQueries, W1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' W6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' [MASK] Ws W4 s S S S S W3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' [MASK] W2 W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='W7 W5 W6 f f W2 [MASK] W4 S S S S S S W3 [MASK] W1W7 W5 W6 f s f s f s f S s W2 [MASK] W4 s W3 [MASK] W1Figure 2: Sparse queries in the attention layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' Only the masked positions are contextualized as query vectors in subsequent s layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' The inputs are contextualized once by the first s layer and f layer, and reused as the keys and values in all subsequent attention layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' Pretrain Finetune Inference GLUE Speedup Speedup Speedup MNLI QNLI SST2 STS-B QQP WNLI Baseline BERT ({12,sf}) 1× 1× 1× 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='87 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='56 Funnel Transformer (B4-4-4) 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='87 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='56 SparseQueries: {1,sf}:{11,sf} 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='47× 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='73× 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='64× 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='87 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='56 {2,sf}:{10,sf} 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='59 {3,sf}:{9,sf} 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='15× 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='43× 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='79× 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='87 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='56 {4,sf}:{8,sf} 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='63× 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='13× 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='33× 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='87 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='57 Table 1: Test scores on various GLUE tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' (‘MNLI’ scores refer to the MNLI matched dev set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=') Finetuning and inference speedups refer to speeds on the MNLI task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' only the [CLS] token is used in the queries of the attention layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' Everything after the narrowing : operation only operates on the [CLS] token, which dramatically speeds up the NarrowBERT variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' 3 Experiments We focus on 2 models in our experiments: ContextFirst (sfsf{10,s}:{10,f}) and Sparse- Queries ({1,sf}:{11,sf}, · · · , {4,sf}:{8,sf}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' Our NarrowBERT models all contain 12 self- attention and 12 feedforward layers in total, with the narrowing operation used at different points in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' We compare NarrowBERT with the baseline BERT model and the Funnel Trans- former model (Dai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=', 2020), which is a pre- trained encoder-decoder transformer model where the encoder goes through a sequence of length bot- tlenecks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' In our experiments, we use 15% masking in masked language model (MLM) training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' Fol- lowing Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' (2019), we do not use next sen- tence prediction as a pretraining task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' We use large batch sizes and high learning rates to fully utilize GPU memory, as suggested in Izsak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' Batches are sized to be the largest that fit in GPU memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' We use a learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='0005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' Models are trained for 70k steps, where each step contains 1728 sequences of 512 tokens, and gradient accu- mulation is used to accumulate the minibatches needed per step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' Models were trained on hosts with 8 Nvidia A100 GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' We used the Hugging Face implementations of the baseline BERT and Funnel Transformer models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' We pretrained the baseline BERT, Funnel Transformer, and Narrow- BERT models using the same Wikipedia and Books corpora and total number of steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' In Figure 3, we see the evolution of the develop- ment MLM loss over the course of model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' The BERT and NarrowBERT models all converge to similar values, with the NarrowBERT models reaching a slightly higher MLM loss near the end of training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' We report the accuracy for MNLI (Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=', 2018), QNLI (Rajpurkar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=', 2016), SST2 (Socher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=', 2013), WNLI (Levesque et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=', 2012), IMDB (Maas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=', 2011), and English Amazon reviews (Keung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=', 2020), F1 for QQP (Sharma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=', 2019) and CoNLL-2003 NER (Tjong Kim Sang and De Meulder, 2003), and Spearman correlation for STS-B (Cer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' [MASK] tokens only hs queries queries h f S S h2 h2 h2 ↑ keys, values ↑ keys, values h1 h1 hi W7 ha ha ha W6 hg hs hs [MASK] W4 f h4 h4 h4 W3 h3 h3 hg [MASK] h2 h2 h2 W1 h1 h1 h1(a) All training steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' (b) Near the end of training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' Figure 3: Development MLM loss over the course of pretraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' At the end of training, the BERT, ContextFirst, and SparseQueries ({2,sf}:{10,sf}) dev MLM losses are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='41, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='43, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='47 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' CoNLL NER IMDB Amazon2 Amazon5 Baseline BERT ({12,sf}) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='66 Funnel Transformer 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='87 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='65 ContextFirst (sfsf{10,s}:{10,f}) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='65 SparseQueries: {1,sf}:{11,sf} 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='87 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='65 {2,sf}:{10,sf} 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='65 {3,sf}:{9,sf} 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='65 {4,sf}:{8,sf} 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='65 Table 2: Test scores on CoNLL NER, IMDB, binarized Amazon reviews, and 5-star Amazon reviews tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' For the Amazon reviews corpus, we consider both the usual 5-star prediction task and the binarized (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=', 1-2 stars versus 4-5 stars) task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' In Table 1, we present the results for our extrinsic evaluation on various GLUE tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' The reduction in performance is small or non-existent, and on WNLI, the NarrowBERT variations perform better than the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' For SparseQueries, it is clear that using more layers prior to the narrowing operation improves performance, though the training and in- ference speedups become smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' We note that the Funnel Transformer implementation in Pytorch is slower than the baseline BERT model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' this may be due to the fact that the original implementation was written in Tensorflow and optimized for Google TPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='1 In Table 2, we provide results on the IMDB and Amazon reviews classification tasks and the CoNLL NER task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' Generally, NarrowBERT is close to the baseline in performance, and the SparseQueries performance improves as more lay- ers are used before the narrowing operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' It is well known that the variability in the per- formance of BERT on certain GLUE tasks is ex- treme (Mosbach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' Dodge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=', 2019), where the differences in perfor- 1See https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='com/laiguokun/Funnel-Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' In their paper, the Funnel Transformer authors claim to have a finetuning FLOPs that is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='58× of the BERT baseline’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' mance between finetuning runs can exceed 20% (absolute).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' We have also observed this extreme variability in the course of our own GLUE fine- tuning experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' While many techniques have been proposed to address this issue, it is not the goal of this work to apply finetuning stabilization methods to maximize BERT’s performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' For this reason, we have excluded the RTE, MRPC, and COLA tasks (which are high-variance tasks studied in the aforementioned papers) from our evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' 4 Discussion and Conclusion We have explored two straightforward ways of ex- ploiting the sparsity in the masked language model loss: rearranging the layers of the transformer encoder to allow the feedforward components to avoid computations on the non-masked positions, and sparsifying the queries in the attention mech- anism to only contextualize the masked positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' The NarrowBERT variants can speed up training by a factor of ~2× and inference by a factor of ~3×, while maintaining very similar performance on GLUE, IMDB, Amazon, and CoNLL NER tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' Based on the favorable trade-off between speed and performance seen in Section 3, we recommend that practitioners consider using the SparseQueries NarrowBERT model with 2 or 3 layers before nar- rowing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' Oss BERT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' 3 ContextFirst SparseQueries > 0 2 M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' 0 10000 20000 30000 40000 50000 60000 70000 StepsOss 8 BERT ContextFirst 9 SparseQueries 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' ML 2 40000 45000 50000 55000 60000 65000 70000 StepsLimitations Due to our budget constraint, we only performed pretraining and downstream experiments with base- sized transformer models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' We also only applied the masked language modeling objective, but there are other effective pretraining objectives (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=', Clark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' Nonetheless, since we introduced minimal changes in architecture, we hope that sub- sequent work will benefit from our narrowing oper- ations and conduct a wider range of pretraining and downstream experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' While pretrained models can be applied to even more downstream tasks, we designed a reasonable task suite in this work, con- sisting of both GLUE sentence classification and the CoNLL NER sequential classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' References Dzmitry Bahdanau, Kyunghyun Cho, and Yoshua Ben- gio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' of EMNLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' Erik F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' Tjong Kim Sang and Fien De Meulder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' Introduction to the CoNLL-2003 shared task: Language-independent named entity recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' of CoNLL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' Gomez, Łukasz Kaiser, and Illia Polosukhin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' Attention is all you need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' of NeurIPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' Alex Wang, Yada Pruksachatkun, Nikita Nangia, Amanpreet Singh, Julian Michael, Felix Hill, Omer Levy, and Samuel Bowman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' SuperGLUE: A stickier benchmark for general-purpose language un- derstanding systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' of BlackboxNLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' Sinong Wang, Belinda Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' Li, Madian Khabsa, Han Fang, and Hao Ma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' Linformer: Self-attention with linear complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' Adina Williams, Nikita Nangia, and Samuel R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' Bow- man.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' A broad-coverage challenge corpus for sentence understanding through inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} +page_content=' of NAACL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE3T4oBgHgl3EQf3QsN/content/2301.04761v1.pdf'} diff --git a/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf b/h9AzT4oBgHgl3EQfbPwA/content/2301.01380v1.pdf new file 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Haselschwardt,1, a B.G. Lenardo,2, b T. Daniels,3 S.W. Finch,4 +F.Q.L. Friesen,4 C.R. Howell,4 C.R. Malone,4, c E. Mancil,4 and W. Tornow4 +1Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA +2SLAC National Accelerator Laboratory, 2575 Sand Hill Rd, Menlo Park, CA 94025, USA +3Department of Physics and Physical Oceanography, +University of North Carolina at Wilmington, Wilmington, NC 28403, USA +4Department of Physics, Duke University, and Triangle Universities Nuclear Laboratory (TUNL), Durham, NC 27708, USA +(Dated: January 30, 2023) +We report on new measurements establishing the existence of low-lying isomeric states in 136Cs +using γ rays produced in 136Xe(p,n)136Cs reactions. Two states with O(100) ns lifetimes are placed +in the decay sequence of the 136Cs levels that are populated in charged-current interactions of +solar neutrinos and fermionic dark matter with 136Xe. +Xenon-based experiments can therefore +exploit a delayed-coincidence tag of these interactions, greatly suppressing backgrounds to enable +spectroscopic studies of solar neutrinos and dark matter. +Introduction – Future xenon-based experiments search- +ing for dark matter and neutrinoless double-beta decay +(0νββ) will deploy roughly 5–100 tonnes of target mass. +These detectors will provide unprecedentedly powerful +searches for rare interactions owing to their large tar- +gets, extremely low intrinsic backgrounds, and event re- +construction capabilities. In this work we present mea- +surements which make possible the identification of a new +class of events in these detectors – charged-current (CC) +interactions on 136Xe nuclei – which can enable novel +studies of low-energy solar neutrinos and provide un- +precedented sensitivity to certain models of dark matter. +We measure the precise energies and lifetimes of low-lying +excited states in 136Cs populated by CC interactions and +identify two new isomeric states that will enable these +signals to be tagged using delayed coincidences in mod- +ern experiments. +The CC “neutrino capture” process, famously ex- +ploited by Ray Davis in the first detection of solar neutri- +nos [1], will be observable in real time in xenon detectors +as νe + A +54Xe → A +55Cs(∗) + e−, where a xenon nucleus is +converted into a (possibly excited) cesium nucleus via a +Gamow-Teller transition (∆J = 1). The signal gener- +ated in the detector is the combination of the outgoing +electron and any γ rays/conversion electrons emitted as +the Cs nucleus relaxes to its ground state. Measurement +of the energies of the final state particles in this reac- +tion provides event-by-event neutrino energy reconstruc- +tion, in contrast to the elastic scattering of neutrinos on +electrons or nuclei [2–4]. The total energy deposited in +the detector is Eν − Q, where Q is the reaction thresh- +a scotthaselschwardt@lbl.gov +b Corresponding author: blenardo@slac.stanford.edu +c Present address: Savannah River National Laboratory, Aiken, +SC 29802, USA +old equal to the mass difference between the Xe and Cs +isobars. If Q is low enough, this reaction can be used +to search for neutrinos from the solar carbon-nitrogen- +oxygen (CNO) cycle, an elusive signal which has only +been observed in one experiment to date [5, 6] and which +plays a crucial role in determining the solar metallicity. +This reaction can also provide a unique measurement of +7Be neutrinos, which may enable novel measurements of +temperature of the solar core [7]. A variety of targets and +techniques have been proposed to detect the lower-energy +solar neutrino components in this way [8–13]; however, +none have yet been realized at scale. +The same final state can be used to search for CC ab- +sorption of MeV-scale fermionic dark matter on nuclei: +χ + A +54Xe → A +55Cs(∗) + e− [14, 15]. In this model, χ car- +ries lepton number and interacts with Standard Model +particles via a new gauge boson W ′. A recent analysis of +data from a low-background xenon time projection cham- +ber (TPC) yields constraints that are competitive with +state-of-the-art collider searches [16], and the sensitivity +of future xenon-based searches will improve dramatically +with improved background suppression. +As an isotope which undergoes ββ decay, 136Xe is par- +ticularly well-suited as a CC reaction target [9]. It fea- +tures a low threshold of Q = 90.3 keV and relatively +large cross section due to the sizable Gamow-Teller tran- +sition strengths [17, 18] connecting the 0+ 136Xe ground +state and the lowest-lying 1+ excited states of 136Cs near +590 keV and 850 keV. The possible experimental signa- +tures from neutrino capture on 136Xe and their detection +in liquid xenon (LXe) TPCs are discussed in Ref. [19]. +Using shell-model (SM) calculations, that work predicts +that the 136Cs 1+ states preferentially relax through an +isomeric state, setting up a delayed-coincidence signa- +ture that would allow unambiguous identification of CC +interactions, potentially enabling background-free mea- +surements of these signals in current- and next-generation +arXiv:2301.11893v1 [nucl-ex] 27 Jan 2023 + +2 +detectors. These opportunities depend critically on the +level structure of 136Cs below ∼590 keV, and particularly +on the as-yet-unknown γ-ray emission properties of the +lowest-lying states. +Until recently the low-lying level structure of 136Cs +has been relatively unexplored [20]. Current data stems +mainly from two experimental surveys focused on pro- +viding inputs for the calculation of nuclear matrix ele- +ments for ββ decay of 136Xe. First, the high-resolution +136Xe(3He, t) measurements reported in Ref. [17] estab- +lished the spectrum of 1+ states of interest here and +provide a measurement of the Gamow-Teller transition +strengths to these states from the 0+ 136Xe ground state. +Second, a campaign using the 138Ba(d,α) reaction re- +ported on the existence of several new low-lying states +in 136Cs [21–24]. There is only one study of excitations +near the ground state in 136Cs using γ rays, which estab- +lished a single 4+ level at 104.8(3) keV relevant for CC +events [25]. +In this paper we report on an experiment to charac- +terize the energies and lifetimes of the low-lying states +in 136Cs using γ rays produced in (p,n) reactions on +136Xe. We identify many new nuclear transitions, several +of which have O(100) ns lifetimes. Using γ-γ coincidences +we reconstruct a level scheme which describes the decay +of the 1+ states of interest and identify isomeric states +which will produce the delayed-coincidence signature re- +quired for low-background study of CC neutrino and dark +matter interactions in xenon-based experiments. +Data collection and analysis – Measurements were per- +formed using the tandem accelerator at the Triangle Uni- +versities Nuclear Laboratory (TUNL). A pulsed beam of +7 MeV protons with period of 1.6 µs and typical pulse +width of 2 ns was directed through a target cell contain- +ing xenon gas enriched to 94% in 136Xe. The beam cur- +rent on target was kept between 5–15 nA throughout the +run, giving a Xe(p,n)Cs reaction rate of ∼1/pulse. The +cell volume was a 1.3 cm tall, 1.6 cm diameter cylinder +with 25 µm thick polyethylene naphthalate (PEN) films +serving as beam entrance and exit windows. +The cell +was housed in a cylindrical aluminum vacuum chamber +with inner diameter 30.5 cm and nominal wall thickness +of 6.4 mm. The inner wall of the chamber was lined with +1-mm thick lead, except for the sections through which +γ-ray detectors viewed the target. +Gamma rays were measured by four high-purity ger- +manium (HPGe) detectors placed outside the vacuum +chamber: two 60% relative efficiency coaxial detectors, +which provided high detection efficiency for γ rays be- +tween 50–3000 keV, and two planar low-energy photon +spectrometers (LEPS), which provided O(10) ns time +resolution and sensitivity between 10–600 keV. To en- +sure high-quality measurements of γ rays below 100 keV, +one LEPS detector viewed the target through a 127 µm- +thick Kapton window and was positioned 51.3 cm from +the target center. +All other detectors viewed the tar- +get through the wall of the target chamber at distances +of 24.5 cm (second LEPS), 27.1 cm, and 27.2 cm (two +coaxial detectors) from the target center. Detectors were +shielded from the beam dump and beamline components +upstream of the target using tungsten and lead placed +inside and outside the chamber. A Mesytec MDPP-16 +digitizer was used to record the energy and time for each +γ ray detected by the HPGe detectors. The time refer- +ence was provided by a capacitive sensor through which +the proton beam passed before entering the target cham- +ber. +The energy scale and efficiency of each detector were +calibrated using a combination of standard 133Ba, 137Cs, +60Co γ-ray sources, a mixed source containing 241Am, +57Co, 54Mn, 65Zn, and known transitions from the β− +decay of 136Cs in the target1. The LEPS detectors’ tim- +ing capabilities were calibrated using the 308 keV iso- +meric transition in 48V (τ = 10.26 ± 0.06 ns [27]) in a +dedicated run of 48Ti(p,n) reactions on a natTi foil tar- +get. +We measure a lifetime of 10.6 ± 0.4 ns, in good +agreement with the accepted value. +Drifts in detector gain and resolution were tracked us- +ing a high-intensity peak observed at 239.9 keV. During +the course of data taking the energy scale varied by at +most 0.8%. A correction is applied to each data set to +align this peak with its value measured immediately fol- +lowing source calibrations. +Results – A typical energy spectrum of single-hit events +in one of the LEPS detectors is shown in Fig. 1. We set +an analysis threshold at 50 keV, below which the spec- +trum is dominated by background from proton-induced +X-ray emission in the Xe target. We identify 65 γ ray +lines between 50–1100 keV from previously unobserved +transitions in 136Cs and measure their energies with an +uncertainty of 0.1 keV. A complete list is available in +the Supplementary Material. These signals can be fur- +ther separated into “prompt” – defined here to be within +40 ns of the beam pulse – or “delayed”. Eight of the new +transitions feature time distributions that extend beyond +the prompt window and decay over periods of O(100) ns. +These, along with steady-state backgrounds from target +activation, appear in the delayed spectrum in Fig. 1. In +addition to the transitions reported here for the first time, +we observe the known lines at 104.8(3) and 517.9(1) keV +associated with direct transitions to the ground state [25], +with the latter appearing constant in time due to the long +lifetime of the parent 8− level (τ = 25.2(3) s). +The low-lying level scheme of +136Cs is constructed +through analysis of γ-γ coincidence events. A 1 µs win- +1 During this procedure, it was discovered that the 66.881 keV +γ from the ground state β− decay of 136Cs has an incorrectly +assigned intensity in the present evaluation [20]. +This evalua- +tion provides Iγ = 4.79(20), whereas our data are in excellent +agreement with Iγ = 12.5(1) as reported in Ref. [26]. + +3 +: 66.9, 86.4, 153.2, 163.9, 176.6, 273.6 keV +W & Pb X-rays : 58.0, 59.3, 72.8, 75.0, 84.9 keV +136Cs β− decay γ′ s +FIG. 1: Energy spectrum of events in the near LEPS +detector. The black curve shows the spectrum obtained +for hits at all times relative to the beam. The blue +curve displays those arriving at least 40 ns after the +beam pulse and shows the presence of delayed γ’s. Grey +arrows indicate lines from 136Cs β− decay with +intensity larger than 1/decay and X-rays from W and +Pb shielding. +dow is used to construct two-dimensional coincidence ma- +trices of the energy in each detector. The identical ma- +trix of accidental coincidences for a given detector pair, +formed by delaying one detector’s signal by one beam +pulse period, is subtracted off. The resulting matrix and +associated coincidence gates are analyzed using the Rad- +Ware program [28]. +We primarily make use of coinci- +dences between the two coaxial detectors due to their +similar efficiencies; however, coincidences between the +closest LEPS detector and the coaxial detectors are used +as a cross check. +The proposed level scheme based on our data is shown +in Fig. 2. +For comparison, we show the SM predic- +tions from Ref. [19] as well as the spectrum measured +via 138Ba(d,α) [24] and 136Xe(3He,t) reactions [17, 18]. +Nearly all of our reconstructed levels can be mapped di- +rectly to levels observed in previous experiments. How- +ever, the high-resolution γ-ray measurements in this work +enable us to reduce the uncertainties in the energy levels +by an order of magnitude. In addition, we reconstruct +a doublet in the vicinity of 420 keV which is unresolved +in charged-particle-based experiments and reported here +for the first time. One of these two states plays a cru- +cial role in CC interactions, as discussed in more detail +below. +Of the eight transitions that extend into the delayed +region, the three at 66.6, 73.7, and 105.0 keV are mea- +sured with sufficient strength in γ-γ coincidences to be +placed in our level scheme. Their background-subtracted +time distributions, shown in Fig. 3, show features con- +sistent with their relative placements. The delayed com- +ponents of the 66.6 keV and 105.0 keV transitions are +This +Work +5+ +0.0 +73.7 157(4) ns +4+ +105.0 +140.3 90(5) ns +313.6 +422.1 +424.1 +459.7 +588.8 +8- +517.9 +448.6 +166.7 +517.9 +386.1 +146.1 +350.5 +319.2 +348.6 +108.5 +239.9 +208.6 +66.6 +105.0 +73.7 +Shell +Model +5+ +3+ +4+ +2+ +3+ +2+ +3+ +4+ +8- +4+ +3+ +6- +1+ +6- +138Ba(d,α) +& +136Xe(3He,t) +5+ +3+ +4+ +3+ +(4+) +(4+) +(3+) +(2+), +(3+) +8- +1+ +1+, +FIG. 2: The rightmost column shows our proposed level +scheme for 136Cs with energies given in keV and +measured state lifetimes. Only levels up to 590 keV are +included here: a scheme using all observed coincidences +is given in the Supplemental Material. Blue lines show +measured γ-ray transitions, where those drawn as +dashed lines have a relatively weak coincidence intensity +but are observed in the singles spectrum. The left +column shows the spectrum of states predicted by the +shell model of Ref. [19]. For comparison, we show levels +previously measured using 138Ba(d,α) [24] and +136Xe(3He,t) [17, 18] reactions in the middle column +with their assigned Jπ values shown in green and red, +respectively. +well described by a single exponential distribution with +the same lifetime. The 105.0 component also contains a +significant prompt component. This indicates that the +105.0 → G.S. decay itself is prompt, and that the ob- +served exponential distribution comes from the isomeric +140.3 keV state feeding the 105.0 level through a 35.3 keV +transition which falls below our analysis threshold. Se- +lecting only 105.0 keV γ’s that occur in coincidence with +the 319 keV transition (thereby skipping the 140.3 keV +level) produces a time distribution with no delayed com- +ponent, supporting this conclusion. The 73.7 keV transi- +tion is placed between the lowest-lying excited state and +the ground state. As such, it can either be fed by the +66.6 keV transition (from the long-lived 140.3 keV state) +or by other (prompt) transitions. Components to model +each of these scenarios are included in the fit to its time +distribution. +The fit lifetime for the 73.7 keV state is +τ = 157 ± 4 ns and is the longest observed in our exper- + +4 +iment. +The relative γ-ray intensities can be used to estimate +branching fractions for various decay schemes. The ratio +of observed intensities for the 448.6 keV and 166.7 keV +transitions is 18.2:7, indicating that the first 1+ level +decays via the transitions 588.8 → 140.3 and 588.8 → +422.1 approximately 70% and 30% of the time, respec- +tively, subject to a ∼5% uncertainty due to the un- +known internal conversion (IC) coefficient of each tran- +sition. The former directly feeds the isomeric state at +140.3 keV, guaranteeing that it reaches the ground state +through at least one long-lived state. Of the latter, the +(208.6 keV):(239.9 keV) intensity ratio indicates that +approximately 2% of decays proceed via the 422.1 → +313.6 → 105.0 sequence to the ground state composed +of purely prompt transitions; the remainder proceed +through the isomeric state at 73.7 keV. This leads us to +the conclusion that approximately 99% of events which +populate the lowest-lying 1+ level will relax through at +least one isomeric transition, and possibly two. +The +branching of decays from the 140.3 keV state remains +uncertain; the relative (delayed 105.0 keV):66.6 keV in- +tensities give a ratio of 1:4, however, the multipolarity +and/or IC coefficients of these low-energy transitions will +need to be measured independently to translate this into +a branching fraction. +The proposed level scheme can be compared to pre- +dictions from the SM. The model shown in Fig. 2 uses +the NuShellX@MSU [29] code with the SN100PN effec- +tive interaction from Ref [30]. It correctly describes the +band structures and ordering of yrast states at high spin +(J ≥ 8) [31] and predicts the correct spin-parity of 5+ +for the 136Cs ground state. In the context of CC inter- +actions on 136Xe, a key feature of the SM level structure +is the presence of two 2+ states at 83 and 224 keV, to +which the lowest 1+ state is predicted to decay promptly +with 69% and 31% branching fractions, respectively [19]. +Neither has been conclusively observed to date. One can- +didate is our proposed level at 422.1 keV, which is placed +through strong, two-fold coincidences of γ rays at 108.5, +239.8, and 73.7 keV, and which is fed directly by the +first 1+ state via the 166.7 keV transition. A 2+ level +at this energy would be unresolved in 138Ba(d,α) reac- +tions from the (4+) state observed at 423 ± 3 keV [24], +which we in turn associate with our reconstructed level +at 424.1 keV level due to its strong connection with the +known 4+ state at 105.0 keV via coincidences of the 105.0 +and 319.2 keV γ’s. +Another candidate is the level at +140.2 keV, which is fed directly by the first 1+ state via +the 588.8 → 140.2 → 73.7 → G.S. sequence that is re- +constructed by our observed coincidences of the 448.6, +66.6, and 73.7 keV transitions. +However, the angular +distributions in 138Ba(d,α) reactions favor Jπ = 3+ for +the level observed at 140 ± 3 keV, and we do not find +any evidence for multiple states in the vicinity of this +energy, leaving this interpretation uncertain. +Finally, +FIG. 3: Background-subtracted γ-ray arrival times of +the three delayed transitions found in our level scheme. +The first two corresponding to the 105 keV and 67 keV +γ’s are fit with exponential and constant background +terms only, while the fit to the 74 keV transition +contains a term which accounts for the additional delay +caused by preceding 67 keV transitions, in accordance +with the proposed level scheme. X-rays from Pb result +in the prompt peak observed in the 74 keV timing +distribution. +a state at 432(2) keV, first observed in Ref. [18] and +given a tentative spin-parity assignment of (3+), was +recently identified as a possible (2+) state in Ref. [24]. +This state is not reconstructed by any coincidences in +the present work, and we do not observe any transitions +near 157 keV that would connect this state with the 1+ +state at 588.8 keV. Thus our data are in tension with +the more recent spin-party assignment. Further measure- +ments with both charged-particle and γ-ray detection will +be ideal for fully characterizing this structure. +Discussion – In a xenon-based detector the isomeric +states measured here will produce a unique signature for +the identification of CC interactions. The prompt inter- +action, composed of the emitted electron and the initial +relaxation of the Cs nucleus, will almost always be fol- +lowed by the delayed emission of one or more ∼ 100 keV +γ’s/IC electrons. +In detectors which measure scintil- +lation this will produce a characteristic two- or three- +pulse signal, where the time structure of each pulse is +determined by the convolution of three components: (1) +the photoemission time constant(s) of the scintillating +medium, (2) the photon transport time in the detec- +tor, and (3) the photosensor and electronics response +times. In the case of (1), there are three technologies cur- + +Counts +10 +105.0 keV +t= 90 + 5 ns +10 +10 +66.6 keV +t = 90 ± 5 ns +10° +10 +73.7 keV +t = 157 ± 4 ns +10 +10 +0 +200 +400 +600 +800 +1000 +Time [ns]5 +TABLE I: Event rates for 7Be and CNO solar neutrino +capture in current (marked with an ∗) and planned +experiments which deploy 136Xe and the previously +proposed target isotopes 130Te and 100Mo. Rates are +given in solar neutrino units (SNU) for each isotope and +events per year for each experiment. The mass column +indicates the mass of isotope deployed, as opposed to +the total payload of the experiment. Scattering rates on +130Te and 100Mo are taken from Ref. [41], while those +on 136Xe are from Ref. [19]. +Isotope +Rate (SNU) +Experiment +Mass +Rate (evt/yr) +7Be +CNO +(t) +7Be +CNO +136Xe +42.5 +6.3 +LZ∗ +0.62 +3.7 +0.54 +KamLAND-Zen∗ +0.68 +4.0 +0.59 +nEXO +3.2 +19.0 +2.8 +80t natXe TPC +7.1 +42.1 +6.2 +130Te +23.2 +3.7 +SNO+ (0.5% Te) +1.3 +4.4 +0.70 +100Mo +126 +15 +CUPID +0.25 +6.1 +0.72 +rently being pursued for experiments at the tonne-scale +and beyond: LXe TPCs [32–34], high-pressure gaseous +xenon (GXe) TPCs [35], and loaded liquid scintilla- +tor (LS) detectors [36], which have emission time con- +stants of τLXe = 27 ns [37, 38], τGXe = 4 ns [39], and +τLS = 6 ns [40], respectively. For most modern experi- +ments (2) is O(10) ns and (3) is a design parameter that +can be as low as O(1) ns. Given the lifetimes we have +measured, pile-up of the scintillation pulses is a concern +for robust reconstruction of their energies, and a coinci- +dence analysis would realistically wait some small time +before opening a window in which to search for a forth- +coming pulse. Experiments which minimize the impact +of (3) will benefit from higher signal efficiency. +Other ββ-decay isotopes have been proposed as fa- +vorable targets for solar neutrino capture. +Suggested +schemes for their deployment include metal-loaded LS [8, +9, 12] or solid-state sensors [10, 11]. In Table I, we com- +pare current and proposed xenon-based experiments with +the two most massive experiments of each type planned +for the near future2,3: the SNO+ [43] and CUPID [44] +experiments, which will search for 0νββ in 130Te and +100Mo, respectively. The solar neutrino CC interaction +rates in these next-generation experiments are compa- +rable to those in currently-operating experiments which +deploy 136Xe. In the future, LXe TPC experiments such +as nEXO [32] and a next-generation dark matter experi- +ment [34] offer a promising avenue for spectroscopic stud- +ies of solar neutrinos enabled by the measurements pre- +sented here. +2 See e.g. Ref. [42]. +3 The ββ isotope 76Ge is not considered here as the CC-scattering +rate on this isotope is negligible in the present context. +Within the scope of existing experiments, KamLAND- +Zen recently reported the largest exposure of 136Xe to +date, totaling 1 tonne-year [36]. The fast intrinsic timing +of the experiment’s organic LS is advantageous, though +identification of delayed pulses from the 105, 74, or +67 keV transitions may require dedicated assessment of +experiment’s energy threshold and the rate of 14C back- +ground in this region. The LZ [45] experiment features a +very low (∼1 keV) threshold with a very low background +rate at ∼100 keV and is expected to amass a 1.7 tonne- +year exposure of 136Xe by the end of its 1000-day physics +run. In addition to searching for the roughly 10 expected +solar neutrino events in these experiments, a search for +the delayed-coincidence signature described here would +provide world-leading sensitivity to MeV-scale fermionic +dark matter and could be implemented immediately. +In summary, we have measured the energies and life- +times of low-lying states in 136Cs. The isomeric states +measured here enable the use of a delayed-coincidence +tag to uniquely identify CC interactions in experiments +which deploy 136Xe. 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' CA 94025,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' USA 3Department of Physics and Physical Oceanography,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' University of North Carolina at Wilmington,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' Wilmington,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' NC 28403,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' USA 4Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' Duke University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' and Triangle Universities Nuclear Laboratory (TUNL),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' Durham,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' NC 27708,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' USA (Dated: January 30,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' 2023) We report on new measurements establishing the existence of low-lying isomeric states in 136Cs using γ rays produced in 136Xe(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='n)136Cs reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' Two states with O(100) ns lifetimes are placed in the decay sequence of the 136Cs levels that are populated in charged-current interactions of solar neutrinos and fermionic dark matter with 136Xe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' Xenon-based experiments can therefore exploit a delayed-coincidence tag of these interactions, greatly suppressing backgrounds to enable spectroscopic studies of solar neutrinos and dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' Introduction – Future xenon-based experiments search- ing for dark matter and neutrinoless double-beta decay (0νββ) will deploy roughly 5–100 tonnes of target mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' These detectors will provide unprecedentedly powerful searches for rare interactions owing to their large tar- gets, extremely low intrinsic backgrounds, and event re- construction capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' In this work we present mea- surements which make possible the identification of a new class of events in these detectors – charged-current (CC) interactions on 136Xe nuclei – which can enable novel studies of low-energy solar neutrinos and provide un- precedented sensitivity to certain models of dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' We measure the precise energies and lifetimes of low-lying excited states in 136Cs populated by CC interactions and identify two new isomeric states that will enable these signals to be tagged using delayed coincidences in mod- ern experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' The CC “neutrino capture” process, famously ex- ploited by Ray Davis in the first detection of solar neutri- nos [1], will be observable in real time in xenon detectors as νe + A 54Xe → A 55Cs(∗) + e−, where a xenon nucleus is converted into a (possibly excited) cesium nucleus via a Gamow-Teller transition (∆J = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' The signal gener- ated in the detector is the combination of the outgoing electron and any γ rays/conversion electrons emitted as the Cs nucleus relaxes to its ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' Measurement of the energies of the final state particles in this reac- tion provides event-by-event neutrino energy reconstruc- tion, in contrast to the elastic scattering of neutrinos on electrons or nuclei [2–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' The total energy deposited in the detector is Eν − Q, where Q is the reaction thresh- a scotthaselschwardt@lbl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='gov b Corresponding author: blenardo@slac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='stanford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='edu c Present address: Savannah River National Laboratory, Aiken, SC 29802, USA old equal to the mass difference between the Xe and Cs isobars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' If Q is low enough, this reaction can be used to search for neutrinos from the solar carbon-nitrogen- oxygen (CNO) cycle, an elusive signal which has only been observed in one experiment to date [5, 6] and which plays a crucial role in determining the solar metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' This reaction can also provide a unique measurement of 7Be neutrinos, which may enable novel measurements of temperature of the solar core [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' A variety of targets and techniques have been proposed to detect the lower-energy solar neutrino components in this way [8–13];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' however, none have yet been realized at scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' The same final state can be used to search for CC ab- sorption of MeV-scale fermionic dark matter on nuclei: χ + A 54Xe → A 55Cs(∗) + e− [14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' In this model, χ car- ries lepton number and interacts with Standard Model particles via a new gauge boson W ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' A recent analysis of data from a low-background xenon time projection cham- ber (TPC) yields constraints that are competitive with state-of-the-art collider searches [16], and the sensitivity of future xenon-based searches will improve dramatically with improved background suppression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' As an isotope which undergoes ββ decay, 136Xe is par- ticularly well-suited as a CC reaction target [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' It fea- tures a low threshold of Q = 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='3 keV and relatively large cross section due to the sizable Gamow-Teller tran- sition strengths [17, 18] connecting the 0+ 136Xe ground state and the lowest-lying 1+ excited states of 136Cs near 590 keV and 850 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' The possible experimental signa- tures from neutrino capture on 136Xe and their detection in liquid xenon (LXe) TPCs are discussed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' Using shell-model (SM) calculations, that work predicts that the 136Cs 1+ states preferentially relax through an isomeric state, setting up a delayed-coincidence signa- ture that would allow unambiguous identification of CC interactions, potentially enabling background-free mea- surements of these signals in current- and next-generation arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='11893v1 [nucl-ex] 27 Jan 2023 2 detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' These opportunities depend critically on the level structure of 136Cs below ∼590 keV, and particularly on the as-yet-unknown γ-ray emission properties of the lowest-lying states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' Until recently the low-lying level structure of 136Cs has been relatively unexplored [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' Current data stems mainly from two experimental surveys focused on pro- viding inputs for the calculation of nuclear matrix ele- ments for ββ decay of 136Xe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' First, the high-resolution 136Xe(3He, t) measurements reported in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' [17] estab- lished the spectrum of 1+ states of interest here and provide a measurement of the Gamow-Teller transition strengths to these states from the 0+ 136Xe ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' Second, a campaign using the 138Ba(d,α) reaction re- ported on the existence of several new low-lying states in 136Cs [21–24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' There is only one study of excitations near the ground state in 136Cs using γ rays, which estab- lished a single 4+ level at 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='8(3) keV relevant for CC events [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' In this paper we report on an experiment to charac- terize the energies and lifetimes of the low-lying states in 136Cs using γ rays produced in (p,n) reactions on 136Xe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' We identify many new nuclear transitions, several of which have O(100) ns lifetimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' Using γ-γ coincidences we reconstruct a level scheme which describes the decay of the 1+ states of interest and identify isomeric states which will produce the delayed-coincidence signature re- quired for low-background study of CC neutrino and dark matter interactions in xenon-based experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' Data collection and analysis – Measurements were per- formed using the tandem accelerator at the Triangle Uni- versities Nuclear Laboratory (TUNL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' A pulsed beam of 7 MeV protons with period of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='6 µs and typical pulse width of 2 ns was directed through a target cell contain- ing xenon gas enriched to 94% in 136Xe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' The beam cur- rent on target was kept between 5–15 nA throughout the run, giving a Xe(p,n)Cs reaction rate of ∼1/pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' The cell volume was a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='3 cm tall, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='6 cm diameter cylinder with 25 µm thick polyethylene naphthalate (PEN) films serving as beam entrance and exit windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' The cell was housed in a cylindrical aluminum vacuum chamber with inner diameter 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='5 cm and nominal wall thickness of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='4 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' The inner wall of the chamber was lined with 1-mm thick lead, except for the sections through which γ-ray detectors viewed the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' Gamma rays were measured by four high-purity ger- manium (HPGe) detectors placed outside the vacuum chamber: two 60% relative efficiency coaxial detectors, which provided high detection efficiency for γ rays be- tween 50–3000 keV, and two planar low-energy photon spectrometers (LEPS), which provided O(10) ns time resolution and sensitivity between 10–600 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' To en- sure high-quality measurements of γ rays below 100 keV, one LEPS detector viewed the target through a 127 µm- thick Kapton window and was positioned 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='3 cm from the target center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' All other detectors viewed the tar- get through the wall of the target chamber at distances of 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='5 cm (second LEPS), 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='1 cm, and 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='2 cm (two coaxial detectors) from the target center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' Detectors were shielded from the beam dump and beamline components upstream of the target using tungsten and lead placed inside and outside the chamber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' A Mesytec MDPP-16 digitizer was used to record the energy and time for each γ ray detected by the HPGe detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' The time refer- ence was provided by a capacitive sensor through which the proton beam passed before entering the target cham- ber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' The energy scale and efficiency of each detector were calibrated using a combination of standard 133Ba, 137Cs, 60Co γ-ray sources, a mixed source containing 241Am, 57Co, 54Mn, 65Zn, and known transitions from the β− decay of 136Cs in the target1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' The LEPS detectors’ tim- ing capabilities were calibrated using the 308 keV iso- meric transition in 48V (τ = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='26 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='06 ns [27]) in a dedicated run of 48Ti(p,n) reactions on a natTi foil tar- get.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' We measure a lifetime of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='4 ns, in good agreement with the accepted value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' Drifts in detector gain and resolution were tracked us- ing a high-intensity peak observed at 239.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='9 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' During the course of data taking the energy scale varied by at most 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='8%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' A correction is applied to each data set to align this peak with its value measured immediately fol- lowing source calibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' Results – A typical energy spectrum of single-hit events in one of the LEPS detectors is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' We set an analysis threshold at 50 keV, below which the spec- trum is dominated by background from proton-induced X-ray emission in the Xe target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' We identify 65 γ ray lines between 50–1100 keV from previously unobserved transitions in 136Cs and measure their energies with an uncertainty of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='1 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' A complete list is available in the Supplementary Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' These signals can be fur- ther separated into “prompt” – defined here to be within 40 ns of the beam pulse – or “delayed”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' Eight of the new transitions feature time distributions that extend beyond the prompt window and decay over periods of O(100) ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' These, along with steady-state backgrounds from target activation, appear in the delayed spectrum in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' In addition to the transitions reported here for the first time, we observe the known lines at 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='8(3) and 517.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='9(1) keV associated with direct transitions to the ground state [25], with the latter appearing constant in time due to the long lifetime of the parent 8− level (τ = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='2(3) s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' The low-lying level scheme of 136Cs is constructed through analysis of γ-γ coincidence events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' A 1 µs win- 1 During this procedure, it was discovered that the 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='881 keV γ from the ground state β− decay of 136Cs has an incorrectly assigned intensity in the present evaluation [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' This evalua- tion provides Iγ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='79(20), whereas our data are in excellent agreement with Iγ = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='5(1) as reported in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' 3 : 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='9, 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='4, 153.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='2, 163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='9, 176.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='6, 273.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='6 keV W & Pb X-rays : 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='0, 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='3, 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='8, 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='0, 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='9 keV 136Cs β− decay γ′ s FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' 1: Energy spectrum of events in the near LEPS detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' The black curve shows the spectrum obtained for hits at all times relative to the beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' The blue curve displays those arriving at least 40 ns after the beam pulse and shows the presence of delayed γ’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' Grey arrows indicate lines from 136Cs β− decay with intensity larger than 1/decay and X-rays from W and Pb shielding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' dow is used to construct two-dimensional coincidence ma- trices of the energy in each detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' The identical ma- trix of accidental coincidences for a given detector pair, formed by delaying one detector’s signal by one beam pulse period, is subtracted off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' The resulting matrix and associated coincidence gates are analyzed using the Rad- Ware program [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' We primarily make use of coinci- dences between the two coaxial detectors due to their similar efficiencies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' however, coincidences between the closest LEPS detector and the coaxial detectors are used as a cross check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' The proposed level scheme based on our data is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' For comparison, we show the SM predic- tions from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' [19] as well as the spectrum measured via 138Ba(d,α) [24] and 136Xe(3He,t) reactions [17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' Nearly all of our reconstructed levels can be mapped di- rectly to levels observed in previous experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' How- ever, the high-resolution γ-ray measurements in this work enable us to reduce the uncertainties in the energy levels by an order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' In addition, we reconstruct a doublet in the vicinity of 420 keV which is unresolved in charged-particle-based experiments and reported here for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' One of these two states plays a cru- cial role in CC interactions, as discussed in more detail below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' Of the eight transitions that extend into the delayed region, the three at 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='6, 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='7, and 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='0 keV are mea- sured with sufficient strength in γ-γ coincidences to be placed in our level scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' Their background-subtracted time distributions, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' 3, show features con- sistent with their relative placements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' The delayed com- ponents of the 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='6 keV and 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='0 keV transitions are This Work 5+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='0 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='7 157(4) ns 4+ 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='0 140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='3 90(5) ns 313.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='6 422.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='1 424.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='1 459.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='7 588.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='8 8- 517.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='9 448.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='6 166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='7 517.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='9 386.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='1 146.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='1 350.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='5 319.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='2 348.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='6 108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='5 239.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='9 208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='6 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='6 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='0 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='7 Shell Model 5+ 3+ 4+ 2+ 3+ 2+ 3+ 4+ 8- 4+ 3+ 6- 1+ 6- 138Ba(d,α) & 136Xe(3He,t) 5+ 3+ 4+ 3+ (4+) (4+) (3+) (2+), (3+) 8- 1+ 1+, FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' 2: The rightmost column shows our proposed level scheme for 136Cs with energies given in keV and measured state lifetimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' Only levels up to 590 keV are included here: a scheme using all observed coincidences is given in the Supplemental Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' Blue lines show measured γ-ray transitions, where those drawn as dashed lines have a relatively weak coincidence intensity but are observed in the singles spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' The left column shows the spectrum of states predicted by the shell model of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' For comparison, we show levels previously measured using 138Ba(d,α) [24] and 136Xe(3He,t) [17, 18] reactions in the middle column with their assigned Jπ values shown in green and red, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' well described by a single exponential distribution with the same lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' The 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='0 component also contains a significant prompt component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' This indicates that the 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='0 → G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' decay itself is prompt, and that the ob- served exponential distribution comes from the isomeric 140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='3 keV state feeding the 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='0 level through a 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='3 keV transition which falls below our analysis threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' Se- lecting only 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='0 keV γ’s that occur in coincidence with the 319 keV transition (thereby skipping the 140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='3 keV level) produces a time distribution with no delayed com- ponent, supporting this conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' The 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='7 keV transi- tion is placed between the lowest-lying excited state and the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' As such, it can either be fed by the 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='6 keV transition (from the long-lived 140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='3 keV state) or by other (prompt) transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' Components to model each of these scenarios are included in the fit to its time distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' The fit lifetime for the 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='7 keV state is τ = 157 ± 4 ns and is the longest observed in our exper- 4 iment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' The relative γ-ray intensities can be used to estimate branching fractions for various decay schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' The ratio of observed intensities for the 448.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='6 keV and 166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='7 keV transitions is 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='2:7, indicating that the first 1+ level decays via the transitions 588.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='8 → 140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='3 and 588.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='8 → 422.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='1 approximately 70% and 30% of the time, respec- tively, subject to a ∼5% uncertainty due to the un- known internal conversion (IC) coefficient of each tran- sition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' The former directly feeds the isomeric state at 140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='3 keV, guaranteeing that it reaches the ground state through at least one long-lived state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' Of the latter, the (208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='6 keV):(239.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='9 keV) intensity ratio indicates that approximately 2% of decays proceed via the 422.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='1 → 313.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='6 → 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='0 sequence to the ground state composed of purely prompt transitions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' the remainder proceed through the isomeric state at 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='7 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' This leads us to the conclusion that approximately 99% of events which populate the lowest-lying 1+ level will relax through at least one isomeric transition, and possibly two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' The branching of decays from the 140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='3 keV state remains uncertain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' the relative (delayed 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='0 keV):66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='6 keV in- tensities give a ratio of 1:4, however, the multipolarity and/or IC coefficients of these low-energy transitions will need to be measured independently to translate this into a branching fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' The proposed level scheme can be compared to pre- dictions from the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' The model shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' 2 uses the NuShellX@MSU [29] code with the SN100PN effec- tive interaction from Ref [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' It correctly describes the band structures and ordering of yrast states at high spin (J ≥ 8) [31] and predicts the correct spin-parity of 5+ for the 136Cs ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' In the context of CC inter- actions on 136Xe, a key feature of the SM level structure is the presence of two 2+ states at 83 and 224 keV, to which the lowest 1+ state is predicted to decay promptly with 69% and 31% branching fractions, respectively [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' Neither has been conclusively observed to date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' One can- didate is our proposed level at 422.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='1 keV, which is placed through strong, two-fold coincidences of γ rays at 108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='5, 239.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='8, and 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='7 keV, and which is fed directly by the first 1+ state via the 166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='7 keV transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' A 2+ level at this energy would be unresolved in 138Ba(d,α) reac- tions from the (4+) state observed at 423 ± 3 keV [24], which we in turn associate with our reconstructed level at 424.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='1 keV level due to its strong connection with the known 4+ state at 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='0 keV via coincidences of the 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='0 and 319.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='2 keV γ’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' Another candidate is the level at 140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='2 keV, which is fed directly by the first 1+ state via the 588.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='8 → 140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='2 → 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='7 → G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' sequence that is re- constructed by our observed coincidences of the 448.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='6, 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='6, and 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='7 keV transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' However, the angular distributions in 138Ba(d,α) reactions favor Jπ = 3+ for the level observed at 140 ± 3 keV, and we do not find any evidence for multiple states in the vicinity of this energy, leaving this interpretation uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' Finally, FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' 3: Background-subtracted γ-ray arrival times of the three delayed transitions found in our level scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' The first two corresponding to the 105 keV and 67 keV γ’s are fit with exponential and constant background terms only, while the fit to the 74 keV transition contains a term which accounts for the additional delay caused by preceding 67 keV transitions, in accordance with the proposed level scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' X-rays from Pb result in the prompt peak observed in the 74 keV timing distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' a state at 432(2) keV, first observed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' [18] and given a tentative spin-parity assignment of (3+), was recently identified as a possible (2+) state in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' This state is not reconstructed by any coincidences in the present work, and we do not observe any transitions near 157 keV that would connect this state with the 1+ state at 588.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='8 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' Thus our data are in tension with the more recent spin-party assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' Further measure- ments with both charged-particle and γ-ray detection will be ideal for fully characterizing this structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' Discussion – In a xenon-based detector the isomeric states measured here will produce a unique signature for the identification of CC interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' The prompt inter- action, composed of the emitted electron and the initial relaxation of the Cs nucleus, will almost always be fol- lowed by the delayed emission of one or more ∼ 100 keV γ’s/IC electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' In detectors which measure scintil- lation this will produce a characteristic two- or three- pulse signal, where the time structure of each pulse is determined by the convolution of three components: (1) the photoemission time constant(s) of the scintillating medium, (2) the photon transport time in the detec- tor, and (3) the photosensor and electronics response times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' In the case of (1), there are three technologies cur- Counts 10 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='0 keV t= 90 + 5 ns 10 10 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='6 keV t = 90 ± 5 ns 10° 10 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='7 keV t = 157 ± 4 ns 10 10 0 200 400 600 800 1000 Time [ns]5 TABLE I: Event rates for 7Be and CNO solar neutrino capture in current (marked with an ∗) and planned experiments which deploy 136Xe and the previously proposed target isotopes 130Te and 100Mo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' Rates are given in solar neutrino units (SNU) for each isotope and events per year for each experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' The mass column indicates the mass of isotope deployed, as opposed to the total payload of the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' Scattering rates on 130Te and 100Mo are taken from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' [41], while those on 136Xe are from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' Isotope Rate (SNU) Experiment Mass Rate (evt/yr) 7Be CNO (t) 7Be CNO 136Xe 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='3 LZ∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='62 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='54 KamLAND-Zen∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='68 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='59 nEXO 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='2 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='8 80t natXe TPC 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='1 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='2 130Te 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='7 SNO+ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='5% Te) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='70 100Mo 126 15 CUPID 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='25 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='72 rently being pursued for experiments at the tonne-scale and beyond: LXe TPCs [32–34], high-pressure gaseous xenon (GXe) TPCs [35], and loaded liquid scintilla- tor (LS) detectors [36], which have emission time con- stants of τLXe = 27 ns [37, 38], τGXe = 4 ns [39], and τLS = 6 ns [40], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' For most modern experi- ments (2) is O(10) ns and (3) is a design parameter that can be as low as O(1) ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' Given the lifetimes we have measured, pile-up of the scintillation pulses is a concern for robust reconstruction of their energies, and a coinci- dence analysis would realistically wait some small time before opening a window in which to search for a forth- coming pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' Experiments which minimize the impact of (3) will benefit from higher signal efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' Other ββ-decay isotopes have been proposed as fa- vorable targets for solar neutrino capture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' Suggested schemes for their deployment include metal-loaded LS [8, 9, 12] or solid-state sensors [10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' In Table I, we com- pare current and proposed xenon-based experiments with the two most massive experiments of each type planned for the near future2,3: the SNO+ [43] and CUPID [44] experiments, which will search for 0νββ in 130Te and 100Mo, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' The solar neutrino CC interaction rates in these next-generation experiments are compa- rable to those in currently-operating experiments which deploy 136Xe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' In the future, LXe TPC experiments such as nEXO [32] and a next-generation dark matter experi- ment [34] offer a promising avenue for spectroscopic stud- ies of solar neutrinos enabled by the measurements pre- sented here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' 2 See e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' 3 The ββ isotope 76Ge is not considered here as the CC-scattering rate on this isotope is negligible in the present context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' Within the scope of existing experiments, KamLAND- Zen recently reported the largest exposure of 136Xe to date, totaling 1 tonne-year [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' The fast intrinsic timing of the experiment’s organic LS is advantageous, though identification of delayed pulses from the 105, 74, or 67 keV transitions may require dedicated assessment of experiment’s energy threshold and the rate of 14C back- ground in this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' The LZ [45] experiment features a very low (∼1 keV) threshold with a very low background rate at ∼100 keV and is expected to amass a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='7 tonne- year exposure of 136Xe by the end of its 1000-day physics run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' In addition to searching for the roughly 10 expected solar neutrino events in these experiments, a search for the delayed-coincidence signature described here would provide world-leading sensitivity to MeV-scale fermionic dark matter and could be implemented immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' In summary, we have measured the energies and life- times of low-lying states in 136Cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' The isomeric states measured here enable the use of a delayed-coincidence tag to uniquely identify CC interactions in experiments which deploy 136Xe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' Thus a new channel exists in which to perform novel spectroscopic studies of solar neutrinos and dark matter in current and future xenon-based ex- periments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' Acknowledgments – The authors would like to thank David Radford for helpful discussions and assistance with the RadWare analysis software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' We thank Smarajit Triambak for helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' This work was sup- ported by the U.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' Meth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content=' A 953, 163047 (2020), arXiv:1910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='09124 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} +page_content='ins-det].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FKT4oBgHgl3EQfvC6t/content/2301.11893v1.pdf'} diff --git a/jtE1T4oBgHgl3EQfgQQH/content/tmp_files/2301.03226v1.pdf.txt b/jtE1T4oBgHgl3EQfgQQH/content/tmp_files/2301.03226v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..29411dc9d8c31f6a9113e0c20ebde2ecb7699a17 --- /dev/null +++ b/jtE1T4oBgHgl3EQfgQQH/content/tmp_files/2301.03226v1.pdf.txt @@ -0,0 +1,4844 @@ +Elasticity solution for a 3D hollow cylinder +axially loaded at the end faces +Mario DE MIRANDA‡, Marta DE MIRANDA‡, Alessio FALOCCHI†, +Alberto FERRERO∗, Luca MARININI‡, +‡ Studio De Miranda Associati, Via C. Pisacane 26, Milano, Italy +† Dipartimento di Matematica - Politecnico di Milano, Milano, Italy +∗ Dipartimento di Scienze e Innovazione Tecnologica, Universit`a del Piemonte Orientale, Alessandria, Italy +Abstract +Starting from an applicative problem related to the modeling of an element of a cable-stayed bridge, +we compute the elasticity solution for a hollow cylinder loaded at the end faces with axial loads. We +prove results of symmetry for the solution and we expand it in proper Fourier series; computing the +Fourier coefficients in adapted power series, we provide the explicit solution. We consider an engineering +case of study, applying the corresponding approximate formula and giving some estimates on the error +committed with respect to the truncation of the series. +1 +Introduction +In the recent years the interest of the mathematicians for engineering applications has grown more and +more; this is due to an evolution of the mathematics, thanks for instance to the development of new +techniques to deal with nonlinear problems and the support of automatic calculators to obtain previsions +unthinkable in the past. +Figure 1: From the top on the left a render of a recent cable stayed bridge designed by Studio De Miranda +Associati and a detail of its deck. +The mathematical modeling of specific phenomena is one of the ambitious aims of the applied math- +ematics. Recently some mathematical models for suspension bridges [11] have been developed with the +scope to understand and, then, to prevent instability phenomena; they were studied models for suspension +bridges with geometrical nonlinearities, e.g. see [7, 8], models for partially hinged plates, see [9], models for +1 +arXiv:2301.03226v1 [math.AP] 9 Jan 2023 + +non homogeneous partially hinged plates, see [1, 2, 3], models for homogeneous beams with intermediate +piers [10] and non homogeneous beams [4]. In all these cases the application of analytical methods to real +problems allowed to find suggestions and practical remedies that can be discussed with engineers. +This is the aim also of this paper. Here the problem, suggested by the structural civil engineering Studio +De Miranda Associati, is related to the modeling of the stresses in a constructive detail of a bridge: the +blister. In the cable-stayed bridge the blister is the structural element where the steel forestay anchors to +the deck. In Figure 1 is shown a render of a future cable-stayed bridge, designed by Studio De Miranda +Associati, that will be built in Brazil; a detail of the related blister element is given in Figure 3. +When the deck is built in reinforced concrete, as in this case, the blister is an important point to design; +indeed, the high density of the steel reinforcement may cause zone with low concrete capacity and possible +remarkable cracking. To have an idea of the complexity of this element a detail of the executive draw of +a blister for another stayed bridge, designed by Studio De Miranda Associati, is shown in Figure 2. +Figure 2: Detail of the executive draw of a blister of a recent prestressed concrete bridge. +For all these reasons it is important to estimate with precision the stresses acting on the element, so +that the reinforcing steel in the concrete can be computed without surplus. In engineering literature some +of the best known references related to the distribution of the stresses in prisms of concrete are [6, 14]; +here the authors consider many combinations of load on the prism and for each one the possible strategies +to design the steel bars. These results are obtained from particular solutions of the well known equation +of the linear elasticity, see e.g. [5]; we recall it here briefly in the general 3D case. +Given Ω ⊂ R3 an elastic homogeneous solid body, we denote by u : Ω → R3 the displacement vector at +any point of the reference configuration of the elastic body itself, see the list of notations at the end of the +paper. We denote by Tu the stress tensor and by λ and µ the classical Lam´e constants; it is known that +λ and µ may be expressed in terms of the Young modulus E and Poisson ratio ν ∈ (−1, 1 +2) as +λ = +Eν +(1 + ν)(1 − 2ν) , +µ = +E +2(1 + ν) . +(1.1) +The equation of linear elasticity reads +� +� +� +−µ∆u − (λ + µ)∇(divu) = f +in Ω, +(Tu)n = g +on ∂Ω, +(1.2) +where f and g are respectively the forces per unit volume and the boundary forces per unit surface acting +on Ω, while n is the unit outward normal vector to ∂Ω. +In Section 2 we briefly derive (1.2) from variational principles and we recall the existence and uniqueness +results in Theorem 2.1; these are classical topics in linear elasticity, see e.g. [5, 16], but we recall them in +our framework for completeness since the question about uniqueness of solutions of (1.2) is not trivial at +all and it needs an additional condition to be achieved. +The theoretical solution from which come the applicative cases considered in [6, 14] is given in [13], +where Ω is a rectangular prism under end loads. Thanks to this simple geometry and loading condition +2 + +POST-INSTALLED +2a:2h) +REINFORCEMENT TO BE +BENT AFTER HARDENING +(4a:4p)@16 (tot. _12) +5a:5f @16 (tot. 6) +OF RESIN +=616 +* max +min=423 +645 +STRESSING RECESS +RESIN TYPE HIT-RE 500 V3 +Hole +：Φ12x100mm +SECTION X-X +@2+2@16/100 L=1134 +3a:39) +2000- +1000the authors find explicitly the solution in form of double Fourier series. The result is obtained applying +the Galerkin vector method, a technique allowing to pass from the second order differential equation (1.2) +to a simpler biharmonic equation, see [13]. +We point out that to find the explicit solution of (1.2) for generic Ω and loading conditions is a very +hard task. In this paper we find it for Ω coincident with a hollow cylinder loaded on the opposite faces, +since this geometry fits the modeling of the concrete of the blister, see Figure 3 on the right; indeed, the +forestay of the bridge is circular and passes through the cylindrical hole, applying a distributed load on +the opposite faces due to its tensioning, see Figure 4. +Figure 3: From the left a frontal view of blister elements and the modelization of the element through the +hollow cylinder (in red). +The precise definition of the model is given in Section 4; the application of axial loads leads to a solution +having axial symmetric properties, see Proposition 3.1. In the real blister it is also possible to have non +radial loadings coming from the deck, but this is a first attempt of modeling that may be implemented in +future works; anyway, the solution found here may have general interest beyond this specific application. +The definition of the solution is given by steps: in subsection 3.1 we provide a periodic extension of the +loads in the variable z corresponding to the symmetry axis of the hollow cylinder, in such a way that it +becomes possible to expand the solution in Fourier series with respect to the variable z; then we compute +the Fourier coefficients which come to be functions in the other two variables x and y, corresponding +to directions orthogonal to the symmetry axis of the hollow cylinder; in subsection 3.2 we pass to the +cylindrical coordinates and, exploiting the axial symmetry, we reduce ourself to study a system of ODEs +in the radial polar coordinate ρ; we compute the Fourier coefficients as functions of the variable ρ through +an adapted expansion in power series so that we are able to state Theorem 3.7, collecting the explicit +solution. +In Section 4 we give some hints to truncate the series and we apply the results to an engineering case +of study. As it will be explained in details, it will be necessary to compute numerically the first M terms +in the Fourier series expansion with M to be chosen sufficiently large in order to minimize the truncation +error. The main question in this procedure is that the computation of those Fourier coefficients, which +are solutions of suitable boundary value problems of ODEs, requires the numerical resolution of some +algebraic linear systems in four variables which exhibit a condition number higher and higher as M grows; +if we need a truncation error smaller than ours, we may consider alternative numerical procedures. We +emphasize that the main purpose of this article is to obtain an analytical representation of the unique +symmetric solution of (1.2) in the case of the hollow cylinder with the perspective of reproducing such +method in more general situations with not necessarily symmetric external loads. +As already explained in details, the main analytical and numerical results of the article are stated +in Sections 2-4 and their proofs are given in Section 5. The final part of the paper is devoted to the +conclusions, see Section 6, and a list of notations which can be helpful for the reader. +3 + +2 +The definition of the mathematical model for the linear elasticity +In this Section we derive the differential equations for the linear elasticity from variational arguments and +we state a theorem related to the existence of solutions. Although these results are well known overall in +the engineering field, we review them from a mathematical point of view, applying the Fredholm alternative +to prove existence of solutions. +2.1 +The derivation of the differential equations +We recall that Ω ⊂ R3 is the domain of the elastic body and u is the displacement function with components +u = (u1, u2, u3). We denote by Du the linearized strain tensor, which in the sequel will be simply called +strain tensor, since we only deal with the linear theory; the stress tensor can be written as +Tu = +� +� +σ1 +τ 12 +τ 13 +τ 12 +σ2 +τ 23 +τ 13 +τ 23 +σ3 +� +� . +(2.1) +It is well known that by the Hooke’s Law for isotropic materials it holds +Tu = λtr(Du) I + 2µDu , +(2.2) +where λ and µ are the Lam´e constants. +Combining (1.1), (2.1) and (2.2) we infer +σ1 = +E +(1 + ν)(1 − 2ν) +� +(1 − ν)∂u1 +∂x + ν +�∂u2 +∂y + ∂u3 +∂z +�� +τ 12 = +E +2(1 + ν) +�∂u1 +∂y + ∂u2 +∂x +� +σ2 = +E +(1 + ν)(1 − 2ν) +� +(1 − ν)∂u2 +∂y + ν +�∂u1 +∂x + ∂u3 +∂z +�� +τ 13 = +E +2(1 + ν) +�∂u1 +∂z + ∂u3 +∂x +� +σ3 = +E +(1 + ν)(1 − 2ν) +� +(1 − ν)∂u3 +∂z + ν +�∂u1 +∂x + ∂u2 +∂y +�� +τ 23 = +E +2(1 + ν) +�∂u2 +∂z + ∂u3 +∂y +� +. +(2.3) +The elastic energy related to the internal forces in the configuration corresponding to a generic displace- +ment u is given by +Eel(u) = 1 +2 +� +Ω +Tu : Du dx . +If we assume that on Ω act body forces per unit of volume f = (f1, f2, f3) and boundary forces per unit of +surface g = (g1, g2, g3) we obtain the total energy of the system +E(u) = 1 +2 +� +Ω +Tu : Du dx − +� +Ω +f · u dx − +� +∂Ω +g · u dS . +(2.4) +Thanks to the symmetry of the stress tensor Tu = (Tu)T we infer that for any u, v ∈ H1(Ω; R3) +Tu : ∇v = (Tu)T : (∇v)T = Tu : (∇v)T +so that +2(Tu : ∇v) = Tu : ∇v + Tu : (∇v)T +⇒ +Tu : ∇v = Tu : Dv . +(2.5) +Recalling the Hooke’s law (2.2), we observe that the bilinear form +(u, v) �→ +� +Ω +Tu : Dv dx , +(u, v) ∈ H1(Ω; R3) × H1(Ω; R3) +4 + +is symmetric, since +Tu : Dv = λ (divu) (divv) + 2µ Du : Dv . +(2.6) +By looking at the total energy E in (2.4) as a functional E : H1(Ω; R3) → R and exploiting the symmetry +of the bilinear form above mentioned, we see that a critical point u ∈ H1(Ω; R3) of E solves the variational +problem +� +Ω +Tu : Dv dx = +� +Ω +f · v dx + +� +∂Ω +g · v dS +for any v ∈ H1(Ω; R3) . +(2.7) +By (2.5) and a formal integration by parts, we see that (2.7) is the weak formulation of the boundary +value problem +� +−div(Tu) = f +in Ω, +(Tu)n = g +on ∂Ω. +(2.8) +Inserting (2.2) into (2.8) we find the well known equations of linear elasticity (1.2). +In the next subsection we prove the existence of solution, stating some classical results about functional +spaces of vector valued functions which find a natural application in the theory of linear elasticity. These +results are related to the well known Korn inequality which has a general validity for vector functions from +RN to RN for any N ≥ 1. Clearly, in the present paper we will be mainly interested to the case N = 3, +being R3 the natural space where a solid elastic body can be modelled. For completeness, we will state +those results in the general N-dimensional case. +2.2 +Existence of a solution +Let Ω ⊂ Rn a bounded domain, i.e. an open connected bounded set of RN. Let us introduce the following +Sobolev-type space H1 +D(Ω; RN) defined as the completion of C∞(Ω; RN) with respect to the scalar product +(u, v)H1 +D = +� +Ω +Du : Dv dx + +� +Ω +u · v dx +for any u, v ∈ C∞(Ω; RN) . +(2.9) +Here x = (x1, . . . , xN) denotes the generic variabile of a function defined in a domain of RN and +dx = dx1 . . . dxN denotes the N-dimensional volume integral in RN. From its definition, it is clear that +H1 +D(Ω; RN) ⊂ L2(Ω; RN) becomes a Hilbert space with the extension of the scalar product (2.9). +The Korn inequality states the equivalence on the space C∞(Ω; RN) between the usual scalar product +of H1(Ω; RN), namely +(u, v)H1 = +� +Ω +∇u : ∇v dx + +� +Ω +u · v dx +for any u, v ∈ H1(Ω; RN) +(2.10) +and the scalar product (2.9). More precisely, if Ω ⊂ RN is a bounded domain with Lipschitz boundary, +then there exists C > 0 such that +� +Ω +|∇u|2dx ≤ C +�� +Ω +|u|2dx + +� +Ω +|Du|2dx +� +for any u ∈ H1(Ω; RN) . +(2.11) +Among the others, for a clear and elegant proof of (2.11), we address the reader to [12] by V. A. +Kondrat’ev & O. A. Oleinik. +Thanks to (2.11) we deduce that the Hilbert space H1 +D(Ω; RN) actually coincides with H1(Ω; RN) as +one can deduce from the definition of H1 +D(Ω; RN) and the well known result about density of C∞(Ω; RN) +in H1(Ω; RN) whenever Ω, is a bounded domain with Lipschitzian boundary. The Korn inequality is a +fundamental tool for proving the existence of weak solutions of (1.2). +We fix N = 3 and we observe that by (2.6) and (2.11), the bilinear form defined by +(u, v)T = +� +Ω +Tu : Dv dx + +� +Ω +u · v dx +for any u, v ∈ H1(Ω; R3) +(2.12) +5 + +is a scalar product in H1(Ω; R3) which is equivalent to (2.10). +Let us introduce the space +V0 := +� +v0 ∈ H1(Ω; R3) : +� +Ω +Tv0 : Dv dx = 0 +∀v ∈ H1(Ω; R3) +� +. +(2.13) +We observe that V0 coincides with the eigenspace associated to the first eigenvalue of the following +eigenvalue problem: α is an eigenvalue if there exists a nontrivial function u ∈ H1(Ω; R3), which will be +called eigenfunction associated to α, such that +� +Ω +Tu : Dv dx = α +� +Ω +u · v dx +for any v ∈ H1(Ω; R3). +In particular if α = 0 and v0 is a corresponding eigenfunction we have that +� +Ω +Tv0 : Dv dx = 0 +for any v ∈ H1(Ω; R3) . +(2.14) +After some computation one can verify that V0 is the space of functions v = (v1, v2, v3) admitting the +following representation +� +� +� +� +� +v1(x1, x2, x3) = αx2 + βx3 + δ1 , +v2(x1, x2, x3) = −αx1 + γx3 + δ2 , +v3(x1, x2, x3) = −βx1 − γx2 + δ3 . +(2.15) +where α, β, γ, δ1, δ2, δ3 are arbitrary constants. +Roughly speaking, configurations associated with such functions v ∈ V0 correspond to translations and +rotations of the solid body without deforming it in such a way the elastic energy Eel(v) equals zero. +Actually, assuming for simplicity δ1 = δ2 = δ3 = 0, deformations corresponding to displacements v ∈ V0 +can be considered good approximations of a rotation only for α, β, γ small; when at least one of the +constants α, β, γ is not small the corresponding deformation of the solid body is no more negligible. In +such a case, one may wonder why the elastic energy remains anyway zero; the answer is that in the linear +theory only small deformations are allowed so that large deformations are no more meaningful for our +model. +Let us recall that we are considering in our model the linearized strain tensor Dv which is a good +approximation of the real strain tensor only for small deformations since the last one also contains quadratic +terms in the first order derivatives of v1, v2, v3; these quadratic terms can be neglected when first order +derivatives are small. +In the next theore we state the existence result for problem (1.2). +Theorem 2.1. Let Ω ⊂ R3 a bounded domain with Lipschitzian boundary and let f ∈ L2(Ω; R3) and +g ∈ L2(∂Ω; R3). Let us introduce the following compatibility condition +� +Ω +f · v dx + +� +∂Ω +g · v dS = 0 +for any v ∈ V0 . +(2.16) +Then the following statements hold true: +(i) problem (2.7) admits at least one solution u ∈ H1(Ω; R3), or equivalently the boundary value problem +(1.2) admits at least one weak solution, if and only if (2.16) holds true; +(ii) if u ∈ H1(Ω; R3) is a particular solution of (2.7) then for any v0 ∈ V0 the function u + v0 is still a +solution of (2.7); +6 + +(iii) if u ∈ H1(Ω; R3) is a particular solution of (2.7) and if w ∈ H1(Ω; R3) is any other solution of +(2.7), then there exists v0 ∈ V0 such that w = u + v0; +(iv) letting V ⊥ +0 +be the space orthogonal to V0 with respect to the scalar product (2.12), we have that +problem (2.7) admits a unique solution in V ⊥ +0 . +Remark 2.2. The results of Theorem 2.1 may be extended by replacing the boundary function g ∈ +L2(∂Ω; R3) by an element of the space H−1/2(∂Ω; R3) denoting the dual space of H1/2(∂Ω; R3). In such a +case, if g ∈ H−1/2(∂Ω; R3) the compatibility condition (2.16) has to be replaced by its natural extension +� +Ω +f · v dx + H−1/2⟨g, v⟩H1/2 = 0 +for any v ∈ V0 . +The validity of this fact comes from the trace theory for vector valued functions h admitting a weak +divergence in L2(Ω); for such functions h, it is possible to define the trace of h · n as an element of the +dual space of H1/2(∂Ω), the last one being the space of traces of H1(Ω) functions. Such a result has to be +applied in our case to each line of Tu. +Remark 2.3. We observe that, as a consequence of Theorem 2.1, if u and w are solutions of (2.7), then +the two configurations of the elastic body, corresponding to u and w, generate the same stress state. More +precisely we have Tu = Tw in Ω as a consequence of the Hooke’s law and of the fact that D(u − w) +vanishes in Ω being u − w ∈ V0. Physically, this is completely reasonable since, given the configuration +corresponding to u, the one corresponding to w can be obtained from the first one by means of rotations +and translations of the elastic body, which clearly do not affect the stress state of the solid body itself. +The proof of Theorem 2.1 is given in subsection 5.1 and is based on standard arguments and the +Fredholm alternative. +3 +The hollow cylinder axially loaded at the end faces +We consider a circular, finite, homogeneous, isotropic and elastic cylinder with height h, radius b > 0, +having a coaxial hole of radius a > 0. +In this section we use the usual notation x, y, z for the three +coordinates in R3. We maintain the notation dx to denote the differential volume dxdydz. +Therefore, we introduce the annular domain Ca,b := {(x, y) ∈ R2 : a2 < x2 + y2 < b2} in such a way +that +Ω = Ca,b × +� +− h +2, h +2 +� +. +In the sequel we want to model a hollow cylinder subject to an external load acting on the upper and +lower faces of the cylinder compressing the cylinder itself. Recalling the notations introduced in (1.2), we +will then assume that the volume forces represented by the vector function f vanish everywhere in Ω. +In order to better describe the surface forces represented by the vector function g, we split ∂Ω in four +regular parts +Γ1 := Ca,b × +� +− h +2 +� +, +Γ2 := Ca,b × +� h +2 +� +, +Γ3 := {(x, y) ∈ R2 : x2 + y2 = b2} × +� +− h +2, h +2 +� +, +Γ4 := {(x, y) ∈ R2 : x2 + y2 = a2} × +� +− h +2, h +2 +� +, +having respectively outward unit normal vectors (0, 0, −1), (0, 0, 1), (x/b, y/b, 0) when (x, y, z) ∈ Γ3 and +(−x/a, −y/a, 0) when (x, y, z) ∈ Γ4. In this way, the outward unit normal vector n is well defined on the +whole ∂Ω. +7 + +Figure 4: The domain Ω and in red the load considered. +Exploiting the above notations, the vector function g can be represented in the following way +g(x, y, z) = +� +� +� +� +� +(0, 0, χp(x, y)) +for any (x, y, z) ∈ Γ1, +(0, 0, −χp(x, y)) +for any (x, y, z) ∈ Γ2, +(0, 0, 0) +for any (x, y, z) ∈ Γ3 ∪ Γ4, +(3.1) +where the function χp : Ca,b → R, p ∈ R+, is defined by +χp(x, y) := +� +p +if a2 ≤ x2 + y2 < ϵ2, +0 +if ϵ2 < x2 + y2 ≤ b2, +(3.2) +for some ϵ ∈ (a, b). +Resuming all the assumptions on f and g we are led to consider the problem +� +� +� +� +� +� +� +� +� +� +� +� +� +−µ∆u − (λ + µ)∇(divu) = 0 +in Ω , +(Tu)n = (0, 0, χp) +on Γ1, +(Tu)n = (0, 0, −χp) +on Γ2, +(Tu)n = 0 +on Γ3 ∪ Γ4 . +(3.3) +Among all solutions of (3.3) which can be obtained by a single solution by adding to it a function in +the space V0, we focus our attention on the unique solution u = (u1, u2, u3) of (3.3) in the space V ⊥ +0 +where orthogonality is meant in the sense of the scalar product defined in (2.12), see Theorem 2.1. From a +geometric point of view, condition u ∈ V ⊥ +0 +avoids translations and rotations of the hollow cylinder, being +V0 the space of displacement functions which generate translations and rotations. +In subsection 5.2 we prove a symmetry result for the unique solution of (3.3) in the space V ⊥ +0 , whose +validity is physically evident, but which however needs a rigorous proof: +8 + +-p +h +U +2 +p +h +I4 +I3 +2 +pProposition 3.1. Let u be the unique solution of (3.3) in the space V ⊥ +0 . Then u satisfies the following +symmetry properties: +(i) for any (x, y, z) ∈ Ω we have +u1(x, y, −z) = u1(x, y, z), +u2(x, y, −z) = u2(x, y, z) +u3(x, y, −z) = −u3(x, y, z) , +(3.4) +u1(−x, y, z) = −u1(x, y, z), +u2(−x, y, z) = u2(x, y, z) +u3(−x, y, z) = u3(x, y, z) , +(3.5) +u1(x, −y, z) = u1(x, y, z), +u2(x, −y, z) = −u2(x, y, z) +u3(x, −y, z) = u3(x, y, z) ; +(3.6) +(ii) the third component u3 of the solution u is axially symmetric in the sense that: +u3(x1, y1, z) = u3(x2, y2, z) +∀ (x1, y1, z), (x2, y2, z) ∈ Ω with x2 +1 + y2 +1 = x2 +2 + y2 +2 ; +(iii) the first two components u1, u2 of the solution u form a central vector field in two dimensions in the +sense that +|(u1, u2)||(x1,y1,z) = |(u1, u2)||(x2,y2,z) +∀ (x1, y1, z), (x2, y2, z) ∈ Ω with x2 +1 + y2 +1 = x2 +2 + y2 +2 +and +(u1, u2)|(x,y,z) = |(u1, u2)||(x,y,z) +� +x +� +x2 + y2 , +y +� +x2 + y2 +� +for any (x, y, z) ∈ Ω . +3.1 +Periodic extension of the problem +Our next purpose is to look for and construct a solution u = (u1, u2, u3) of (3.3) admitting a Fourier series +expansion and, hence, admitting a periodic extension defined on the whole Ca,b × R. In order to obtain +this construction, we need to assume that the horizontal displacements u1 and u2 vanish on the upper and +lower faces of the hollow cylinder Ω: +u1 +� +x, y, h +2 +� += u1 +� +x, y, − h +2 +� += 0 +and +u2 +� +x, y, h +2 +� += u2 +� +x, y, − h +2 +� += 0 . +(3.7) +We find a solution satisfying (3.7) and, a posteriori, we show that it necessarily coincides with the unique +solution of (3.3) belonging to V ⊥ +0 , see the end of the proof of Theorem 3.7. +As a first step, since u ∈ H1(Ω; R3), we define a function, still denoted for simplicity by u, on the +domain Ca,b × +� +− h +2, 3h +2 +� +by extending it in suitable way: the new function u coincides with the original +function u on Ca,b × +� +− h +2, h +2 +� +and +u1(x, y, z) = −u1(x, y, h − z) , +u2(x, y, z) = −u2(x, y, h − z) , +u2(x, y, z) = u3(x, y, h − z) , +(3.8) +for any (x, y, z) ∈ Ca,b × +� h +2, 3h +2 +� +. This means that u1 and u2 are antisymmetric with respect to z = h +2 +and u3 is symmetric with respect to z = h +2. This symmetric extension with respect to z = h +2 produces a +function u ∈ H1 � +Ca,b × +� +− h +2, 3h +2 +� +; R3� +thanks to condition (3.7). +The second step is to extend the new function u : Ca,b × +� +− h +2, 3h +2 +� +→ R to the whole Ca,b × R as a +2h-periodic function in the variable z. It is easy to understand that the periodic extension, still denoted +for simplicity by u, is a function satisfying u ∈ H1(Ca,b × I; R3) for any open bounded interval I. +The periodic extension of the boundary data can be achieved according to the next lemma, proved in +subsection 5.3. We state here some lemmas in order to understand the main steps in the construction of +the solution of (3.3), given in the final theorem. +9 + +Lemma 3.2. Let u be the periodic extension of the solution of (3.3) defined as above and let Λ be the +distribution defined by +−div(Tu) = Λ +in D′(Ca,b × R; R3) . +(3.9) +Then Λ admits the following Fourier series expansion +Λ = (Λ1, Λ2, Λ3) = +� +0, 0, χp(x, y) ++∞ +� +m=0 +(−1)m+1 4 +h sin +�π +h(2m + 1)z +�� +. +(3.10) +The symmetry properties (3.8) and the construction of the periodic extension, allow expanding u = +(u1, u2, u3) in Fourier series with respect to the variable z: +u1(x, y, z) = ++∞ +� +k=0 +ϕ1 +k(x, y) cos +� π +h kz +� +, +u2(x, y, z) = ++∞ +� +k=0 +ϕ2 +k(x, y) cos +� π +h kz +� +, +(3.11) +u3(x, y, z) = ++∞ +� +k=0 +ϕ3 +k(x, y) sin +� π +h kz +� +. +In subsection 5.4 we prove the following lemma. +Lemma 3.3. For any k ≥ 1 odd, there exists a unique (ϕ1 +k, ϕ2 +k, ϕ3 +k) ∈ H1(Ca,b; R3), satisfying (3.3) and +(3.11). For any k ≥ 2 even, there exists a unique trivial (ϕ1 +k, ϕ2 +k, ϕ3 +k) ≡ (0, 0, 0) in Ca,b, satisfying (3.3) +and (3.11). +Remark 3.4. We observe that for k = 0, the boundary value problem (3.3), or equivalently (5.23)-(5.24) +and (5.26), see the proof in subsection 5.4, admits an infinite number of solutions. More precisely, these +solutions are in form (ϕ1 +0, ϕ2 +0, ϕ3 +0) = (c1, c2, c3) where c1, c2, c3 are three arbitrary constants. +We may +choose ϕ3 +0 ≡ 0 being irrelevant in the Fourier expansion of u3. Concerning the other two components, we +have necessarily ϕ1 +0 ≡ ϕ2 +0 ≡ 0 in Ca,b due to the odd symmetry of u1 and u2 with respect to the variables x +and y, as stated in (3.5) and (3.6). +3.2 +Cylindrical coordinates exchange +The symmetry properties of u stated in Proposition 3.1 imply that ϕ3 +k is a radial function and the vector +field (ϕ1 +k, ϕ2 +k) is a central vector field in the plane, in the sense that it is oriented toward the origin and +its modulus is a function only of the distance from the origin. This implies that for any k ≥ 1 odd, there +exist two radial functions Yk = Yk(ρ) and Zk = Zk(ρ) such that in polar coordinates we may write +ϕ1 +k(ρ, θ) = Yk(ρ) cos θ , +ϕ2 +k(ρ, θ) = Yk(ρ) sin θ , +ϕ3 +k(ρ, θ) = Zk(ρ) , +(3.12) +with ρ ∈ [a, b] and θ ∈ [0, 2π). +In Section 5 we show that Yk and Zk solve a proper boundary value problem. More precisely, this fact +will be shown in subsection 5.5 which is devoted to the proof of the next lemma, where we state existence +and uniqueness for solutions of the boundary value problem mentioned above. +Lemma 3.5. Let Ψk : Ca,b → R be defined as +Ψk(x, y) := +� +� +� +(−1) +k+1 +2 4 +h χp(x, y) +if k is odd, +0 +if k is even, +∀(x, y) ∈ Ca,b. +(3.13) +10 + +For any k ≥ 1 odd, the boundary value problem +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +Y ′′ +k (ρ) + Y ′ +k(ρ) +ρ +− Yk(ρ) +ρ2 +− +µ +λ + 2µ +π2k2 +h2 +Yk(ρ) + λ + µ +λ + 2µ +πk +h Z′ +k(ρ) = 0 +in (a, b) , +Z′′ +k(ρ) + Z′ +k(ρ) +ρ +− λ + 2µ +µ +π2k2 +h2 Zk(ρ) − λ + µ +µ +πk +h +� +Y ′ +k(ρ) + Yk(ρ) +ρ +� += − 1 +µ Ψk(ρ) +in (a, b) , +(λ + 2µ)Y ′ +k(ρ) + λ +ρ Yk(ρ) + λ πk +h Zk(ρ) = 0 , +ρ ∈ {a, b} +Z′ +k(ρ) − πk +h Yk(ρ) = 0 , +ρ ∈ {a, b} +(3.14) +admits a unique solution (Yk, Zk) ∈ H1(a, b; R2). +About existence and uniqueness of solutions of (3.14), in subsection 5.5 we only give an idea of the +proof since it can be proved exactly as Lemma 3.3 of which Lemma 3.5 is the radial version. +Now we need a more explicit representation for the unique solution (Yk, Zk) of (3.14). This will be +done by performing a power series expansion in which the coefficients will be characterized explicitly in +terms of a suitable iterative scheme. As a byproduct of this result in Section 4 we also obtain a numerical +approximation of the exact solution and we estimate the corresponding error. Being a linear problem, we +proceed by applying the superposition principle and we provide the explicit formula in the next lemma. +Lemma 3.6. For any k ≥ 1, odd, let Υk = (Yk, Zk) the unique solution of (3.14). Omitting for brevity +the k-index, we have a unique (C1, C2, C3, C4) ∈ R4 such that +Υ(ρ) = C1Υ1(ρ) + C2Υ2(ρ) + C3Υ3(ρ) + C4Υ4(ρ) + Υ(ρ), +(3.15) +where Υj = (Y j, Zj) with j = 1, . . . , 4 are four linear independent solutions of the corresponding homoge- +nous system and Υ = (Y , Z) solves +� +Y (ρ) +Y +′(ρ) +Z(ρ) +Z +′(ρ) +�T += W(ρ) +� ρ +a +(W(r))−1 � +0 +0 +0 +− 1 +µΨk(r) +�T +dr , +ρ > 0 , +(3.16) +being W(ρ) the wronskian obtained through Υj(ρ) (j = 1, . . . , 4). Each of the linear independent solutions +of the homogeneous system can be written as +� +� +� +� +� +� +� +� +� +� +� +� +� +Y j(ρ) = ++∞ +� +n=−1 +aj +n ρn + (ln ρ) ++∞ +� +n=0 +bj +n ρn , +Zj(ρ) = ++∞ +� +n=0 +cj +n ρn + (ln ρ) ++∞ +� +n=0 +dj +n ρn +(j = 1, . . . , 4), +(3.17) +where the coefficients are uniquely determined. +In the proof of the Lemma we give all the details related to the computation of the constants Cj in (3.15) +and of the coefficients in the series (3.17), see subsection 5.6. As a consequence of Lemmas 3.2-3.3-3.5-3.6 +we state the main theorem, whose proof can be found in subsection 5.7. +Theorem 3.7. Let u be the unique solution of (3.3) satisfying u ∈ V ⊥ +0 +and let (Yk, Zk), k ≥ 1 odd, +be the unique solution of (3.14). Then, in cylindrical coordinates, u = (u1, u2, u3) admits the following +11 + +representation: +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +u1(ρ, θ, z) = ++∞ +� +m=0 +Y2m+1 (ρ) cos θ cos +� +(2m+1)π +h +z +� +, +u2(ρ, θ, z) = ++∞ +� +m=0 +Y2m+1 (ρ) sin θ cos +� +(2m+1)π +h +z +� +, +u3(ρ, θ, z) = ++∞ +� +m=0 +Z2m+1 (ρ) sin +� +(2m+1)π +h +z +� +, +(3.18) +with ρ ∈ (a, b), θ ∈ [0, 2π), z ∈ +� +− h +2, h +2 +� +where the three series in (3.18) converge weakly in H1(Ω) and +strongly in L2(Ω). +Moreover, letting UM = (U 1 +M, U2 +M, U3 +M) be the sequence of vector partial sums corresponding to the series +expansions in (3.18), we have for any M ≥ 1 +∥U 1 +M − u1∥L2(Ω) ≤ pb2 +µ +� +h(b − a) +2aπ +1 +√ +M +, +∥U 2 +M − u2∥L2(Ω) ≤ pb2 +µ +� +h(b − a) +2aπ +1 +√ +M +, +(3.19) +∥U 3 +M − u3∥L2(Ω) ≤ +p +µaπ2 +� +h3b3(b − a) +24 +1 +√ +M3 . +4 +An engineering application +In this section we consider a case of study: a hollow cylinder having the features of a blister for the bridge +in Figure 1. In Table 1 we give the mechanical parameters, see also Figure 4. We consider stays composed +of 19 strands, see Figure 5, suitable to bear the concentrated load P in Table 1. P is computed from +the executive project, while the diameter 2a is taken from the catalogue of Protende ABS-2021 [15], a +company producing such elements, see in Figure 5 the diameter φD1 for 19 strands anchorage; hence, the +h +3.00 +m +Height of the cylinder +2a +273 mm +Diameter of the cylindrical hollow +2b +800 mm +External diameter of the cylinder +2ϵ +425 mm +External diameter of the load +P +1900 kN +Concentrated load +E +35000 MPa +Young modulus of the concrete +ν +0.2 +Poisson ratio of the concrete +Table 1: Mechanical parameters assumed. +distributed load in (3.2) is given by p = +P +π(ϵ2−a2) = 22.80 MPa. +Our purpose is to obtain a good approximation of the functions Υj = (Y j, Zj), j ∈ {1, 2, 3, 4} introduced +in Lemma 3.6. For j = 1, . . . , 4 we consider the approximate solution (N ≥ 1) +Y j +N(ρ) = +N +� +n=−1 +an ρn + (ln ρ) +N +� +n=0 +bn ρn +and +Zj +N(ρ) = +N−1 +� +n=0 +cn ρn + (ln ρ) +N−1 +� +n=0 +dn ρn . +(4.1) +The reason for in (4.1) we have n = 0, . . . , N − 1 in the expansion of Zj +N will be clarified in the proof +in subsection 5.8 of the next proposition about an estimate of the truncating error. +12 + +Figure 5: Detail of the strands anchorage and in table the geometric features for a 19 strands element, +from the commercial catalogue [15]. +Proposition 4.1. Let k > 1 , k ∈ N odd, and let N ≥ 3, odd integer, be the truncating index of the series +as in (4.1). Then, letting +Ek,N := +max +j∈{1,2,3,4} +� +max +� +max +ρ∈[a,b] |Y j +N(ρ) − Y j(ρ)|, max +ρ∈[a,b] |Zj +N(ρ) − Zj(ρ)| +�� +, +we have that +Ek,N ≤ C(a, b, k)3(2λ + 5µ)(λ + µ)2 +16µ3 +�πkb +h +�N+2 +e( πkb +h ) +2 (N + 3)(3N3 + 21N2 + 42N + 32) +2N �� N+1 +2 +� +! +�2 +, +(4.2) +where +C(a, b, k) = max +� +1, +h +πkb +� +max +� +1, +��ln +� πka +h +��� , | ln +� πkb +h +� +| +� +max +� +πk +h , +µ +λ+µ +πk +h ln +� πk +h +� +, 2(λ+2µ) +λ+µ +h +πk ln +� πk +h +� � +. +Once we have (4.2), one may choose N in such a way that +Ek,N +min +j∈{1,2,3,4} +� +min +� +max +ρ∈[a,b] |Y j +N(ρ)|, max +t∈[a,b] |Zj +N(ρ)| +�� < ε +(4.3) +with ε small enough. Condition (4.3) means that the truncation error is relatively small compared to the +order of magnitude of both functions Y j +N and Zj +N for all j ∈ {1, 2, 3, 4}. +In our numerical simulation the condition (4.3) is verified by making use of estimate (4.2) on the +truncation error Ek,N, i.e. the program verifies at each step the validity of (4.3) in which the numerator +of the fraction is replaced by the majorant in (4.2). The program runs until the value of N is sufficiently +large to guarantee (4.3). +In Figure 6 we plot a vertical section of the cylinder and the corresponding more stressed horizontal +section. +We show the vertical displacement u3 and the following components of the stress tensor in +13 + +Variavel +H +E1 +OD1 +OF +R +I +B1 0C +QA1 +op +中 +L1 min. +VariavelNUMERO +DE +0A1 +B1 +OC +OD1 +E1 +OF +H +L1min. +OP +CORDOALHAS +4 +110 +270 +160 +141 +20 +89 +30 +430 +63 +7 +150 +300 +200 +168 +30 +114 +40 +550 +75 +13 +200 +340 +256 +219 +36 +168 +55 +860 +110 +19 +228 +380 +300 +273 +53 +168 +75 +960 +140 +31 +280 +470 +350 +323 +60 +219 +90 +1280 +180 +37 +280 +490 +365 +323 +70 +273 +100 +1410 +180 +55 +323 +570 +415 +355 +85 +273 +125 +1720 +225 +61 +349 +600 +440 +355 +85 +323 +130 +1870 +250 +73 +410 +680 +525 +457 +95 +323 +145 +2030 +280 +91 +450 +750 +570 +508 +110 +355 +160 +2280 +315 +109 +470 +780 +580 +508 +120 +355 +175 +2400 +315 +127 +500 +810 +620 +559 +130 +406 +190 +2700 +355 +169 +570 +006 +680 +609 +145 +457 +225 +3120 +400 +Valores sujeitos a variacoes de acordo com os requisitos especiais do projetoFigure 6: From the the left the vertical displacement u3 in mm, the vertical stress σz in MPa, the radial +stress σr in MPa, the angular stress σθ in MPa and the tangential stress τ rz in MPa. +cylindrical coordinates +σz = +2µ +1 − 2ν +� +(1 − ν)∂u3 +∂z + ν +�ur +ρ + ∂ur +∂ρ +�� +σr = +2µ +1 − 2ν +� +(1 − ν)∂ur +∂ρ + ν +�ur +ρ + ∂u3 +∂z +�� +σθ = +2µ +1 − 2ν +� +(1 − ν)ur +ρ + ν +�∂ur +∂ρ + ∂u3 +∂z +�� +τ rz = µ +�∂ur +∂z + ∂u3 +∂ρ +� +, +(4.4) +where ur = +� +u2 +1 + u2 +2 is the radial displacement. We point out that putting n = (cos θ, sin θ, 0), t = +(− sin θ, cos θ, 0) and k = (0, 0, 1), the four components introduced in (4.4) are defined by σz := (Tu)k · k, +σr := (Tu)n · n, σθ := (Tu)t · t and τ rz := (Tu)n · k and the representation (4.4) can be deduced by (2.1) +and (2.3). +We consider an approximate solution UM as stated in Theorem 3.7 truncating the Fourier series at +M = 29 with ε < 10−3 in (4.3), implying N = 123 in (4.1) and ∥U 1 +29 − u1∥L2(Ω) ≤ 4.46 · 10−5 m5/2, +∥U 3 +29 − u3∥L2(Ω) ≤ 1.02 · 10−6 m5/2 in (3.19). In Table 2 we give the maximum absolute values of the +variables involved, including the coordinate of the point (ρ, z) where they are assumed (for all θ ∈ [0, 2π) +thanks to the radial symmetry of the problem). +As expected the vertical displacement u3 achieves its maximum absolute value at z = ± h +2. From the +plots we see that there are two (symmetric) critical zones where we observe the loading diffusion; they +are close to the upper and bottom faces of the cylinder and involve approximately the 20% of the closest +volume, i.e. the volume of Ω such that z ∈ (− h +2, − 2h +5 ) ∪ ( 2h +5 , h +2). +14 + +.0 +W3 +_rz +0 +1.5 +1.5 +1.5 +1.5 +1.5 +0 +2 +-2 +C +0.2 +1.5 +1.5 +-0.5 +-4 +0.15 +1 +7 +1 +1 +1 +-6 +-1 +0.1 +0.5 +0.5 +0.5 +0.5 +0.5 +-8 +0.5 +0.5 +-1.5 +0.05 +0 +-10 +0 +0 +-2 +0 +0 +0 +0 +-0.5 +-12 +-0.5 +-0.05 +-2.5 +-1 +-14 +-1 +0.5 +-0.5 +-0.5 +-0.5 +-0.5 +-0.1 +-3 +-1.5 +-16 +-1.5 +-0.15 +-3.5 +-1 +-1 +-1 +-1 +-18 +-2 +-2 +-0.2 +-4 +-20 +-2.5 +-2.5 +-1.5 +-1.5 +-1.5 +-1.5 +-1.5 +0.20.4 +-0.4 -0.200.20.47 +-0.4 -0.20 +-0.4-0.2 +0.20.4 +-0.4 -0.20 +[mm] +MPal +MPal +z= 1.50m +z=±1.34m +Z=±1.42m +z= ±1.43m +z=±1.42m +0.4 +0.4 +0.4 +0.4 +0.4 +0.22 +O +0.2 +0.2 +0.2 +0.2 +0.2 +-1 +0.2 +-10 +0 +2 +0.18 +-15 +-3 +0.2 +-0.2 +0.2 +-0.2 +-0.2 +2 +0.16 +-20 +-0.4. +-0.4-0.2 +0.2 +0.4 +0.4 +0.2 +0.2 +0 +0.2 +0.4 +0.2 +0.4 +0.4max | · | +ρ +z [m] +u3 +0.24 mm +a +± 1.50 +σz +20.77 MPa +a +± 1.42 +σr +4.23 MPa +a +±1.42 +σθ +2.62 MPa +a +± 1.43 +τ rz +2.52 MPa +ϵ +± 1.34 +Table 2: Maximum absolute values and points of Ω in which they are assumed. +5 +Proofs of the results +5.1 +Proof of Theorem 2.1 +By identity (2.6), the estimates (divu)2 ≤ |∇u|2 and |Du|2 ≤ |∇u|2 and the H¨older inequality we infer +that for any u, v ∈ H1(Ω; R3) +���� +� +Ω +Tu : Dv dx +���� = +����2µ +� +Ω +Du : Dv dx + λ +� +Ω +(divu)(divv) dx +���� ≤ (λ + 2µ) ∥u ∥H1 ∥v ∥H1 . +(5.1) +Estimate (5.1) proves the continuity of the bilinear form +a(u, v) = +� +Ω +Tu : Dv dx , +u, v ∈ H1(Ω; R3) . +By Korn inequality we also see that a(·, ·) is weakly coercive; indeed for any 0 < ε < 4µ we have +a(u, u) + ε∥u ∥2 +L2 ≥ 2µ +� +1 − ε +4µ +� � +Ω +|Du|2dx + ε +2 +� 1 +C +� +Ω +|∇u|2dx − +� +Ω +|u|2dx +� ++ ε +� +Ω +|u|2dx +(5.2) +≥ ε +2C +� +Ω +|∇u|2dx + ε +2 +� +Ω +|u|2dx ≥ min +� ε +2C , ε +2 +� +∥u ∥2 +H1 . +On the other hand, it is easy to check that the linear functional Λ : H1(Ω; R3) → R defined by +Λ(v) = +� +Ω +f · v dx + +� +∂Ω +g · v dx , +v ∈ H1(Ω; R3) +is continuous thanks to the H¨older inequality and the classical trace inequality for H1-functions. Hence, +we may write Λ ∈ (H1(Ω; R3))′. +With the notations introduced in this proof, the variational problem (2.7) may be written in the form +a(u, v) = ⟨Λ, v⟩ +for any v ∈ H1(Ω; R3) . +Introducing the linear continuous operator L : H1(Ω; R3) → (H1(Ω; R3))′ defined by +⟨Lu, v⟩ = a(u, v) +for any u, v ∈ H1(Ω; R3) , +we may write (2.7) in the form +Lu = Λ +(5.3) +as an identity between elements of the dual space (H1(Ω; R3))′. +The next step is to introduce the following operator Rε : (H1(Ω; R3))′ → H1(Ω; R3) which maps each +element h ∈ (H1(Ω; R3))′ into the unique solution w of the variational problem +a(w, v) + ε(w, v)L2 = ⟨h, v⟩ +for any v ∈ H1(Ω; R3) . +15 + +This problem admits a unique solution by the continuity and coercivity estimates (5.1), (5.2) combined +with the Lax-Milgram Theorem. In particular Rε is well defined and continuous. Moreover, Rε is invertible +and by the Open Mapping Theorem its inverse is also continuous. +In the rest of the proof we denote by J : H1(Ω; R3) → (H1(Ω; R3))′ the linear operator defined by +⟨Ju, v⟩ = +� +Ω +u · v dx +for any u, v ∈ H1(Ω; R3) , +which is compact as a consequence of the compact embedding H1(Ω; R3) ⊂ L2(Ω; R3). +We now introduce on H1(Ω; R3) the following scalar product +(u, v)ε = a(u, v) + ε(u, v)L2 +for any u, v ∈ H1(Ω; R3) , +which is equivalent to the natural scalar product of H1(Ω; R3) thanks to (5.1) and (5.2). +In this way we may now define the compact self-adjoint linear operator Tε : H1(Ω; R3) → H1(Ω; R3) +given by Rε ◦ J where by self-adjoint we mean (Tεu, v)ε = (u, Tεv)ε for any u, v ∈ H1(Ω; R3). Indeed, +from the definition of Rε, J and (·, ·)ε we see that +(Tεu, v)ε = (u, v)L2 = (u, Tεv)ε +for any u, v ∈ H1(Ω; R3) . +By definition of Rε and J we have that L = R−1 +ε +− εJ. In particular u ∈ H1(Ω; R3) is a solution of +(5.3) if and only if +−ε−1 Rε(R−1 +ε +− εJ)u = −ε−1 RεΛ +or equivalently Tεu − ε−1 u = w once we put +w = −ε−1 RεΛ. Then, applying the Fredholm alternative to the operator Tε we deduce that (5.3), or +equivalently (2.7), admits a solution u ∈ H1(Ω; R3) if and only if +w ∈ +� +Ker +� +T ∗ +ε − ε−1 IH1 +��⊥ = +� +Ker +� +Tε − ε−1 IH1 +��⊥ +(5.4) +where T ∗ +ε denotes the adjoint operator of Tε, IH1 denotes the identity map in H1(Ω; R3) and the orthogonal +spaces are defined in the sense of the scalar product (·, ·)ε. +We observe that a function v ∈ Ker +� +Tε − ε−1 IH1 +� +if and only if RεJv = ε−1 v and by the definition of +Rε this is equivalent to +a +� +ε−1 v, φ +� ++ ε +� +ε−1 v, φ +� +L2 = ⟨Jv, φ⟩ +for any φ ∈ H1(Ω; R3) +and, in turn, recalling the definition of J this is equivalent to a(v, φ) = 0 for any φ ∈ H1(Ω; R3). This +shows that Ker +� +Tε − ε−1 IH1 +� += V0 as we deduce by (2.13). +Let us proceed by proving (i)-(iv). +The proof of (i) is complete once we show that (5.4) is equivalent to condition (2.16). Condition (5.4) +is equivalent to +a(w, v) + ε(w, v)L2 = 0 +for any v ∈ V0, +(5.5) +being Ker +� +Tε − ε−1 IH1 +� += V0. But w = −ε−1 RεΛ so that by definition of Rε, we infer +a(w, v) + ε(w, v)L2 = −ε−1 [a(RεΛ, v) + ε(RεΛ, v)L2] = −ε−1 ⟨Λ, v⟩ +for any v ∈ H1(Ω; R3) . +(5.6) +Combining (5.5) and (5.6) we finally obtain ⟨Λ, v⟩ = 0 for any v ∈ V0, which is exactly (2.16) in view +of the definition of the functional Λ. +For the proof of (ii) we observe that by (2.2), (2.7), (2.13) and (2.14) we have for any v ∈ H1(Ω; R3) +� +Ω +T(u + v0) : Dv dx = +� +Ω +Tu : Dv dx + +� +Ω +Tv0 : Dv dx = +� +Ω +Tu : Dv dx = +� +Ω +f · v dx + +� +∂Ω +g · v dS +which shows that u + v0 is a solution of (2.7). +16 + +For the proof of (iii) we consider two solutions u and w of (2.7) and let v0 = w − u. By (2.2) and (2.7) +we obtain +� +Ω +Tv0 : Dv dx = +� +Ω +Tw : Dv dx − +� +Ω +Tu : Dv dx += +� +Ω +f · v dx + +� +∂Ω +g · v dS − +� +Ω +f · v dx − +� +∂Ω +g · v dS = 0 +for any v ∈ H1(Ω; R3) +which immediately gives v0 ∈ V0 thanks to (2.13). +Finally, let us proceed with the proof of (iv). First we prove the existence of a solution of (2.7) in V ⊥ +0 . +Let u be a generic solution of (2.7) and consider its orthogonal decomposition u = u0 + u1 ∈ V0 ⊕ V ⊥ +0 +with respect to the scalar product (2.12). Then, u1 = u − u0 ∈ V ⊥ +0 +and by part (ii) we deduce that u1 is +still a solution of (2.7). +Once we have proved existence, let us prove uniqueness. Let u, w ∈ V ⊥ +0 be two solutions of (2.7). Then, +on one hand we have that u − w ∈ V ⊥ +0 +and on the other hand u − v ∈ V0 thanks to part (iii). Therefore, +u − w ∈ V0 ∩ V ⊥ +0 = {0} and this readily implies u = w thus completing the proof of (iv). +□ +5.2 +Proof of Proposition 3.1 +Concerning part (i) of the Proposition we only give the proof of (3.4) since the proof of (3.5)-(3.6) can +be obtained with a similar procedure. For any function v ∈ H1(Ω; R3) we denote by ¯v = (¯v1, ¯v2, ¯v2) ∈ +H1(Ω; R3) the function defined by +¯v1(x, y, z) = v1(x, y, −z) , +¯v2(x, y, z) = v2(x, y, −z) , +¯v3(x, y, z) = −v3(x, y, −z) , +∀ (x, y, z) ∈ Ω. (5.7) +Let u be the unique solution of (3.3) in V ⊥ +0 +and let ¯u be the corresponding function defined by (5.7). +We start by showing that ¯u solves problem (3.3). In doing this we show that it solves the variational +problem (2.7) where in the present case f = 0 and g is the function defined in (3.1). +By direct computation one can see that for any test function v ∈ H1(Ω; R3) we have for any (x, y, z) ∈ Ω +(D¯u : Dv)|(x,y,z) = (Du : D¯v)|(x,y,−z) , +[(div ¯u)(div v]|(x,y,z) = [(div u)(div ¯v]|(x,y,−z) . +(5.8) +By (2.7), (2.2), (5.8), (3.1) and a change of variables, we obtain +2µ +� +Ω +D¯u : Dv dx + λ +� +Ω +(div ¯u)(div v) dx +(5.9) += 2µ +� +Ω +Du : D¯v dx + λ +� +Ω +(div u)(div ¯v) dx = +� +∂Ω +g · ¯v dS = +� +∂Ω +g · v dS . +By (5.9) we deduce that ¯u is a solution of (2.7) and hence a weak solution of (3.3). We now prove that +¯u ∈ V ⊥ +0 . Indeed, proceeding as in (5.9) one can easily show that (¯u, v)T = (u, ¯v)T = 0 for any v ∈ V0 +since u ∈ V ⊥ +0 and ¯v ∈ V0 whenever v ∈ V0, as one can deduce by (2.15). This completes the proof of (3.4). +Let us proceed with the proof of part (ii) and (iii) of the proposition. For any θ ∈ (−2π, 2π) we denote +by Rθ : R2 → R2 the anticlockwise rotation of an angle θ and by Aθ the associate matrix. Clearly we have +that the inverse map of Rθ is given by R−θ and A−1 +θ += A−θ. +We use the notation u = (u′, u3) ∈ R2 × R with u′ = (u1, u2) and we denote by +∇′u′ = +� ∂u1 +∂x +∂u1 +∂y +∂u2 +∂x +∂u2 +∂y +� +17 + +its Jacobian matrix in the x and y variables, and by D′u′ the corresponding symmetric gradient given by +1 +2 +� +∇′u′ + (∇′u′)T � +; more in general, throughout this proof we will use the symbol ∇′ for denoting the +gradient with respect to the x and y variables. +We now define +uθ(x, y, z) = +� +R−θ +� +u1(Rθ(x, y), z), u2(Rθ(x, y), z) +� +, u3(Rθ(x, y), z) +� +for any (x, y, z) ∈ Ω . +Then, the Jacobian matrix ∇uθ ∈ R3×3 and in turn the matrix Duθ admit a representation in terms of +four blocks of dimensions 2×2, 2×1, 1×2, 1×1 respectively. We proceed directly with the representation +of Duθ: +Duθ = +� +� +� +� +A−θ D′u′� +Rθ(x, y), z +� +Aθ +A−θ ∂u′ +∂z +� +Rθ(x, y), z +� ++ +� +∇′u3 +� +Rθ(x, y), z +� +Aθ +�T +� +A−θ ∂u′ +∂z +� +Rθ(x, y), z +��T ++∇′u3 +� +Rθ(x, y), z +� +Aθ +∂u3 +∂z +� +Rθ(x, y), z +� +� +� +� +� +(5.10) +In the same way, for any test function v ∈ H1(Ω; R3) and any θ ∈ (−2π, 2π) we may define the correspond- +ing function vθ. Looking at v as (v−θ)θ and applying (5.10) to v−θ we claim that for any (x, y, z) ∈ Ω +Duθ(x, y, z) : Dv(x, y, z) = Du +� +Rθ(x, y), z +� +: Dv−θ +� +Rθ(x, y), z +� +. +(5.11) +This is a consequence of the fact that Aθ is orthogonal and the linear map Lθ : R2×2 → R2×2, Lθ(X) = +A−θXAθ is an isometry in R2×2 as one can see by verifying the orthogonality of the associated matrix +Mθ ∈ R4×4. This implies +(A−θ XAθ) : Y = Lθ(X) : Y = Lθ(X) : Lθ(L−1 +θ (Y )) = X : L−1 +θ (Y ) = X : (AθY A−θ) +for any X, Y ∈ R2×2. This arguments allow to treat the scalar products between the 2×2 block appearing +in the representation (5.10). Even easier is to treat the scalar products between the 2 × 1 and 1 × 2 blocks +thanks to the orthogonality of Aθ. This proves the claim (5.11). +The invariance of the trace of a matrix X under maps of the form X �→ A−1XA combined with (5.10) +shows that div uθ(x, y, z) = div u +� +Rθ(x, y), z +� +and in particular for any (x, y, z) ∈ Ω we have +(div uθ(x, y, z))(div v(x, y, z)) = +� +div u +� +Rθ(x, y), z +�� � +div v−θ +� +Rθ(x, y), z +�� +. +(5.12) +By (2.7), (3.1), (3.2), (5.11), (5.12), two changes of variables and the definitions of v−θ and g, we obtain +2µ +� +Ω +Duθ : Dv dx + λ +� +Ω +(div uθ)(div v) dx +(5.13) += 2µ +� +Ω +Du : Dv−θ dx + λ +� +Ω +(div u) (div v−θ) dx = +� +∂Ω +g · v−θ dS = +� +∂Ω +g · v dS . +We have just proved that uθ is still a weak solution of (3.3). We now show that uθ ∈ V ⊥ +0 as a consequence +of the fact that u ∈ V ⊥ +0 . Proceeding as in (5.13), we infer +(uθ, v)T = (u, v−θ)T +for any v ∈ H1(Ω; R3) . +(5.14) +We need to prove that if v ∈ V0 then v−θ ∈ V0. For any θ ∈ (−2π, 2π), let Bθ be the 3 × 3 matrix +corresponding to an anticlockwise rotation of an angle θ around the z axis. Clearly Bθ is orthogonal and +B−1 +θ += B−θ. With this notation we may write +v−θ(x) = Bθ v(B−θ x) +for any x ∈ R3 +(5.15) +18 + +where both x and v have to be considered vector columns in the right hand side of the identity. +If v ∈ V0, then by (2.15) we have that v admits the following matrix representation +v(x) = Mx + δ +for any x ∈ R3 +(5.16) +where M is an antisymmetric matrix and δ = (δ1 δ2 δ3)T . +Combining (5.15) and (5.16) we obtain v−θ(x) = BθMB−θ x + Bθδ where the matrix BθMB−θ is +antisymmetric since +(BθMB−θ)T = BT +−θ MT BT +θ = B−1 +−θ(−M)B−1 +θ += −BθMB−θ . +This proves that also v−θ ∈ V0 since it admits a representation like in (2.15). +Now, if we choose v ∈ V0 in (5.14), we readily see that (uθ, v)T = 0 being u ∈ V ⊥ +0 +and v−θ ∈ V0. This +proves that uθ ∈ V ⊥ +0 . +By the uniqueness result stated in Theorem 2.1 (iv) we infer that uθ = u for any θ ∈ (−2π, 2π). +Now the validity of (ii) and of the first part of (iii) follows immediately from the definition of uθ. +It remains to observe that the vector field u′ is oriented radially in the xy-plane. To do this, it is +sufficient to combine the identity u = uθ with the identity u2(x, 0, z) = 0, valid for any a < x < b and +z ∈ +� +− h +2, h +2 +� +, as a consequence of (3.6). +□ +5.3 +Proof of Lemma 3.2 +Let us introduce the sequence of intervals Ik := +� +− h +2 + kh, h +2 + kh +� +, the corresponding sequence of domains +Ωk := Ca,b × Ik and the sequence of functions gk : ∂Ωk → R3 +gk(x, y, z) := +� +� +� +� +� +� +� +� +� +(0, 0, (−1)k χp(x, y)) +if (x, y, z) ∈ Ca,b × +� +− h +2 + kh +� +, +(0, 0, (−1)k+1 χp(x, y)) +if (x, y, z) ∈ Ca,b × +� h +2 + kh +� +, +(0, 0, 0) +if (x, y, z) ∈ ∂Ca,b × Ik . +(5.17) +We know that the original function u is a weak solution of problem (3.3) in the sense that +� +Ω +Tu : Dv dx = +� +∂Ω +g · v dS +for any v ∈ H1(Ω; R3) . +(5.18) +We need to find, starting from (5.18), the equation solved, in the sense of distributions, by the periodic +extension. First of all, we observe that by (3.8), (5.17), (5.18) and some computations, we have +� +Ωk +Tu : Dv dx = +� +∂Ωk +gk · v dS +for any v ∈ H1(Ωk; R3) . +(5.19) +Now, letting φ = (φ1, φ2, φ3) ∈ D(Ca,b × R; R3), by (5.19) we infer +� +Ca,b×R +Tu : Dφ dx = +� +k∈Z +� +Ωk +Tu : Dφ dx = +� +k∈Z +� +∂Ωk +gk · φ dS += +� +k∈Z +2(−1)k+1 +� +Ca,b +χp(x, y) φ3 +� +x, y, h +2 + kh +� +dxdy . +This proves (3.9), where Λ is the distribution defined by +⟨Λ, φ⟩ := +� +Ca,b +χp(x, y) +� +k∈Z +2(−1)k+1φ3 +� +x, y, h +2 + kh +� +dxdy +(5.20) +19 + +for any φ = (φ1, φ2, φ3) ∈ D(Ca,b × R; R3). +The distribution Λ admits a sort of factorization as a product of a function in the variables x and y +and of a distribution acting on functions of the variable z: +Λ = (Λ1, Λ2, Λ3) = +� +0, 0, 2 χp +� +k∈Z +(−1)k+1 δ h +2 +kh +� +where Λ1, Λ2, Λ3 ∈ D′(Ca,b × R; R) are the scalar distributions defined by +⟨Λi, φ⟩ := ⟨Λ, φ ei⟩ +for any φ ∈ D(Ca,b × R; R) , +with e1 = i, e2 = j, e3 = k, and δ h +2 +kh are Dirac delta distributions concentrated at z = h +2 + kh. +Expanding in Fourier series the periodic distribution � +k∈Z 2(−1)k+1 δ h +2 +kh we obtain (3.10), where +the Fourier series converges in the sense of distributions. For more details on this convergence see the +arguments introduced in subsection 5.7. +□ +5.4 +Proof of Lemma 3.3 +First of all we insert (3.11) into (3.9); recalling the Hooke’s law (2.2) and exploiting (3.10), we obtain +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +−µ∆ϕ1 +k + µπ2k2 +h2 +ϕ1 +k − (λ + µ) +�∂2ϕ1 +k +∂x2 + ∂2ϕ2 +k +∂x∂y + πk +h +∂ϕ3 +k +∂x +� += 0 +in Ca,b , +−µ∆ϕ2 +k + µπ2k2 +h2 +ϕ2 +k − (λ + µ) +� ∂2ϕ1 +k +∂x∂y + ∂2ϕ2 +k +∂y2 + πk +h +∂ϕ3 +k +∂y +� += 0 +in Ca,b , +−µ∆ϕ3 +k + µπ2k2 +h2 +ϕ3 +k + πk +h (λ + µ) +�∂ϕ1 +k +∂x + ∂ϕ2 +k +∂y + πk +h ϕ3 +k +� += Ψk +in Ca,b , +(5.21) +where the forcing term is defined in (3.13). We observe that in (5.21), the operator ∆ stands for the +Laplace operator in the variables x and y, i.e. ∆ = ∂2/∂x2 + ∂2/∂y2. +Putting Φk := (ϕ1 +k, ϕ2 +k) and ¯n ∈ R2 the outward unit normal to ∂Ca,b, system (5.21) may be rewritten +in the following form +� +� +� +−µ∆Φk + µ π2k2 +h2 Φk − (λ + µ)∇(div Φk) − (λ + µ) πk +h ∇ϕ3 +k = 0 +in Ca,b , +−µ∆ϕ3 +k + µ π2k2 +h2 ϕ3 +k + πk +h (λ + µ) +� +div Φk + πk +h ϕ3 +k +� += Ψk +in Ca,b , +(5.22) +or equivalently in the following form +� +� +� +−div(λ(div Φk)I + 2µDΦk) + µ π2k2 +h2 Φk − (λ + µ) πk +h ∇ϕ3 +k = 0 +in Ca,b , +−µ∆ϕ3 +k + (λ + 2µ) π2k2 +h2 ϕ3 +k + (λ + µ) πk +h div Φk = Ψk +in Ca,b , +(5.23) +where D represents here the symmetric gradient in the two-dimensional case and I is the 2 × 2 identity +matrix. +We also recall that by (3.3), (Tu)n = 0 on ∂Ca,b × R so that by the Hooke’s law (2.2) we obtain +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +(λ + 2µ)x∂u1 +∂x + λx∂u2 +∂y + λx∂u3 +∂z + µy +�∂u1 +∂y + ∂u2 +∂x +� += 0 +on ∂Ca,b × R , +λy∂u1 +∂x + (λ + 2µ)y∂u2 +∂y + λy∂u3 +∂z + µx +�∂u1 +∂y + ∂u2 +∂x +� += 0 +on ∂Ca,b × R , +µx +�∂u1 +∂z + ∂u3 +∂x +� ++ µy +�∂u2 +∂z + ∂u3 +∂y +� += 0 +on ∂Ca,b × R , +20 + +and by (3.11) we obtain +� +� +� +λ(div Φk)¯n + 2µ(DΦk)¯n + λ πk +h ϕ3 +k ¯n = 0 +on ∂Ca,b , +µ∇ϕ3 +k · ¯n − µ πk +h Φk · ¯n = 0 +on ∂Ca,b . +(5.24) +Let us derive the weak formulation of (5.22)-(5.24). Testing (5.23) with (w1, w2, w3), putting W = (w1, w2) +and integrating by parts we obtain +− +� +∂Ca,b +[(λ(div Φk)I + 2µDΦk) ¯n] · W ds + +� +Ca,b +(λ(div Φk)I + 2µDΦk) : ∇W dxdy +(5.25) ++ µπ2k2 +h2 +� +Ca,b +Φk · W dxdy − µπk +h +� +Ca,b +∇ϕ3 +k · W dxdy − λπk +h +� +∂Ca,b +ϕ3 +k ¯n · W ds + λπk +h +� +Ca,b +ϕ3 +k div W dxdy +− +� +∂Ca,b +µ∂ϕ3 +k +∂¯n w3ds + µ +� +Ca,b +∇ϕ3 +k · ∇w3 dxdy + (λ + 2µ)π2k2 +h2 +� +Ca,b +ϕ3 +k w3 dxdy ++ µπk +h +� +∂Ca,b +w3Φk · ¯n ds − µπk +h +� +Ca,b +Φk · ∇w3 dxdy + λπk +h +� +Ca,b +div Φk w3 dxdy = +� +Ca,b +Ψk w3 dxdy . +We observe that by (5.24) the boundary integrals in (5.25) disappear; on the other hand collecting the +double integrals and recalling that DΦk : ∇W = DΦk : DW, we may write (5.25) in the form +2µ +� +Ca,b +DΦk : DW dxdy + λ +� +Ca,b +� +div Φk + πk +h ϕ3 +k +� � +div W + πk +h w3� +dxdy +(5.26) ++ µ +� +Ca,b +(∇ϕ3 +k − πk +h Φk) · (∇w3 − πk +h W)dxdy + 2µ π2k2 +h2 +� +Ca,b +ϕ3 +kw3dxdy= +� +Ca,b +Ψkw3dxdy +for any w ∈ H1(Ca,b; R3), where W = (w1, w2). This represents the weak form of (5.22)-(5.24). +For any k ≥ 2 even we observe that, by (3.13), ϕ1 +k ≡ ϕ2 +k ≡ ϕ3 +k ≡ 0 in Ca,b, as one can deduce by testing +(5.26) with (w1, w2, w2) = (ϕ1 +k, ϕ2 +k, ϕ3 +k). +For any k ≥ 1 odd, we define the following bilinear form +¯ak(ϕ, w) := 2µ +� +Ca,b +DΦ : DW dxdy + λ +� +Ca,b +� +div Φ + πk +h ϕ3 +k +� � +div W + πk +h w3� +dxdy +(5.27) ++ µ +� +Ca,b +(∇ϕ3 − πk +h Φ) · (∇w3 − πk +h W) dxdy + 2µ π2k2 +h2 +� +Ca,b +ϕ3 w3 dxdy +for any ϕ, w ∈ H1(Ca,b; R3) +where ϕ = (ϕ1, ϕ2, ϕ3), Φ := (ϕ1, ϕ2), w = (w1, w2, w3) and W = (w1, w2). +For the uniqueness issue we claim that for any ε > 0 there exists Cε > 0 such that +¯ak(ϕ, ϕ) + ε∥ϕ∥2 +L2 ≥ Cε∥ϕ∥2 +H1 +for any ϕ ∈ H1(Ca,b; R3) . +(5.28) +Suppose by contradiction that there exists ε > 0 such that for any m ≥ 1 there exists ϕm ∈ H1(Ca,b; R3) +such that +¯ak(ϕm, ϕm) + ε∥ϕm∥2 +L2 ≤ 1 +m∥ϕm∥2 +H1 . +(5.29) +Up to normalization, it is not restrictive to assume that the sequence {ϕm} satisfies ∥ϕm∥H1 = 1 for +any m ≥ 1, so that by (5.27) and (5.29) we infer +ϕm → 0 +in L2(Ca,b; R3) , +� +Ca,b +|DΦm|2dxdy → 0 , +∇ϕ3 +m − kΦm → 0 +in L2(Ca,b; R2) +(5.30) +21 + +as m → +∞. Applying (2.11) in the two-dimensional case we obtain +� +Ca,b +|∇Φm|2dxdy ≤ C +�� +Ca,b +|DΦm|2dxdy + +� +Ca,b +|Φm|2 dxdy +� +for some constant C > 0. This, combined with (5.30), proves that +∇Φm → 0 +in L2(Ca,b; R2×2) , +∇ϕ3 +m → 0 +in L2(Ca,b; R2) +and, in turn, that ϕm → 0 in H1(Ca,b; R3). This contradicts the assumption ∥ϕm∥H1 = 1. We have +completed the proof of the claim (5.28). +Thanks to (5.28), we may proceed as in the proof of Theorem 2.1 and apply the Fredholm alternative +to show that (5.23)-(5.24) admits a solution if and only if +� +Ca,b +Ψk w3 dxdy = 0 +for any w = (w1, w2, w3) ∈ ¯Vk +(5.31) +where ¯Vk := {w ∈ H1(Ca,b; R3) : ¯ak(w, v) = 0 for any v ∈ H1(Ca,b; R3)}. Testing the variational identity +in the definition of ¯Vk with v = w, we readily see that for any k ≥ 1 we ¯Vk = {0} and hence, condition +(5.31) is always satisfied. This completes the proof of the lemma. +□ +5.5 +Proof of Lemma 3.5 +Before proceeding with the proof of the lemma, we devote the first part of this subsection to show that +the functions Yk and Zk introduced in (3.12) really satisfy (3.14). +In order to simplify the notations we denote by Y and Z the unknown functions, omitting the index k. +Testing (5.26) with a test function (w1, w2, w3) admitting in polar coordinates the following representation +w1(ρ, θ) = H(ρ) cos θ , +w2(ρ, θ) = H(ρ) sin θ , +w3(ρ, θ) = K(ρ) , +by (3.12) we obtain +2µ +� b +a +� +ρY ′(ρ)H′(ρ) + Y (ρ)H(ρ) +ρ +� +dρ + λ +� b +a +ρ +� +Y ′(ρ) + Y (ρ) +ρ ++ πk +h Z(ρ) +� � +H′(ρ) + H(ρ) +ρ ++ πk +h K(ρ) +� +dρ +(5.32) ++ µ +� b +a +ρ +� +Z′(ρ) − πk +h Y (ρ) +� � +K′(ρ) − πk +h H(ρ) +� +dρ + 2µπ2k2 +h2 +� b +a +ρZ(ρ)K(ρ) dρ = +� b +a +ρΨk(ρ)K(ρ) dρ +with obvious meaning of the notation Ψk(ρ) being it a radial function. +Collecting in a proper way the terms of (5.32), we may rewrite it in the form +� b +a +� +(λ + 2µ)ρY ′(ρ) + λY (ρ) + λπk +h ρZ(ρ) +� +H′(ρ) dρ +(5.33) ++ +� b +a +� +λY ′(ρ) + (λ + 2µ)Y (ρ) +ρ ++ µπ2k2 +h2 ρY (ρ) − µπk +h ρZ′(ρ) + λπk +h Z(ρ) +� +H(ρ) dρ ++ +� b +a +� +µρZ′(ρ) − µπk +h ρY (ρ) +� +K′(ρ) dρ + +� b +a +� +(λ + 2µ)π2k2 +h2 ρZ(ρ) + λπk +h ρY ′(ρ) + λπk +h Y (ρ) +� +K(ρ) dρ += +� b +a +ρΨk(ρ)K(ρ) dρ +Integrating by parts the terms in (5.33) containing H′(ρ) and K′(ρ), we see that (5.33) is the variational +formulation of (3.14). +22 + +Let us proceed now with the proof of the lemma which is the main point of this section. Actually, we +give here only a sketch of the proof since it essentially follows the ideas already introduced in the proof of +Lemma 3.3. +About the uniqueness issue, on the space H1(a, b; R2) it sufficient to define the bilinear form +bk : H1(a, b; R2) × H1(a, b; R2) → R +corresponding to the left hand side of (5.32) and prove for it an estimate of the type (5.28). +Then, following again the proof of Lemma 3.3, one finds that the compatibility condition for Ψk is given +by +� b +a +ρΨk(ρ)K(ρ) dρ = 0 +(5.34) +for any (H, K) ∈ H1(a, b; R2) satisfying bk +� +(H, K), (H, K) +� += 0. A simple check shows that (H, K) ≡ +(0, 0) so that (5.34) is trivially satisfied. +The Fredholm alternative then implies the existence of a solution. +□ +5.6 +Proof of Lemma 3.6 +We omit for simplicity the dependence from the index k in the unknowns Yk and Zk. For more clarity we +divide the construction of this representation of Y and Z into different steps each of them is contained in +the next subsections. +5.6.1 +The solution of the homogeneous system +We consider the homogeneous version of the system in (3.14) +� +� +� +� +� +� +� +� +� +Y ′′(ρ) + Y ′(ρ) +ρ +− Y (ρ) +ρ2 +− α k2 Y (ρ) + β kZ′(ρ) = 0 +ρ > 0 , +Z′′(ρ) + Z′(ρ) +ρ +− γ k2Z(ρ) − δ k +� +Y ′(ρ) + Y (ρ) +ρ +� += 0 +ρ > 0 , +(5.35) +where we put for simplicity +α = +π2µ +h2(λ + 2µ) , +β = π(λ + µ) +h(λ + 2µ) , +γ = π2(λ + 2µ) +h2µ +, +δ = π(λ + µ) +hµ +. +We look for a solution admitting the following expansion +� +� +� +� +� +� +� +� +� +� +� +� +� +Y (ρ) = ++∞ +� +n=−1 +an ρn + (ln ρ) ++∞ +� +n=0 +bn ρn , +Z(ρ) = ++∞ +� +n=0 +cn ρn + (ln ρ) ++∞ +� +n=0 +dn ρn . +(5.36) +Inserting the representation (5.36) in the system (5.35), we obtain for each of the two equations the +following identities: +23 + ++∞ +� +n=−1 +n(n − 1)an ρn + ++∞ +� +n=0 +(n − 1)bn ρn + ++∞ +� +n=0 +nbn ρn + (ln ρ) ++∞ +� +n=0 +n(n − 1)bn ρn ++ ++∞ +� +n=−1 +nan ρn + ++∞ +� +n=0 +bn ρn + (ln ρ) ++∞ +� +n=0 +nbn ρn +−αk2 ++∞ +� +n=1 +an−2 ρn − αk2(ln ρ) ++∞ +� +n=2 +bn−2 ρn − ++∞ +� +n=−1 +an ρn − (ln ρ) ++∞ +� +n=0 +bn ρn ++βk ++∞ +� +n=1 +(n − 1)cn−1 ρn + βk ++∞ +� +n=1 +dn−1 ρn + βk(ln ρ) ++∞ +� +n=1 +(n − 1)dn−1 ρn = 0 , +(5.37) ++∞ +� +n=0 +n(n − 1)cn ρn + ++∞ +� +n=0 +(n − 1)dn ρn + ++∞ +� +n=0 +ndn ρn + (ln ρ) ++∞ +� +n=0 +n(n − 1)dn ρn ++ ++∞ +� +n=0 +ncn ρn + ++∞ +� +n=0 +dn ρn + (ln ρ) ++∞ +� +n=0 +ndn ρn − γk2 ++∞ +� +n=2 +cn−2 ρn − γk2(ln ρ) ++∞ +� +n=2 +dn−2 ρn +−δk ++∞ +� +n=0 +(n − 1)an−1 ρn − δk ++∞ +� +n=1 +bn−1 ρn − δk(ln ρ) ++∞ +� +n=1 +(n − 1)bn−1 ρn +−δk ++∞ +� +n=0 +an−1 ρn − δk(ln ρ) ++∞ +� +n=1 +bn−1 ρn = 0 . +(5.38) +To determine the values of the coefficients an, bn, cn, dn we need an iterative scheme starting from the +values of the coefficients a−1, a0, a1, b0, b1, c0, c1, d0, d1. The values of these nine parameters have to be +determined collecting the coefficients of the terms ρ−1, ρ0, ρ0 ln ρ, ρ, ρ ln ρ appearing in (5.37)-(5.38) and +equating them to zero. +As a result of this procedure we obtain the following constraint: +� +a0 = b0 = c1 = d1 = 0 , +2b1 + βkd0 = αk2a−1 . +(5.39) +Among the left five parameters a−1, a1, b1, c0, d0 that may be possibly different from zero, a1, c0 and two +among a−1, b1, d0 can be chosen arbitrarily, while the remaining one is determined by the equation in the +second line of (5.39); for example, we may choose arbitrarily a−1, a1, b1, c0 and put d0 = αk +β a−1 − +2 +βk b1. +In particular, we are interested in finding the general solution of (5.35) as a linear combination of four +linearly independent special solutions, denoted by Υj = (Y j, Zj) with j = 1, . . . , 4. A possible choice +for the independent solutions is given respectively by the assumption on the following combinations of +coefficients: +Υ1 : (a−1, a1, b1, c0) = (1, 0, 0, 0) , +Υ2 : (a−1, a1, b1, c0) = (0, 1, 0, 0) , +Υ3 : (a−1, a1, b1, c0) = (0, 0, 1, 0) , +Υ4 : (a−1, a1, b1, c0) = (0, 0, 0, 1). +(5.40) +By (5.37)-(5.38) we deduce the following linear system in the unknowns an, bn, cn−1, dn−1 with data +24 + +expressed in terms of an−2, bn−2, cn−3, dn−3: +� +� +� +� +� +� +� +� +� +� +� +(n2 − 1)an + 2nbn + βk(n − 1)cn−1 + βkdn−1 = αk2an−2 +(n2 − 1)bn + βk(n − 1)dn−1 = αk2bn−2 +(n − 1)2cn−1 + 2(n − 1)dn−1 = δk(n − 1)an−2 + δkbn−2 + γk2cn−3 +(n − 1)2dn−1 = δk(n − 1)bn−2 + γk2dn−3 +(n ≥ 3). +(5.41) +We observe that the matrix of coefficients associated to system (5.41) is given by +� +� +� +� +� +� +n2 − 1 +2n +β(n − 1)k +βk +0 +n2 − 1 +0 +β(n − 1)k +0 +0 +(n − 1)2 +2(n − 1) +0 +0 +0 +(n − 1)2 +� +� +� +� +� +� +whose determinant is given by (n−1)6(n+1)2 ̸= 0, thus showing that the system is not singular for n ≥ 2 +and hence admits a unique solution. +With the restriction n ≥ 3 the coefficients a2, b2 remained excluded, but their calculation can be +obtained from the first two equations of (5.41) by choosing n = 2; this gives a2 = b2 = 0. +The linear independence of Υ1, Υ2, Υ3, Υ4 can be verified by looking at the asymptotic behavior of +Y j(ρ), j = 1, 2, 3, 4 as ρ → 0+ in the four cases (5.40): +case 1: +Y 1(ρ) ∼ ρ−1 +as ρ → 0+ ; +case 2: +Y 2(ρ) ∼ ρ +as ρ → 0+ ; +case 3: +Y 3(ρ) ∼ ρ ln ρ +as ρ → 0+ ; +case 4: +Y 4(ρ) = O(ρ2 ln ρ) +as ρ → 0+ . +Remark 5.1. We observe that, after a suitable scaling, the dependence of system (5.35) from the parameter +k can be dropped: given a solution (Y, Z) of (5.35), we may define the functions �Y (t) = Y +� h +πk t +� +and +�Z(t) = Z +� h +πk t +� +in such a way that the couple (�Y , �Z) solves system +� +� +� +� +� +� +� +� +� +�Y ′′(t) + +�Y ′(t) +t +− +�Y (t) +t2 +− �α �Y (t) + �β �Z′(t) = 0 +t > 0 , +�Z′′(t) + +�Z′(t) +t +− �γ �Z(t) − �δ +� +�Y ′(t) + +�Y (t) +t +� += 0 +t > 0 , +(5.42) +where �α = µ/(λ + 2µ), �β = (λ + µ)/(λ + 2µ), �γ = (λ + 2µ)/µ and �δ = (λ + µ)/µ. +5.6.2 +The particular solution +We write the nonhomogeneous system in the matrix form +� +� +� +� +� +� +Y (ρ) +Y ′(ρ) +Z(ρ) +Z′(ρ) +� +� +� +� +� +� +′ += +� +� +� +� +� +� +0 +1 +0 +0 +1 +ρ2 + αk2 +− 1 +ρ +0 +−βk +0 +0 +0 +1 +δk +ρ +δk +γk2 +− 1 +ρ +� +� +� +� +� +� +� +� +� +� +� +� +Y (ρ) +Y ′(ρ) +Z(ρ) +Z′(ρ) +� +� +� +� +� +� ++ +� +� +� +� +� +� +0 +0 +0 +− 1 +µΨk(ρ) +� +� +� +� +� +� +, +ρ > 0 , +(5.43) +where the function Ψk = Ψk(ρ) is extended trivially outside the interval (a, b). +25 + +Maintaining the order of the components, we may write the Wronskian matrix associated with Υ1, Υ2, +Υ3, Υ4 in the form +W(ρ) = +� +� +� +� +� +� +Y 1(ρ) +Y 2(ρ) +Y 3(ρ) +Y 4(ρ) +(Y 1(ρ))′ +(Y 2(ρ))′ +(Y 3(ρ))′ +(Y 4(ρ))′ +Z1(ρ) +Z2(ρ) +Z3(ρ) +Z4(ρ) +(Z1(ρ))′ +(Z2(ρ))′ +(Z3(ρ))′ +(Z4(ρ))′ +� +� +� +� +� +� +, +so that a particular solution Υ = (Y , Z) of (5.43) is given by (3.16). +5.6.3 +The unique solution of (3.14) +Applying the superposition principle we get (3.15). In order to obtain the unique solution (Y, Z) of the +boundary value problem (3.14), it remains to determine the constants C1, C2, C3, C4 so that the boundary +conditions at ρ = a and ρ = b are satisfied. +We check that the constants C1, C2, C3, C4 are uniquely determined. They solve the system +A +� +� +� +� +C1 +C2 +C3 +C4 +� +� +� +� = +� +� +� +� +� +� +� +� +� +− +� +(λ + 2µ)Y +′(a) + λ +a Y (a) + λ πk +h Z(a) +� +− +� +(λ + 2µ)Y +′(b) + λ +b Y (b) + λ πk +h Z(b) +� +− +� +Z +′(a) − πk +h Y (a) +� +− +� +Z +′(b) − πk +h Y (b) +� +� +� +� +� +� +� +� +� +� +where the matrix A = (aij), i, j ∈ {1, 2, 3, 4}, is given by +a1j = (λ + 2µ)(Y j)′(a) + λ +a Y j(a) + λ πk +h Zj(a) , +a2j = (λ + 2µ)(Y j)′(b) + λ +b Y j(b) + λ πk +h Zj(b) , +a3j = (Zj)′(a) − πk +h Y j(a) , +a4j = (Zj)′(b) − πk +h Y j(b) . +We claim that the matrix A is not singular. Consider the homogeneous linear system Ad = 0 with +d = (D1, D2, D3, D4)T . Then the function Γ = (G, H) given by +Γ(ρ) = D1Υ1(ρ) + D2Υ2(ρ) + D3Υ3(ρ) + D4Υ4(ρ) +solves system (5.35) coupled with the boundary conditions +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +(λ + 2µ)G′(a) + λ +a G(a) + λ πk +h H(a) = 0 , +(λ + 2µ)G′(b) + λ +b G(b) + λ πk +h H(b) = 0 , +H′(a) − πk +h G(a) = 0 , +H′(b) − πk +h G(b) = 0 . +By Lemma 3.5 we then have that Γ ≡ (0, 0) in (a, b) but being Γ also a solution of system (5.35) for +ρ ∈ (0, +∞), by local uniqueness for Cauchy problems, Γ ≡ (0, 0) in (0, +∞). The linear independence of +the functions Υ1, Υ2, Υ3, Υ4, then implies D1 = D2 = D3 = D4 = 0. +We just proved that the linear system Ad = 0 admits only the trivial solution, thus completing the +proof. +□ +26 + +5.7 +Proof of Theorem 3.7 +The formal series contained in (3.18) are a consequence of (3.11), Lemma 3.3 and Lemma 3.5. +It remains to show how those series converge. We start by proving the weak convergence in H1(Ω). Let +F be the linear functional defined by F(v) := +� +∂Ω g · v dS for any v ∈ H1(Ω; R3) with g as in (3.1). We +observe that thanks to the H¨older inequality and the trace inequality F ∈ (H1(Ω; R3))′: +��(H1(Ω;R3))′⟨F, v⟩H1(Ω;R3) +�� ≤ 2p +� +π(b2 − a2) C(Ω) ∥v∥H1(Ω;R3) +for any v ∈ H1(Ω; R3) , +where C(Ω) is such that ∥trace(v)∥L2(∂Ω) ≤ C(Ω)∥v∥H1(Ω) for any v ∈ H1(Ω). +Writing F = (F1, F2, F3) we have that F1, F2, F3 ∈ (H1(Ω))′ with F1 = F2 are the null functionals and +(H1(Ω))′⟨F3, v⟩H1(Ω) = − +� +Ca,b +χp(x, y) +� +v +� +x, y, h +2 +� +− v +� +x, y, − h +2 +�� +dxdy +for any v ∈ H1(Ω) . +Let us define the sequence of partial sums corresponding to the Fourier expansion in (3.10): +SM(x, y, z) := χp(x, y) +M +� +m=0 +(−1)m+1 4 +h sin +�π +h(2m + 1)z +� +. +We claim that SM ⇀ F3 weakly in (H1(Ω))′ as M → +∞. We first prove that the sequence {SM} is +bounded in (H1(Ω))′. In the next estimate we use the following notations: we put �Ω := Ca,b × +� +− h +2, 3h +2 +� +, +we still denote by v the symmetric and 2h-periodic extension of a function v ∈ H1(Ω) (see Section 3.1) +and by v +� +x, y, 3h +2 +� +and v +� +x, y, − h +2 +� +, the traces of a function v ∈ H1(�Ω) on the upper and lower faces of +the hollow cylinder �Ω, respectively. +��(H1(Ω))′⟨SM, v⟩H1(Ω) +�� = +����� +M +� +m=0 +(−1)m+1 4 +h +� +Ω +χp(x, y) sin +�π +h(2m + 1)z +� +v(x, y, z) dxdydz +����� += +����� +M +� +m=0 +(−1)m+1 2 +h +� +�Ω +χp(x, y) sin +�π +h(2m + 1)z +� +v(x, y, z) dxdydz +����� +(integration by parts) += +����� +M +� +m=0 +(−1)m+1 2 +h +�� +Ca,b +−h cos +� 3π +2 (2m + 1) +� +χp(x, y) +π(2m + 1) +v +� +x, y, 3h +2 +� +dxdy ++ +� +Ca,b +h cos +� +− π +2 (2m + 1) +� +χp(x, y) +π(2m + 1) +v +� +x, y, −h +2 +� +dxdy ++ +� +�Ω +hχp(x, y) +π(2m + 1) cos +�π +h(2m + 1)z +� ∂v +∂z (x, y, z) dxdydz +����� +(2h-periodicity of v) += +����� +M +� +m=0 +(−1)m+1 2 +h +�� +�Ω +hχp(x, y) +π(2m + 1) cos +�π +h(2m + 1)z +� ∂v +∂z (x, y, z) dxdydz +������ += +����� +2 +π +� +Ca,b +� M +� +m=0 +(−1)m+1χp(x, y) +2m + 1 +� +3h +2 +− h +2 +cos +�π +h(2m + 1)z +� ∂v +∂z (x, y, z) dz +� +dxdy +����� +(Cauchy-Schwarz inequality in Rn+1) +≤ 2p +π +� M +� +m=0 +1 +(2m + 1)2 +� 1 +2 � +Ca,b +� +� +M +� +m=0 +�� +3h +2 +− h +2 +cos +�π +h(2m + 1)z +� ∂v +∂z (x, y, z) dz +�2� +� +1 +2 +dxdy +27 + +(Bessel inequality) +≤ 2p +π +� +∞ +� +m=0 +1 +(2m + 1)2 +� 1 +2 � +Ca,b +� +1 +h +� +3h +2 +− h +2 +�∂v +∂z (x, y, z) +�2 +dz +� 1 +2 +dxdy +(H¨older inequality) +≤ 2p +π +� +∞ +� +m=0 +1 +(2m + 1)2 +� 1 +2� +π(b2 − a2) +�� +Ca,b +� +1 +h +� +3h +2 +− h +2 +�∂v +∂z (x, y, z) +�2 +dz +� +dxdy +� 1 +2 += 2p +� +b2 − a2 +πh +� +∞ +� +m=0 +1 +(2m + 1)2 +� 1 +2 ���� +∂v +∂z +���� +L2(�Ω) +≤ 4p +� +b2 − a2 +πh +� +∞ +� +m=0 +1 +(2m + 1)2 +� 1 +2 +∥v∥H1(Ω) . +This readily implies +∥SM∥(H1(Ω))′ ≤ 4p +� +b2 − a2 +πh +� +∞ +� +m=0 +1 +(2m + 1)2 +� 1 +2 +(5.44) +and boundedness of {SM} in (H1(Ω))′ is proved. +Now we claim that +� +Ω +SMφ dx → − +� +Ca,b +χp(x, y) +� +φ +� +x, y, h +2 +� +− φ +� +x, y, − h +2 +�� +dxdy =(H1(Ω))′ ⟨F3, φ⟩H1(Ω) +(5.45) +as M → +∞, for any φ ∈ C∞(Ω). First of all, by using the classical results about pointwise convergence +of the Fourier Series applied to suitable 2h-periodic extensions of the functions +z �→ φ(x, y, z) , z ∈ +� +− h +2, 0 +� +and +z �→ φ(x, y, z) , z ∈ +� +0, h +2 +� +one can show that for any (x, y) ∈ Ca,b +� +h +2 +− h +2 +SM(x, y, z)φ(x, y, z)dz → −χp(x, y) +� +φ +� +x, y, h +2 +� +− φ +� +x, y, − h +2 +�� +. +(5.46) +Then applying to the test function φ the estimates used for proving boundedness of {SM} in (H1(Ω))′, +one can show that for any M +����� +� +h +2 +− h +2 +SM(x, y, z)φ(x, y, z)dz +����� ≤ 2 +√ +2p +π +� +∞ +� +m=0 +1 +(2m + 1)2 +� 1 +2 ���� +∂φ +∂z +���� +L∞(Ω) +for any (x, y) ∈ Ca,b . +(5.47) +By (5.46), (5.47) and the Dominated Convergence Theorem the proof of (5.45) follows. With an essentially +similar procedure one can prove that SM converges in the sense of distributions to Λ3 where Λ3 is the +third component of the vector distribution Λ defined in (5.20). +Since (H1(Ω))′ is a reflexive Banach space, by (5.44) we infer that along suitable subsequences, the +partial sums are weakly convergent in (H1(Ω))′. Thanks to (5.45), we deduce that the weak limits of +this subsequences coincide on the space C∞(Ω) and they equal F3 on it. By density of C∞(Ω) in H1(Ω), +they actually coincide on the whole H1(Ω). This proves that all weakly convergent subsequences weakly +converge to F3 and hence the sequence SM is itself weakly convergent to F3 in (H1(Ω))′. We can now +denote by SM = (0, 0, SM) the sequence of vector partial sums in such a way that SM ⇀ F weakly in +(H1(Ω; R3))′ as M → +∞. +Now, let us consider the linear continuous operator L introduced in the proof of Theorem 2.1 and its +restriction to V ⊥ +0 , where we recall that orthogonality is with respect to the scalar product (2.12). Then, +by Theorem 2.1 (iv) we deduce that L|V ⊥ +0 +: V ⊥ +0 +→ (H1(Ω; R3))′ is invertible and by the Open Mapping +Theorem it follows that its inverse is continuous. +28 + +If we define UM := L−1 +|V ⊥ +0 SM, then UM is the vector partial sum corresponding to the Fourier expansion +(3.18). Since SM is weakly convergent in (H1(Ω; R3))′ to F, then the continuity of L|V ⊥ +0 implies that UM +is weakly convergent in H1(Ω; R3) to the unique solution u of (3.3) as M → +∞. +The strong convergence UM → u in L2(Ω; R3) as M → +∞ is a consequence of the compactness of the +embedding H1(Ω; R3) ⊂ L2(Ω; R3). +It remains to prove (3.19). In order to emphasize the dependence on k we reintroduce it for denoting +the functions Yk and Zk appearing in the proof of Lemma 3.5. Testing (5.32) with (H, K) = (Yk, Zk) we +have +2µπ2 +h2 +k2 +� b +a +ρ(Zk(ρ))2dρ ≤ +� b +a +ρΨk(ρ)Zk(ρ) dρ +from which we obtain +�� b +a +(Zk(ρ))2dρ +� 1 +2 +≤ 2pbh +√ +b − a +µaπ2 +1 +k2 . +(5.48) +Testing again (5.32) with (H, K) = (Yk, Zk) we also have +2µ +� b +a +(Yk(ρ))2 +ρ +dρ ≤ +� b +a +ρΨk(ρ)Zk(ρ) dρ +from which we obtain +� b +a +(Yk(ρ))2 dρ ≤ 2pb2√ +b − a +µh +�� b +a +(Zk(ρ))2dρ +� 1 +2 +≤ 4p2b3(b − a) +µ2aπ2 +1 +k2 +(5.49) +where in the last inequality we used (5.48). +Let us proceed by considering the difference between the partial sum for u1 and u1 itself: +∥U 1 +M − u1∥2 +L2(Ω) = +� +Ca,b +����� ++∞ +� +m=M+1 +Y2m+1(ρ) cos θ cos +�(2m + 1)π +h +z +������ +2 +L2(− h +2 , h +2) +dxdy += h +2 ++∞ +� +m=M+1 +� +Ca,b +(cos2 θ)(Y2m+1(ρ))2dxdy = πh +2 ++∞ +� +m=M+1 +� b +a +(Y2m+1(ρ))2ρ dρ +≤ 2p2b4(b − a)h +µ2aπ ++∞ +� +m=M+1 +1 +(2m + 1)2 ≤ p2b4(b − a)h +2µ2aπ ++∞ +� +m=M+1 +1 +m2 ≤ p2b4(b − a)h +2µ2aπ +1 +M , +where we also used (5.49). The estimate for ∥U 2 +n − u2∥L2(Ω) gives the same result for obvious reasons. +With a completely similar procedure by exploiting this time (5.48), we obtain +∥U 3 +n − u3∥2 +L2(Ω) ≤ 2p2b3h3(b − a) +µ2a2π4 ++∞ +� +m=M+1 +1 +(2m + 1)4 ≤ p2b3h3(b − a) +8µ2a2π4 ++∞ +� +m=M+1 +1 +m4 ≤ p2b3h3(b − a) +24µ2a2π4 +1 +M3 . +The solution we found by means of the Fourier series expansion satisfies (3.7) in the sense of traces of +H1-functions. We conclude the proof of the theorem by observing that this solution coincides with the +unique solution of (3.3) belonging to V ⊥ +0 . To see this, denote by u the solution found by means of the +Fourier series expansion and by w the solution in V ⊥ +0 . Both u and w possesses the symmetry properties +stated in Proposition 3.1 as it occurs to their difference u − w. But from Theorem 2.1 we have that +u − w ∈ V0 and it is readily seen from (2.15) that functions in V0 satisfying those symmetry properties +are necessarily the null function. This proves that u = w and completes the proof of the theorem. +□ +29 + +5.8 +Proof of Proposition 4.1 +We rewrite the homogeneous system (5.35) as in (5.42) so that the corresponding series expansion can be +written in the form +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +�Y (t) = ++∞ +� +n=−1 +�an tn + (ln t) ++∞ +� +n=0 +�bn tn , +�Z(t) = ++∞ +� +n=0 +�cn tn + (ln t) ++∞ +� +n=0 +�dn tn . +(5.50) +The coefficients �an,�bn, �cn, �dn are related to the corresponding coefficients an, bn, cn, dn appearing in (5.36), +by the formulas +�a−1 = πk +h a−1 , +�an = +� h +πk +�n � +an − ln +�πk +h +� +bn +� +, +�bn = +� h +πk +�n +bn , +(5.51) +�cn = +� h +πk +�n � +cn − ln +�πk +h +� +dn +� +, +�dn = +� h +πk +�n +dn . +Inserting (5.50) into (5.42) or alternatively combining (5.51) and (5.41), we see that �an,�bn, �cn, �dn solve the +system +� +� +� +� +� +� +� +� +� +� +� +(n2 − 1)�an + 2n�bn + β(n − 1)�cn−1 + �β �dn−1 = �α�an−2 +(n2 − 1)�bn + �β(n − 1) �dn−1 = �α�bn−2 +(n − 1)2�cn−1 + 2(n − 1) �dn−1 = �δ(n − 1)�an−2 + �δ�bn−2 + �γ�cn−3 +(n − 1)2 �dn−1 = �δ(n − 1)�bn−2 + �γ �dn−3 +(5.52) +for n ≥ 3; moreover �a0 = �b0 = �c1 = �d1 = �a2 = �b2 = 0, the coefficients �a−1, �a1,�b1, �c0 may be chosen +arbitrarily and �d0 = �α +�β �a−1 − 2 +�β �b1. +By direct computation one can verify that the unique solution of system (5.52) can be written in form +� +� +� +� +�an +�bn +�cn−1 +�dn−1 +� +� +� +� = +� +� +� +� +� +� +� +� +� +� +− λ +µ +1 +(n+1)(n−1) +2λ +µ +n +(n+1)2(n−1)2 +− λ+µ +µ +1 +(n+1)(n−1)2 +λ+µ +µ +3n+1 +(n+1)2(n−1)3 +0 +− λ +µ +1 +(n+1)(n−1) +0 +− λ+µ +µ +1 +(n+1)(n−1)2 +λ+µ +µ +1 +n−1 +− λ+µ +µ +1 +(n−1)2 +λ+2µ +µ +1 +(n−1)2 +− 2(λ+2µ) +µ +1 +(n−1)3 +0 +λ+µ +µ +1 +n−1 +0 +λ+2µ +µ +1 +(n−1)2 +� +� +� +� +� +� +� +� +� +� +� +� +� +� +�an−2 +�bn−2 +�cn−3 +�dn−3 +� +� +� +� +(5.53) +for any n ≥ 3. +We are interested in the case n odd since when n is even, thanks to (5.52), we know that �an = �bn = +�cn−1 = �dn−1 = 0. Looking at (5.53), for any n ≥ 3 odd, we introduce the matrices +Sn := +� +� +− λ +µ +1 +(n+1)(n−1) +− λ+µ +µ +1 +(n+1)(n−1)2 +λ+µ +µ +1 +n−1 +λ+2µ +µ +1 +(n−1)2 +� +� , +Tn := +� +� +2λ +µ +n +(n+1)2(n−1)2 +λ+µ +µ +3n+1 +(n+1)2(n−1)3 +− λ+µ +µ +1 +(n−1)2 +− 2(λ+2µ) +µ +1 +(n−1)3 +� +� . +In this way, system (5.53) may written in the form +� �bn +�dn−1 +� += Sn +��bn−2 +�dn−3 +� +, +� +�an +�cn−1 +� += Sn +� +�an−2 +�cn−3 +� +− Tn +��bn−2 +�dn−3 +� +. +30 + +After an iterative procedure we may write +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� �bn +�dn−1 +� += +� +� +(n−3)/2 +� +m=0 +Sn−2m +� +� +��b1 +�d0 +� +, +� +�an +�cn−1 +� += +� +� +(n−3)/2 +� +m=0 +Sn−2m +� +� +� +�a1 +�c0 +� +− +(n−3)/2 +� +j=0 +� +� +� +j� +m=1 +Sn−2m+2 +� +Tn−2j +� +� +(n−3)/2 +� +m=j+1 +Sn−2m +� +� +��b1 +�d0 +�� +� , +(5.54) +for any n ≥ 3 odd, with the convention that for any sequence of matrices Am ∈ R2×2 +m2 +� +m=m1 +Am = +�1 +0 +0 +1 +� +and +m2 +� +m=m1 +Am = +�0 +0 +0 +0 +� +whenever m1 > m2. +By induction one can verify that for any j ≤ n−3 +2 +j� +m=0 +Sn−2m = +� +� +� +− +(j+1)λ+jµ +µ(n+1)[n+1−2(j+1)] �j +m=1(n+1−2m)2 +− +(j+1)(λ+µ) +µ(n+1) �j+1 +m=1(n+1−2m)2 +(j+1)(λ+µ) +µ[n+1−2(j+1)] �j +m=1(n+1−2m)2 +(j+1)λ+(j+2)µ +µ �j+1 +m=1(n+1−2m)2 +� +� +� +and, in turn, by (6.2) we infer +����� +j� +m=0 +Sn−2m +����� +∞ +≤ (λ + µ)(n − 2j)(j + 2) +µ �j+1 +m=1(n + 1 − 2m)2 . +(5.55) +In particular, with appropriate choices of the minimum and the maximum values of the index in the +product (5.55) and with appropriate changes of index, for any n ≥ 3 odd, we obtain the estimates +������ +(n−3)/2 +� +m=0 +Sn−2m +������ +∞ +≤ 3(λ + µ)(n + 1) +µ2n �� n−1 +2 +� +! +�2 , +����� +j� +m=1 +Sn−2m+2 +����� +∞ +≤ (λ + µ)(n − 2j + 2)(j + 1) +µ �j +m=1(n + 1 − 2m)2 +, +(5.56) +������ +(n−3)/2 +� +m=j+1 +Sn−2m +������ +∞ +≤ +3(λ + µ)(n − 2j − 1) +2µ �(n−1)/2 +m=j+2 (n + 1 − 2m)2 . +On the other hand, we observe that for the components of the matrices Sn and Tn the following +inequalities hold true: +|(Tn)ij| ≤ +3 +n−1 |(Sn)ij| +for any i, j ∈ {1, 2} and n ≥ 3, +which, in turn, implies ∥Tn∥∞ ≤ +3 +n−1 ∥Sn∥∞ = 3(λ + µ) +2n +µ(n−1)3 ; the last inequality is obtained by (5.55) +with j = 0. +31 + +Therefore, combining (5.55) and (5.56), for any n ≥ 3 odd, we obtain +������ +(n−3)/2 +� +j=0 +� +� +� +j� +m=1 +Sn−2m+2 +� +Tn−2j +� +� +(n−3)/2 +� +m=j+1 +Sn−2m +� +� +� +� +������ +∞ +(5.57) +≤ +(n−3)/2 +� +j=0 +����� +j� +m=1 +Sn−2m+2 +����� +∞ +∥Tn−2j∥∞ +������ +(n−3)/2 +� +m=j+1 +Sn−2m +������ +∞ +≤ +(n−3)/2 +� +j=0 +18(λ + µ)3(n − 2j + 2)(n − 2j)(j + 1) +µ3 2n �� n−1 +2 +� +! +�2 +≤ 9(λ + µ)3 n(n + 2)(n2 − 1) +4µ3 2n �� n−1 +2 +� +! +�2 +, +where in the last inequality we used the estimate (n − 2j + 2)(n − 2j) ≤ n(n + 2) and the identity +�(n−3)/2 +j=0 +(j + 1) = n2−1 +8 +. +Combining (6.1) with (5.54), (5.56) and (5.57), for any n ≥ 3 odd, we obtain +���� +� �an +�cn−1 +����� +∞ +≤ 3(λ + µ)(n + 1) +µ2n �� n−1 +2 +� +! +�2 +���� +��a1 +�c0 +����� +∞ ++ 9(λ + µ)3 n(n + 2)(n2 − 1) +4µ3 2n �� n−1 +2 +� +! +�2 +����� +� +�b1 +�d0 +������ +∞ +(5.58) +≤ 3(2λ + 5µ)(λ + µ)2 (n + 1)(3n3 + 3n2 − 6n + 4) +4µ3 2n �� n−1 +2 +� +! +�2 +max{�a−1, �a1,�b1, �c0} +and +����� +� +�bn +�dn−1 +������ +∞ +≤ 3(λ + µ)(n + 1) +µ2n �� n−1 +2 +� +! +�2 +����� +� +�b1 +�d0 +������ +∞ +≤ 3(2λ + 5µ)(n + 1) +µ2n �� n−1 +2 +� +! +�2 +max{�a−1, �a1,�b1, �c0} +(5.59) +where we exploited the fact that �d0 = �α +�β �a−1 − 2 +�β �b1, accordingly with what already explained in the lines +below (5.52), so that +|�d0| ≤ �α + 2 +�β +max{�a−1,�b1} = 2λ + 5µ +λ + µ max{�a−1,�b1} , +from which it follows that +max +����� +��a1 +�c0 +����� +∞ +, +����� +� +�b1 +�d0 +������ +∞ +� +≤ 2λ + 5µ +λ + µ max{�a−1, �a1,�b1, �c0} . +Since we are interested to the restrictions of the functions Y and Z to the interval [a, b], we have to +evaluate the series expansion (5.50) of the functions �Y and �Z for t ∈ +� πk +h a, πk +h b +� +. +Let N odd be the number at which we want to truncate the series expansions in (5.50). Recalling that +the coefficients �an,�bn, �cn−1, �dn−1 vanish for n even, we may write +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +�Y (t) = +� +N +� +n=−1 +�an tn + (ln t) +N +� +n=0 +�bn tn +� ++ +� ++∞ +� +n=N+2 +�an tn + (ln t) ++∞ +� +n=N+2 +�bn tn +� +, +�Z(t) = +� N−1 +� +n=0 +�cn tn + (ln t) +N−1 +� +n=0 +�dn tn +� ++ +� ++∞ +� +n=N+1 +�cn tn + (ln t) ++∞ +� +n=N+1 +�dn tn +� +, +and define the truncation error as +Ek,N = max +� +max +t∈[ πk +h a, πk +h b] +����� ++∞ +� +n=N+2 +�an tn + (ln t) ++∞ +� +n=N+2 +�bn tn +����� , +max +t∈[ πk +h a, πk +h b] +����� ++∞ +� +n=N+1 +�cn tn + (ln t) ++∞ +� +n=N+1 +�dn tn +����� +� +32 + +By (5.58) and (5.59), we see that for any t ∈ +� πk +h a, πk +h b +� +we have +0 ≤ Ek,N ≤ �C(a, b, k) +� ++∞ +� +n=N+2 +�πkb +h +�n ����� +� +�an +�cn−1 +������ +∞ ++ ++∞ +� +n=N+2 +�πkb +h +�n ����� +� �bn +�dn−1 +������ +∞ +� +≤ �C(a, b, k) max{�a−1, �a1,�b1, �c0} ++∞ +� +n=N+2 +n odd +�πkb +h +�n 3(2λ + 5µ)(λ + µ)2 (n + 1)(3n3 + 3n2 − 6n + 8) +4µ3 2n �� n−1 +2 +� +! +�2 +where we put �C(a, b, k) = max +� +1, +h +πkb +� +max +� +1, | ln +� πka +h +� +|, | ln +� πkb +h +� +| +� +. +Since we are interested to truncation of the series expansion with a sufficiently large number of terms, +letting P(n) := (n + 1)(3n3 + 3n2 − 6n + 8), it is not restrictive to assume N ≥ 3 in such a way that the +sequence n �→ 2−nP(n) becomes decreasing for n ≥ N + 2 ≥ 5. +In this way, for N ≥ 3 odd, we obtain for all t ∈ +� πk +h a, πk +h b +� +0 ≤ Ek,N ≤ �C(a, b, k) max{�a−1, �a1,�b1, �c0}3(2λ + 5µ)(λ + µ)2 P(N + 2) +4µ3 2N+2 � N+1 +2 +� +! ++∞ +� +m= N+1 +2 +� πkb +h +�2m+1 +m! +(5.60) +≤ �C(a, b, k) max{�a−1, �a1,�b1, �c0} +�πkb +h +�N+2 +e( πkb +h ) +2 3(2λ + 5µ)(λ + µ)2 P(N + 2) +16µ3 2N �� N+1 +2 +� +! +�2 +, +where in the last estimate we used the Lagrange form of the reminder in the Taylor formula for the +exponential function and P(N + 2) = (N + 3)(3N3 + 21N2 + 42N + 32). +According to the rescaling introduced in (5.42) one may define the functions �Υj, whose series expan- +sions are given by (5.50) with coefficients in (5.51) and with a−1, a1, b1, c0 given by (5.40) in the cases +corresponding to j ∈ {1, 2, 3, 4}. +In these four cases, the quantity max{�a−1, �a1,�b1, �c0}, appearing in the right hand side of (5.60), admits +the following estimates: +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +max{�a−1, �a1,�b1, �c0} = πk +h max +� +1, +µ +λ+µ ln +� πk +h +�� +if j = 1, +max{�a−1, �a1,�b1, �c0} = +h +πk +if j = 2, +max{�a−1, �a1,�b1, �c0} = +h +πk max +� +1, 2(λ+2µ) +λ+µ +ln +� πk +h +�� +if j = 3, +max{�a−1, �a1,�b1, �c0} = 1 +if j = 4. +(5.61) +For k > 1 is easy to see that all the maximum in (5.61) are less or equal than +max +� +πk +h , +µ +λ+µ +πk +h ln +� πk +h +� +, 2(λ+2µ) +λ+µ +h +πk ln +� πk +h +� � +, +so that (4.2) follows. +□ +6 +Conclusions +In this work we started from an applicative problem, suggested by Studio De Miranda Associati, an +engineering company expertized in building long span bridges. They proposed to study the blister, a +structural element in bridges where the steel forestay anchors to the deck. The aim is to obtain an explicit +formula to estimate the tensions in the blister, useful for the practical design of bridges. +33 + +The problem can be solved through the resolution of the elasticity equation with a specific geometry +and load configuration. Hence, the first step was to define the geometry of the element. Through some +simplifications we end up with a hollow circular cylinder axially loaded at the end faces; the volume of +the cylinder represents the portion of the deck concrete where the stresses diffusion happens, while the +applied load is given by the force that the stay has to transfer to the deck. Clearly this geometry and +load configuration can be refined in order to model a real blister, but this is a first step in this way and +we leave more sophisticated models to future works. +As matter of fact, from literature we learn that the elasticity equation was explicitly solved only for very +particular domains and load conditions, e.g. in prisms [13]. In this paper we provide the explicit solution +for the hollow cylinder axially loaded, proceeding by steps: first of all we provide a periodic extension +of the load in z direction, so that we expand the solution in Fourier series with respect to the variable +z. Then we compute the Fourier coefficients in x and y passing to cylindrical coordinates and expanding +such functions in power series. In Theorem 3.7 we write the explicit solution for the problem, written in +series expansion. We point out that this solution may have an own interest in the construction science +field, beyond the application to the blister. +To employ directly the formula in real situations, such as the blister design, it is necessary to consider +approximated solutions, giving some estimates on the errors due to the truncating of the series. In Section +4 we proposed a case of study, where, fixing the parameters involved in the problem, we are able to find +the distribution of the stresses in the cylinder. These plots can be obtained through a simple code, written +in MATLAB® or GNU Octave®, running in brief time, e.g. 1-3 minutes, depending on the number where +we truncate the series. +From these results it is possible to find the maximum and the minimum of the different stresses acting +on the cylinder, their position on the element and an estimate on the error due to the truncation of the +series. Knowing these values, the engineering designer can choice for instance the most appropriate strand +anchorage from the commercial catalogue, see Figure 5, in order to not exceed specific limit stresses in +the reinforced concrete. Since the map of the tensions is given, see e.g. Figure 6, the engineer can design +the steel reinforcements in the concrete, at least on a pre-dimensioning level, and can check the concrete +cracking stresses. +As we explained, to get more precise results on realistic blisters we should modify the geometry of the +element and the configuration of the loads; this may be a future work, but we point out that, more the +geometry and the distribution of the loads are complex more the expectations to find explicit solutions +are few, so that the finite element analysis may be preferred. +Notations We give some notations that will be used throughout this paper about functional spaces +and differential operators acting on scalar functions, vector valued functions, matrix valued functions. We +denote by Ω a general domain in RN, N ≥ 1 where by domain we mean a connected open set in RN. +• Given two vectors x = (x1, . . . , xN), y = (y1, . . . , yN) ∈ RN we denote by x · y = �N +i=1 xiyi their +Euclidean scalar product and by |x| = √x · x the Euclidean modulus of x; +• the ∞-norm of vectors is |x|∞ := max +1≤i≤N |xi|; +• RM×N: space of M × N matrices; +• if A ∈ RM×N and x ∈ RN is a vector, Ax denotes the usual product of matrices where x has to be +seen as a vector column; +• letting A = (aij), B = (bij) ∈ RN×N we denote by A : B = �N +i,j=1 aijbij their Euclidean scalar +product and by |A| = +√ +A : A its Euclidean modulus; +• given A ∈ RM×N we denote by AT ∈ RN×M its transpose; +34 + +• given A ∈ RN×N we introduce the operator ∞-norm of matrices by ∥A∥∞ := +sup +x∈RN\{0} +|Ax|∞ +|x|∞ so that +we have in particular +|Ax|∞ ≤ ∥A∥∞ |x|∞ +for any x ∈ RN. +(6.1) +Letting A = (aij), ∈ RN, the following characterization of ∥ · ∥∞ holds: +∥A∥∞ = max +1≤i≤N +N +� +j=1 +|aij|; +(6.2) +being ∥ · ∥∞ an operator norm, it is sub-multiplicative in the sense that ∥AB∥∞ ≤ ∥A∥∞ ∥B∥∞ for +any A, B ∈ RN×N. +• some well known functional spaces of functions defined from on an open set Ω ⊂ RN to a vector +space V which could be RM or a space of matrices: Ck(Ω; V ), Lp(Ω; V ), Hk(Ω; V ) with 0 ≤ k ≤ ∞ +integer and 1 ≤ p ≤ ∞; +• for 0 ≤ k ≤ ∞ integer, Ck(Ω; V ) denotes the space of restrictions to Ω of functions in Ck(RN; V ); +• D(Ω; V ): space of C∞(Ω; V ) with compact support in Ω; +• D′(Ω; V ): space of vector distributions, i.e. the dual space of D(Ω; V ); +• given a scalar function g ∈ C1(Ω; R), we denote by ∇g ∈ C0(Ω; Rn) its gradient; +• given a vector valued function u ∈ C1(Ω; RM), we denote by ∇u ∈ C0(Ω; RM×N) its Jacobian +matrix; +• given a vector valued function u ∈ C1(Ω; RN), Ω ⊆ RN, we denote by Du ∈ C0(Ω; RN×N) its +symmetric gradient defined by Du = ∇u + ∇uT +2 +(linearized strain tensor when N = 3); +• given U ∈ C1(Ω; RM×N), Ω ⊆ RN, we denote by div U ∈ C0(Ω; RM) the vector field v = (v1, . . . , vM) +such that vi = �N +j=1 +∂Uij +∂xj , i = 1, . . . , M; +• given u = (u1, . . . , uM) ∈ C2(Ω; RM), we denote by ∆u ∈ C0(Ω; RM) the Laplacian of u defined +component by component, i.e. ∆u = (∆u1, . . . , ∆uM) where in the last identity ∆ denotes the usual +Laplacian of a real valued function. +Acknowledgments The two authors are members of the Gruppo Nazionale per l’Analisi Matematica, +la Probabilit`a e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). +The second author acknowledges partial financial support from the PRIN project 2017 “Direct and inverse +problems for partial differential equations: theoretical aspects and applications”. The authors acknowledge +partial financial support from the INdAM - GNAMPA project 2022 “Modelli del 4° ordine per la dinamica +di strutture ingegneristiche: aspetti analitici e applicazioni”. +The second author acknowledges partial financial support from the research project “Metodi e modelli +per la matematica e le sue applicazioni alle scienze, alla tecnologia e alla formazione” Progetto di Ateneo +2019 of the University of Piemonte Orientale “Amedeo Avogadro”. +35 + +References +[1] E. Berchio, A. Falocchi, About symmetry in partially hinged composite plates, Appl. Math. Optim. 84, 2645– +2669, (2021). +[2] E. Berchio, A. Falocchi, Maximizing the ratio of eigenvalues of non-homogeneous partially hinged plates, J. +Spectr. Theory 11, 743–780, (2021). +[3] E. Berchio, A. Falocchi, A. Ferrero, D. Ganguly, On the first frequency of reinforced partially hinged plates, +Commun. Contemp. Math., 1950074, 37 pp. (2019). +[4] E. Berchio, A. Falocchi, M. Garrione, On the stability of a nonlinear non homogeneous multiply hinged beam, +SIAM J. Appl. Dyn. Syst. 20(2), 908–940, (2021). +[5] P. G. Ciarlet, Mathematical elasticity: Three-Dimensional Elasticity, Society for Industrial and Applied Math- +ematics, (2021). +[6] J.L. Clarke, Guide to the design of anchor blocks for post-tensioned prestressed concrete members, Construction +Industry Research and Information Association (1976). +[7] G. Crasta, A. Falocchi, F. Gazzola, A new model for suspension bridges involving the convexification of the +cables, Z. Angew. Math. Phys. 71, 93, (2020). +[8] A. Falocchi, Torsional instability in a nonlinear isolated model for suspension bridges with fixed cables and +extensible hangers, IMA Journal of Applied Mathematics 83, 1007–1036, (2018). +[9] A. Ferrero, F. Gazzola, A partially hinged rectangular plate as a model for suspension bridges, Disc. Cont. +Dyn. Syst. A 35, 5879–5908 (2015). +[10] M. Garrione, F. Gazzola, Linear theory for beams with intermediate piers, Commun. Contemp. Math. 22, +1950081, 41 pp. (2020). +[11] F. Gazzola, Mathematical models for suspension bridges, MS&-A Vol.15, Springer (2015). +[12] V. A. Kondrat’ev, O.A. Oleinik,Boundary-value problems for the system of elasticity theory in unbounded +domains. Korn’s inequalities Uspekhi Mat. Nauk 43, n. 5, 55-98, (1988) (in Russian). English translation in +Russian Mathematical Surveys 43, n. 65, (1988). +[13] K.T.S.R. Iyengar, M. K. Prabhakara, A three dimensional elasticity solution for rectangular prism under end +loads, ZAMM 49, 321-332 (1969) +[14] F. Leonhardt, E. M¨onnig, C. A. & C. A. P. Calcolo di progetto e tecniche costruttive vol.2, Casi speciali di +dimensionamento nelle costruzioni in c.a. e c.a.p., Edizioni di Scienza e Tecnica (1979) +[15] Protende ABS, Commercial Catalogue 2021, see https://protendeabs.com.br/sobre. +[16] H. M. Westergaartd, Theory of Elasticity and Plasticity, American Journal of Physics 34, 545 (1966). +36 + diff --git a/jtE1T4oBgHgl3EQfgQQH/content/tmp_files/load_file.txt b/jtE1T4oBgHgl3EQfgQQH/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5fcbdae6363c20f993b41fcef655e396e8930af4 --- /dev/null +++ b/jtE1T4oBgHgl3EQfgQQH/content/tmp_files/load_file.txt @@ -0,0 +1,1547 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf,len=1546 +page_content='Elasticity solution for a 3D hollow cylinder axially loaded at the end faces Mario DE MIRANDA‡, Marta DE MIRANDA‡, Alessio FALOCCHI†, Alberto FERRERO∗, Luca MARININI‡, ‡ Studio De Miranda Associati, Via C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Pisacane 26, Milano, Italy † Dipartimento di Matematica - Politecnico di Milano, Milano, Italy ∗ Dipartimento di Scienze e Innovazione Tecnologica, Universit`a del Piemonte Orientale, Alessandria, Italy Abstract Starting from an applicative problem related to the modeling of an element of a cable-stayed bridge, we compute the elasticity solution for a hollow cylinder loaded at the end faces with axial loads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' We prove results of symmetry for the solution and we expand it in proper Fourier series;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' computing the Fourier coefficients in adapted power series, we provide the explicit solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' We consider an engineering case of study, applying the corresponding approximate formula and giving some estimates on the error committed with respect to the truncation of the series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' 1 Introduction In the recent years the interest of the mathematicians for engineering applications has grown more and more;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' this is due to an evolution of the mathematics, thanks for instance to the development of new techniques to deal with nonlinear problems and the support of automatic calculators to obtain previsions unthinkable in the past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Figure 1: From the top on the left a render of a recent cable stayed bridge designed by Studio De Miranda Associati and a detail of its deck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' The mathematical modeling of specific phenomena is one of the ambitious aims of the applied math- ematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Recently some mathematical models for suspension bridges [11] have been developed with the scope to understand and, then, to prevent instability phenomena;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' they were studied models for suspension bridges with geometrical nonlinearities, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' see [7, 8], models for partially hinged plates, see [9], models for 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='03226v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='AP] 9 Jan 2023 non homogeneous partially hinged plates, see [1, 2, 3], models for homogeneous beams with intermediate piers [10] and non homogeneous beams [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' In all these cases the application of analytical methods to real problems allowed to find suggestions and practical remedies that can be discussed with engineers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' This is the aim also of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Here the problem, suggested by the structural civil engineering Studio De Miranda Associati, is related to the modeling of the stresses in a constructive detail of a bridge: the blister.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' In the cable-stayed bridge the blister is the structural element where the steel forestay anchors to the deck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' In Figure 1 is shown a render of a future cable-stayed bridge, designed by Studio De Miranda Associati, that will be built in Brazil;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' a detail of the related blister element is given in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' When the deck is built in reinforced concrete, as in this case, the blister is an important point to design;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' indeed, the high density of the steel reinforcement may cause zone with low concrete capacity and possible remarkable cracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' To have an idea of the complexity of this element a detail of the executive draw of a blister for another stayed bridge, designed by Studio De Miranda Associati, is shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Figure 2: Detail of the executive draw of a blister of a recent prestressed concrete bridge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' For all these reasons it is important to estimate with precision the stresses acting on the element, so that the reinforcing steel in the concrete can be computed without surplus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' In engineering literature some of the best known references related to the distribution of the stresses in prisms of concrete are [6, 14];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' here the authors consider many combinations of load on the prism and for each one the possible strategies to design the steel bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' These results are obtained from particular solutions of the well known equation of the linear elasticity, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' [5];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' we recall it here briefly in the general 3D case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Given Ω ⊂ R3 an elastic homogeneous solid body, we denote by u : Ω → R3 the displacement vector at any point of the reference configuration of the elastic body itself, see the list of notations at the end of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' We denote by Tu the stress tensor and by λ and µ the classical Lam´e constants;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' it is known that λ and µ may be expressed in terms of the Young modulus E and Poisson ratio ν ∈ (−1, 1 2) as λ = Eν (1 + ν)(1 − 2ν) , µ = E 2(1 + ν) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='1) The equation of linear elasticity reads � � � −µ∆u − (λ + µ)∇(divu) = f in Ω, (Tu)n = g on ∂Ω, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='2) where f and g are respectively the forces per unit volume and the boundary forces per unit surface acting on Ω, while n is the unit outward normal vector to ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' In Section 2 we briefly derive (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='2) from variational principles and we recall the existence and uniqueness results in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' these are classical topics in linear elasticity, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' [5, 16], but we recall them in our framework for completeness since the question about uniqueness of solutions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='2) is not trivial at all and it needs an additional condition to be achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' The theoretical solution from which come the applicative cases considered in [6, 14] is given in [13], where Ω is a rectangular prism under end loads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Thanks to this simple geometry and loading condition 2 POST-INSTALLED 2a:2h) REINFORCEMENT TO BE BENT AFTER HARDENING (4a:4p)@16 (tot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' _12) 5a:5f @16 (tot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' 6) OF RESIN =616 max min=423 645 STRESSING RECESS RESIN TYPE HIT-RE 500 V3 Hole ：Φ12x100mm SECTION X-X @2+2@16/100 L=1134 3a:39) 2000- 1000the authors find explicitly the solution in form of double Fourier series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' The result is obtained applying the Galerkin vector method, a technique allowing to pass from the second order differential equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='2) to a simpler biharmonic equation, see [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' We point out that to find the explicit solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='2) for generic Ω and loading conditions is a very hard task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' In this paper we find it for Ω coincident with a hollow cylinder loaded on the opposite faces, since this geometry fits the modeling of the concrete of the blister, see Figure 3 on the right;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' indeed, the forestay of the bridge is circular and passes through the cylindrical hole, applying a distributed load on the opposite faces due to its tensioning, see Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Figure 3: From the left a frontal view of blister elements and the modelization of the element through the hollow cylinder (in red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' The precise definition of the model is given in Section 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' the application of axial loads leads to a solution having axial symmetric properties, see Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' In the real blister it is also possible to have non radial loadings coming from the deck, but this is a first attempt of modeling that may be implemented in future works;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' anyway, the solution found here may have general interest beyond this specific application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' The definition of the solution is given by steps: in subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='1 we provide a periodic extension of the loads in the variable z corresponding to the symmetry axis of the hollow cylinder, in such a way that it becomes possible to expand the solution in Fourier series with respect to the variable z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' then we compute the Fourier coefficients which come to be functions in the other two variables x and y, corresponding to directions orthogonal to the symmetry axis of the hollow cylinder;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' in subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='2 we pass to the cylindrical coordinates and, exploiting the axial symmetry, we reduce ourself to study a system of ODEs in the radial polar coordinate ρ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' we compute the Fourier coefficients as functions of the variable ρ through an adapted expansion in power series so that we are able to state Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='7, collecting the explicit solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' In Section 4 we give some hints to truncate the series and we apply the results to an engineering case of study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' As it will be explained in details, it will be necessary to compute numerically the first M terms in the Fourier series expansion with M to be chosen sufficiently large in order to minimize the truncation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' The main question in this procedure is that the computation of those Fourier coefficients, which are solutions of suitable boundary value problems of ODEs, requires the numerical resolution of some algebraic linear systems in four variables which exhibit a condition number higher and higher as M grows;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' if we need a truncation error smaller than ours, we may consider alternative numerical procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' We emphasize that the main purpose of this article is to obtain an analytical representation of the unique symmetric solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='2) in the case of the hollow cylinder with the perspective of reproducing such method in more general situations with not necessarily symmetric external loads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' As already explained in details, the main analytical and numerical results of the article are stated in Sections 2-4 and their proofs are given in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' The final part of the paper is devoted to the conclusions, see Section 6, and a list of notations which can be helpful for the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' 3 2 The definition of the mathematical model for the linear elasticity In this Section we derive the differential equations for the linear elasticity from variational arguments and we state a theorem related to the existence of solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Although these results are well known overall in the engineering field, we review them from a mathematical point of view, applying the Fredholm alternative to prove existence of solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='1 The derivation of the differential equations We recall that Ω ⊂ R3 is the domain of the elastic body and u is the displacement function with components u = (u1, u2, u3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' We denote by Du the linearized strain tensor, which in the sequel will be simply called strain tensor, since we only deal with the linear theory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' the stress tensor can be written as Tu = � � σ1 τ 12 τ 13 τ 12 σ2 τ 23 τ 13 τ 23 σ3 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='1) It is well known that by the Hooke’s Law for isotropic materials it holds Tu = λtr(Du) I + 2µDu , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='2) where λ and µ are the Lam´e constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Combining (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='1), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='2) we infer σ1 = E (1 + ν)(1 − 2ν) � (1 − ν)∂u1 ∂x + ν �∂u2 ∂y + ∂u3 ∂z �� τ 12 = E 2(1 + ν) �∂u1 ∂y + ∂u2 ∂x � σ2 = E (1 + ν)(1 − 2ν) � (1 − ν)∂u2 ∂y + ν �∂u1 ∂x + ∂u3 ∂z �� τ 13 = E 2(1 + ν) �∂u1 ∂z + ∂u3 ∂x � σ3 = E (1 + ν)(1 − 2ν) � (1 − ν)∂u3 ∂z + ν �∂u1 ∂x + ∂u2 ∂y �� τ 23 = E 2(1 + ν) �∂u2 ∂z + ∂u3 ∂y � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='3) The elastic energy related to the internal forces in the configuration corresponding to a generic displace- ment u is given by Eel(u) = 1 2 � Ω Tu : Du dx .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' If we assume that on Ω act body forces per unit of volume f = (f1, f2, f3) and boundary forces per unit of surface g = (g1, g2, g3) we obtain the total energy of the system E(u) = 1 2 � Ω Tu : Du dx − � Ω f · u dx − � ∂Ω g · u dS .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='4) Thanks to the symmetry of the stress tensor Tu = (Tu)T we infer that for any u, v ∈ H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3) Tu : ∇v = (Tu)T : (∇v)T = Tu : (∇v)T so that 2(Tu : ∇v) = Tu : ∇v + Tu : (∇v)T ⇒ Tu : ∇v = Tu : Dv .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='5) Recalling the Hooke’s law (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='2), we observe that the bilinear form (u, v) �→ � Ω Tu : Dv dx , (u, v) ∈ H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3) × H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3) 4 is symmetric, since Tu : Dv = λ (divu) (divv) + 2µ Du : Dv .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='6) By looking at the total energy E in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='4) as a functional E : H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3) → R and exploiting the symmetry of the bilinear form above mentioned, we see that a critical point u ∈ H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3) of E solves the variational problem � Ω Tu : Dv dx = � Ω f · v dx + � ∂Ω g · v dS for any v ∈ H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='7) By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='5) and a formal integration by parts, we see that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='7) is the weak formulation of the boundary value problem � −div(Tu) = f in Ω, (Tu)n = g on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='8) Inserting (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='2) into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='8) we find the well known equations of linear elasticity (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' In the next subsection we prove the existence of solution, stating some classical results about functional spaces of vector valued functions which find a natural application in the theory of linear elasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' These results are related to the well known Korn inequality which has a general validity for vector functions from RN to RN for any N ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Clearly, in the present paper we will be mainly interested to the case N = 3, being R3 the natural space where a solid elastic body can be modelled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' For completeness, we will state those results in the general N-dimensional case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='2 Existence of a solution Let Ω ⊂ Rn a bounded domain, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' an open connected bounded set of RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Let us introduce the following Sobolev-type space H1 D(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' RN) defined as the completion of C∞(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' RN) with respect to the scalar product (u, v)H1 D = � Ω Du : Dv dx + � Ω u · v dx for any u, v ∈ C∞(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' RN) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='9) Here x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' , xN) denotes the generic variabile of a function defined in a domain of RN and dx = dx1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' dxN denotes the N-dimensional volume integral in RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' From its definition, it is clear that H1 D(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' RN) ⊂ L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' RN) becomes a Hilbert space with the extension of the scalar product (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' The Korn inequality states the equivalence on the space C∞(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' RN) between the usual scalar product of H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' RN), namely (u, v)H1 = � Ω ∇u : ∇v dx + � Ω u · v dx for any u, v ∈ H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' RN) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='10) and the scalar product (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' More precisely, if Ω ⊂ RN is a bounded domain with Lipschitz boundary, then there exists C > 0 such that � Ω |∇u|2dx ≤ C �� Ω |u|2dx + � Ω |Du|2dx � for any u ∈ H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' RN) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='11) Among the others, for a clear and elegant proof of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='11), we address the reader to [12] by V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Kondrat’ev & O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Oleinik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Thanks to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='11) we deduce that the Hilbert space H1 D(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' RN) actually coincides with H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' RN) as one can deduce from the definition of H1 D(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' RN) and the well known result about density of C∞(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' RN) in H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' RN) whenever Ω, is a bounded domain with Lipschitzian boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' The Korn inequality is a fundamental tool for proving the existence of weak solutions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' We fix N = 3 and we observe that by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='6) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='11), the bilinear form defined by (u, v)T = � Ω Tu : Dv dx + � Ω u · v dx for any u, v ∈ H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='12) 5 is a scalar product in H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3) which is equivalent to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Let us introduce the space V0 := � v0 ∈ H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3) : � Ω Tv0 : Dv dx = 0 ∀v ∈ H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='13) We observe that V0 coincides with the eigenspace associated to the first eigenvalue of the following eigenvalue problem: α is an eigenvalue if there exists a nontrivial function u ∈ H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3), which will be called eigenfunction associated to α, such that � Ω Tu : Dv dx = α � Ω u · v dx for any v ∈ H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' In particular if α = 0 and v0 is a corresponding eigenfunction we have that � Ω Tv0 : Dv dx = 0 for any v ∈ H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='14) After some computation one can verify that V0 is the space of functions v = (v1, v2, v3) admitting the following representation � � � � � v1(x1, x2, x3) = αx2 + βx3 + δ1 , v2(x1, x2, x3) = −αx1 + γx3 + δ2 , v3(x1, x2, x3) = −βx1 − γx2 + δ3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='15) where α, β, γ, δ1, δ2, δ3 are arbitrary constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Roughly speaking, configurations associated with such functions v ∈ V0 correspond to translations and rotations of the solid body without deforming it in such a way the elastic energy Eel(v) equals zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Actually, assuming for simplicity δ1 = δ2 = δ3 = 0, deformations corresponding to displacements v ∈ V0 can be considered good approximations of a rotation only for α, β, γ small;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' when at least one of the constants α, β, γ is not small the corresponding deformation of the solid body is no more negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' In such a case, one may wonder why the elastic energy remains anyway zero;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' the answer is that in the linear theory only small deformations are allowed so that large deformations are no more meaningful for our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Let us recall that we are considering in our model the linearized strain tensor Dv which is a good approximation of the real strain tensor only for small deformations since the last one also contains quadratic terms in the first order derivatives of v1, v2, v3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' these quadratic terms can be neglected when first order derivatives are small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' In the next theore we state the existence result for problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Let Ω ⊂ R3 a bounded domain with Lipschitzian boundary and let f ∈ L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3) and g ∈ L2(∂Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Let us introduce the following compatibility condition � Ω f · v dx + � ∂Ω g · v dS = 0 for any v ∈ V0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='16) Then the following statements hold true: (i) problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='7) admits at least one solution u ∈ H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3), or equivalently the boundary value problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='2) admits at least one weak solution, if and only if (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='16) holds true;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' (ii) if u ∈ H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3) is a particular solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='7) then for any v0 ∈ V0 the function u + v0 is still a solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='7);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' 6 (iii) if u ∈ H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3) is a particular solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='7) and if w ∈ H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3) is any other solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='7), then there exists v0 ∈ V0 such that w = u + v0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' (iv) letting V ⊥ 0 be the space orthogonal to V0 with respect to the scalar product (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='12), we have that problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='7) admits a unique solution in V ⊥ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' The results of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='1 may be extended by replacing the boundary function g ∈ L2(∂Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3) by an element of the space H−1/2(∂Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3) denoting the dual space of H1/2(∂Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' In such a case, if g ∈ H−1/2(∂Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3) the compatibility condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='16) has to be replaced by its natural extension � Ω f · v dx + H−1/2⟨g, v⟩H1/2 = 0 for any v ∈ V0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' The validity of this fact comes from the trace theory for vector valued functions h admitting a weak divergence in L2(Ω);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' for such functions h, it is possible to define the trace of h · n as an element of the dual space of H1/2(∂Ω), the last one being the space of traces of H1(Ω) functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Such a result has to be applied in our case to each line of Tu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' We observe that, as a consequence of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='1, if u and w are solutions of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='7), then the two configurations of the elastic body, corresponding to u and w, generate the same stress state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' More precisely we have Tu = Tw in Ω as a consequence of the Hooke’s law and of the fact that D(u − w) vanishes in Ω being u − w ∈ V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Physically, this is completely reasonable since, given the configuration corresponding to u, the one corresponding to w can be obtained from the first one by means of rotations and translations of the elastic body, which clearly do not affect the stress state of the solid body itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' The proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='1 is given in subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='1 and is based on standard arguments and the Fredholm alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' 3 The hollow cylinder axially loaded at the end faces We consider a circular, finite, homogeneous, isotropic and elastic cylinder with height h, radius b > 0, having a coaxial hole of radius a > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' In this section we use the usual notation x, y, z for the three coordinates in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' We maintain the notation dx to denote the differential volume dxdydz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Therefore, we introduce the annular domain Ca,b := {(x, y) ∈ R2 : a2 < x2 + y2 < b2} in such a way that Ω = Ca,b × � − h 2, h 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' In the sequel we want to model a hollow cylinder subject to an external load acting on the upper and lower faces of the cylinder compressing the cylinder itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Recalling the notations introduced in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='2), we will then assume that the volume forces represented by the vector function f vanish everywhere in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' In order to better describe the surface forces represented by the vector function g, we split ∂Ω in four regular parts Γ1 := Ca,b × � − h 2 � , Γ2 := Ca,b × � h 2 � , Γ3 := {(x, y) ∈ R2 : x2 + y2 = b2} × � − h 2, h 2 � , Γ4 := {(x, y) ∈ R2 : x2 + y2 = a2} × � − h 2, h 2 � , having respectively outward unit normal vectors (0, 0, −1), (0, 0, 1), (x/b, y/b, 0) when (x, y, z) ∈ Γ3 and (−x/a, −y/a, 0) when (x, y, z) ∈ Γ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' In this way, the outward unit normal vector n is well defined on the whole ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' 7 Figure 4: The domain Ω and in red the load considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Exploiting the above notations, the vector function g can be represented in the following way g(x, y, z) = � � � � � (0, 0, χp(x, y)) for any (x, y, z) ∈ Γ1, (0, 0, −χp(x, y)) for any (x, y, z) ∈ Γ2, (0, 0, 0) for any (x, y, z) ∈ Γ3 ∪ Γ4, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='1) where the function χp : Ca,b → R, p ∈ R+, is defined by χp(x, y) := � p if a2 ≤ x2 + y2 < ϵ2, 0 if ϵ2 < x2 + y2 ≤ b2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='2) for some ϵ ∈ (a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Resuming all the assumptions on f and g we are led to consider the problem � � � � � � � � � � � � � −µ∆u − (λ + µ)∇(divu) = 0 in Ω , (Tu)n = (0, 0, χp) on Γ1, (Tu)n = (0, 0, −χp) on Γ2, (Tu)n = 0 on Γ3 ∪ Γ4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='3) Among all solutions of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='3) which can be obtained by a single solution by adding to it a function in the space V0, we focus our attention on the unique solution u = (u1, u2, u3) of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='3) in the space V ⊥ 0 where orthogonality is meant in the sense of the scalar product defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='12), see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' From a geometric point of view, condition u ∈ V ⊥ 0 avoids translations and rotations of the hollow cylinder, being V0 the space of displacement functions which generate translations and rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' In subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='2 we prove a symmetry result for the unique solution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='3) in the space V ⊥ 0 , whose validity is physically evident, but which however needs a rigorous proof: 8 p h U 2 p h I4 I3 2 pProposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Let u be the unique solution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='3) in the space V ⊥ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Then u satisfies the following symmetry properties: (i) for any (x, y, z) ∈ Ω we have u1(x, y, −z) = u1(x, y, z), u2(x, y, −z) = u2(x, y, z) u3(x, y, −z) = −u3(x, y, z) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='4) u1(−x, y, z) = −u1(x, y, z), u2(−x, y, z) = u2(x, y, z) u3(−x, y, z) = u3(x, y, z) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='5) u1(x, −y, z) = u1(x, y, z), u2(x, −y, z) = −u2(x, y, z) u3(x, −y, z) = u3(x, y, z) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='6) (ii) the third component u3 of the solution u is axially symmetric in the sense that: u3(x1, y1, z) = u3(x2, y2, z) ∀ (x1, y1, z), (x2, y2, z) ∈ Ω with x2 1 + y2 1 = x2 2 + y2 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' (iii) the first two components u1, u2 of the solution u form a central vector field in two dimensions in the sense that |(u1, u2)||(x1,y1,z) = |(u1, u2)||(x2,y2,z) ∀ (x1, y1, z), (x2, y2, z) ∈ Ω with x2 1 + y2 1 = x2 2 + y2 2 and (u1, u2)|(x,y,z) = |(u1, u2)||(x,y,z) � x � x2 + y2 , y � x2 + y2 � for any (x, y, z) ∈ Ω .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='1 Periodic extension of the problem Our next purpose is to look for and construct a solution u = (u1, u2, u3) of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='3) admitting a Fourier series expansion and, hence, admitting a periodic extension defined on the whole Ca,b × R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' In order to obtain this construction, we need to assume that the horizontal displacements u1 and u2 vanish on the upper and lower faces of the hollow cylinder Ω: u1 � x, y, h 2 � = u1 � x, y, − h 2 � = 0 and u2 � x, y, h 2 � = u2 � x, y, − h 2 � = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='7) We find a solution satisfying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='7) and, a posteriori, we show that it necessarily coincides with the unique solution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='3) belonging to V ⊥ 0 , see the end of the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' As a first step, since u ∈ H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3), we define a function, still denoted for simplicity by u, on the domain Ca,b × � − h 2, 3h 2 � by extending it in suitable way: the new function u coincides with the original function u on Ca,b × � − h 2, h 2 � and u1(x, y, z) = −u1(x, y, h − z) , u2(x, y, z) = −u2(x, y, h − z) , u2(x, y, z) = u3(x, y, h − z) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='8) for any (x, y, z) ∈ Ca,b × � h 2, 3h 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' This means that u1 and u2 are antisymmetric with respect to z = h 2 and u3 is symmetric with respect to z = h 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' This symmetric extension with respect to z = h 2 produces a function u ∈ H1 � Ca,b × � − h 2, 3h 2 � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3� thanks to condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' The second step is to extend the new function u : Ca,b × � − h 2, 3h 2 � → R to the whole Ca,b × R as a 2h-periodic function in the variable z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' It is easy to understand that the periodic extension, still denoted for simplicity by u, is a function satisfying u ∈ H1(Ca,b × I;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3) for any open bounded interval I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' The periodic extension of the boundary data can be achieved according to the next lemma, proved in subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' We state here some lemmas in order to understand the main steps in the construction of the solution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='3), given in the final theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' 9 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Let u be the periodic extension of the solution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='3) defined as above and let Λ be the distribution defined by −div(Tu) = Λ in D′(Ca,b × R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='9) Then Λ admits the following Fourier series expansion Λ = (Λ1, Λ2, Λ3) = � 0, 0, χp(x, y) +∞ � m=0 (−1)m+1 4 h sin �π h(2m + 1)z �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='10) The symmetry properties (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='8) and the construction of the periodic extension, allow expanding u = (u1, u2, u3) in Fourier series with respect to the variable z: u1(x, y, z) = +∞ � k=0 ϕ1 k(x, y) cos � π h kz � , u2(x, y, z) = +∞ � k=0 ϕ2 k(x, y) cos � π h kz � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='11) u3(x, y, z) = +∞ � k=0 ϕ3 k(x, y) sin � π h kz � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' In subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='4 we prove the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' For any k ≥ 1 odd, there exists a unique (ϕ1 k, ϕ2 k, ϕ3 k) ∈ H1(Ca,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3), satisfying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='3) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' For any k ≥ 2 even, there exists a unique trivial (ϕ1 k, ϕ2 k, ϕ3 k) ≡ (0, 0, 0) in Ca,b, satisfying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='3) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' We observe that for k = 0, the boundary value problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='3), or equivalently (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='23)-(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='24) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='26), see the proof in subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='4, admits an infinite number of solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' More precisely, these solutions are in form (ϕ1 0, ϕ2 0, ϕ3 0) = (c1, c2, c3) where c1, c2, c3 are three arbitrary constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' We may choose ϕ3 0 ≡ 0 being irrelevant in the Fourier expansion of u3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Concerning the other two components, we have necessarily ϕ1 0 ≡ ϕ2 0 ≡ 0 in Ca,b due to the odd symmetry of u1 and u2 with respect to the variables x and y, as stated in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='5) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='2 Cylindrical coordinates exchange The symmetry properties of u stated in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='1 imply that ϕ3 k is a radial function and the vector field (ϕ1 k, ϕ2 k) is a central vector field in the plane, in the sense that it is oriented toward the origin and its modulus is a function only of the distance from the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' This implies that for any k ≥ 1 odd, there exist two radial functions Yk = Yk(ρ) and Zk = Zk(ρ) such that in polar coordinates we may write ϕ1 k(ρ, θ) = Yk(ρ) cos θ , ϕ2 k(ρ, θ) = Yk(ρ) sin θ , ϕ3 k(ρ, θ) = Zk(ρ) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='12) with ρ ∈ [a, b] and θ ∈ [0, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' In Section 5 we show that Yk and Zk solve a proper boundary value problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' More precisely, this fact will be shown in subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='5 which is devoted to the proof of the next lemma, where we state existence and uniqueness for solutions of the boundary value problem mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Let Ψk : Ca,b → R be defined as Ψk(x, y) := � � � (−1) k+1 2 4 h χp(x, y) if k is odd, 0 if k is even, ∀(x, y) ∈ Ca,b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='13) 10 For any k ≥ 1 odd, the boundary value problem � � � � � � � � � � � � � � � � � � � � � � � � � Y ′′ k (ρ) + Y ′ k(ρ) ρ − Yk(ρ) ρ2 − µ λ + 2µ π2k2 h2 Yk(ρ) + λ + µ λ + 2µ πk h Z′ k(ρ) = 0 in (a, b) , Z′′ k(ρ) + Z′ k(ρ) ρ − λ + 2µ µ π2k2 h2 Zk(ρ) − λ + µ µ πk h � Y ′ k(ρ) + Yk(ρ) ρ � = − 1 µ Ψk(ρ) in (a, b) , (λ + 2µ)Y ′ k(ρ) + λ ρ Yk(ρ) + λ πk h Zk(ρ) = 0 , ρ ∈ {a, b} Z′ k(ρ) − πk h Yk(ρ) = 0 , ρ ∈ {a, b} (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='14) admits a unique solution (Yk, Zk) ∈ H1(a, b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' About existence and uniqueness of solutions of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='14), in subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='5 we only give an idea of the proof since it can be proved exactly as Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='3 of which Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='5 is the radial version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Now we need a more explicit representation for the unique solution (Yk, Zk) of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' This will be done by performing a power series expansion in which the coefficients will be characterized explicitly in terms of a suitable iterative scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' As a byproduct of this result in Section 4 we also obtain a numerical approximation of the exact solution and we estimate the corresponding error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Being a linear problem, we proceed by applying the superposition principle and we provide the explicit formula in the next lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' For any k ≥ 1, odd, let Υk = (Yk, Zk) the unique solution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Omitting for brevity the k-index, we have a unique (C1, C2, C3, C4) ∈ R4 such that Υ(ρ) = C1Υ1(ρ) + C2Υ2(ρ) + C3Υ3(ρ) + C4Υ4(ρ) + Υ(ρ), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='15) where Υj = (Y j, Zj) with j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' , 4 are four linear independent solutions of the corresponding homoge- nous system and Υ = (Y , Z) solves � Y (ρ) Y ′(ρ) Z(ρ) Z ′(ρ) �T = W(ρ) � ρ a (W(r))−1 � 0 0 0 − 1 µΨk(r) �T dr , ρ > 0 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='16) being W(ρ) the wronskian obtained through Υj(ρ) (j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' , 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Each of the linear independent solutions of the homogeneous system can be written as � � � � � � � � � � � � � Y j(ρ) = +∞ � n=−1 aj n ρn + (ln ρ) +∞ � n=0 bj n ρn , Zj(ρ) = +∞ � n=0 cj n ρn + (ln ρ) +∞ � n=0 dj n ρn (j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' , 4), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='17) where the coefficients are uniquely determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' In the proof of the Lemma we give all the details related to the computation of the constants Cj in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='15) and of the coefficients in the series (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='17), see subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' As a consequence of Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='2-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='3-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='5-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='6 we state the main theorem, whose proof can be found in subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Let u be the unique solution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='3) satisfying u ∈ V ⊥ 0 and let (Yk, Zk), k ≥ 1 odd, be the unique solution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Then, in cylindrical coordinates, u = (u1, u2, u3) admits the following 11 representation: � � � � � � � � � � � � � � � � � � � � � � � � � u1(ρ, θ, z) = +∞ � m=0 Y2m+1 (ρ) cos θ cos � (2m+1)π h z � , u2(ρ, θ, z) = +∞ � m=0 Y2m+1 (ρ) sin θ cos � (2m+1)π h z � , u3(ρ, θ, z) = +∞ � m=0 Z2m+1 (ρ) sin � (2m+1)π h z � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='18) with ρ ∈ (a, b), θ ∈ [0, 2π), z ∈ � − h 2, h 2 � where the three series in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='18) converge weakly in H1(Ω) and strongly in L2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Moreover, letting UM = (U 1 M, U2 M, U3 M) be the sequence of vector partial sums corresponding to the series expansions in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='18), we have for any M ≥ 1 ∥U 1 M − u1∥L2(Ω) ≤ pb2 µ � h(b − a) 2aπ 1 √ M , ∥U 2 M − u2∥L2(Ω) ≤ pb2 µ � h(b − a) 2aπ 1 √ M , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='19) ∥U 3 M − u3∥L2(Ω) ≤ p µaπ2 � h3b3(b − a) 24 1 √ M3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' 4 An engineering application In this section we consider a case of study: a hollow cylinder having the features of a blister for the bridge in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' In Table 1 we give the mechanical parameters, see also Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' We consider stays composed of 19 strands, see Figure 5, suitable to bear the concentrated load P in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' P is computed from the executive project, while the diameter 2a is taken from the catalogue of Protende ABS-2021 [15], a company producing such elements, see in Figure 5 the diameter φD1 for 19 strands anchorage;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' hence, the h 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='00 m Height of the cylinder 2a 273 mm Diameter of the cylindrical hollow 2b 800 mm External diameter of the cylinder 2ϵ 425 mm External diameter of the load P 1900 kN Concentrated load E 35000 MPa Young modulus of the concrete ν 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='2 Poisson ratio of the concrete Table 1: Mechanical parameters assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' distributed load in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='2) is given by p = P π(ϵ2−a2) = 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='80 MPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Our purpose is to obtain a good approximation of the functions Υj = (Y j, Zj), j ∈ {1, 2, 3, 4} introduced in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' For j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' , 4 we consider the approximate solution (N ≥ 1) Y j N(ρ) = N � n=−1 an ρn + (ln ρ) N � n=0 bn ρn and Zj N(ρ) = N−1 � n=0 cn ρn + (ln ρ) N−1 � n=0 dn ρn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='1) The reason for in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='1) we have n = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' , N − 1 in the expansion of Zj N will be clarified in the proof in subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='8 of the next proposition about an estimate of the truncating error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' 12 Figure 5: Detail of the strands anchorage and in table the geometric features for a 19 strands element, from the commercial catalogue [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Let k > 1 , k ∈ N odd, and let N ≥ 3, odd integer, be the truncating index of the series as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Then, letting Ek,N := max j∈{1,2,3,4} � max � max ρ∈[a,b] |Y j N(ρ) − Y j(ρ)|, max ρ∈[a,b] |Zj N(ρ) − Zj(ρ)| �� , we have that Ek,N ≤ C(a, b, k)3(2λ + 5µ)(λ + µ)2 16µ3 �πkb h �N+2 e( πkb h ) 2 (N + 3)(3N3 + 21N2 + 42N + 32) 2N �� N+1 2 � !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' �2 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='2) where C(a, b, k) = max � 1, h πkb � max � 1, ��ln � πka h ��� , | ln � πkb h � | � max � πk h , µ λ+µ πk h ln � πk h � , 2(λ+2µ) λ+µ h πk ln � πk h � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Once we have (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='2), one may choose N in such a way that Ek,N min j∈{1,2,3,4} � min � max ρ∈[a,b] |Y j N(ρ)|, max t∈[a,b] |Zj N(ρ)| �� < ε (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='3) with ε small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Condition (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='3) means that the truncation error is relatively small compared to the order of magnitude of both functions Y j N and Zj N for all j ∈ {1, 2, 3, 4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' In our numerical simulation the condition (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='3) is verified by making use of estimate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='2) on the truncation error Ek,N, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' the program verifies at each step the validity of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='3) in which the numerator of the fraction is replaced by the majorant in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' The program runs until the value of N is sufficiently large to guarantee (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' In Figure 6 we plot a vertical section of the cylinder and the corresponding more stressed horizontal section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' We show the vertical displacement u3 and the following components of the stress tensor in 13 Variavel H E1 OD1 OF R I B1 0C QA1 op 中 L1 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' VariavelNUMERO DE 0A1 B1 OC OD1 E1 OF H L1min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='OP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='CORDOALHAS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='110 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='006 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='680 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='609 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='145 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='457 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='225 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='3120 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='Valores sujeitos a variacoes de acordo com os requisitos especiais do projetoFigure 6: From the the left the vertical displacement u3 in mm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' the vertical stress σz in MPa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' the radial stress σr in MPa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' the angular stress σθ in MPa and the tangential stress τ rz in MPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' cylindrical coordinates σz = 2µ 1 − 2ν � (1 − ν)∂u3 ∂z + ν �ur ρ + ∂ur ∂ρ �� σr = 2µ 1 − 2ν � (1 − ν)∂ur ∂ρ + ν �ur ρ + ∂u3 ∂z �� σθ = 2µ 1 − 2ν � (1 − ν)ur ρ + ν �∂ur ∂ρ + ∂u3 ∂z �� τ rz = µ �∂ur ∂z + ∂u3 ∂ρ � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='4) where ur = � u2 1 + u2 2 is the radial displacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' We point out that putting n = (cos θ, sin θ, 0), t = (− sin θ, cos θ, 0) and k = (0, 0, 1), the four components introduced in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='4) are defined by σz := (Tu)k · k, σr := (Tu)n · n, σθ := (Tu)t · t and τ rz := (Tu)n · k and the representation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='4) can be deduced by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' We consider an approximate solution UM as stated in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='7 truncating the Fourier series at M = 29 with ε < 10−3 in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='3), implying N = 123 in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='1) and ∥U 1 29 − u1∥L2(Ω) ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='46 · 10−5 m5/2, ∥U 3 29 − u3∥L2(Ω) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='02 · 10−6 m5/2 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' In Table 2 we give the maximum absolute values of the variables involved, including the coordinate of the point (ρ, z) where they are assumed (for all θ ∈ [0, 2π) thanks to the radial symmetry of the problem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' As expected the vertical displacement u3 achieves its maximum absolute value at z = ± h 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' From the plots we see that there are two (symmetric) critical zones where we observe the loading diffusion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' they are close to the upper and bottom faces of the cylinder and involve approximately the 20% of the closest volume, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' the volume of Ω such that z ∈ (− h 2, − 2h 5 ) ∪ ( 2h 5 , h 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' 14 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='0 W3 _rz 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='5 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='2 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='16 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='4-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='4max | · | ρ z [m] u3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='24 mm a ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='50 σz 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='77 MPa a ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='42 σr 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='23 MPa a ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='42 σθ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='62 MPa a ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='43 τ rz 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='52 MPa ϵ ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='34 Table 2: Maximum absolute values and points of Ω in which they are assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' 5 Proofs of the results 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='1 Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='1 By identity (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='6), the estimates (divu)2 ≤ |∇u|2 and |Du|2 ≤ |∇u|2 and the H¨older inequality we infer that for any u, v ∈ H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3) ���� � Ω Tu : Dv dx ���� = ����2µ � Ω Du : Dv dx + λ � Ω (divu)(divv) dx ���� ≤ (λ + 2µ) ∥u ∥H1 ∥v ∥H1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='1) Estimate (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='1) proves the continuity of the bilinear form a(u, v) = � Ω Tu : Dv dx , u, v ∈ H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' By Korn inequality we also see that a(·, ·) is weakly coercive;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' indeed for any 0 < ε < 4µ we have a(u, u) + ε∥u ∥2 L2 ≥ 2µ � 1 − ε 4µ � � Ω |Du|2dx + ε 2 � 1 C � Ω |∇u|2dx − � Ω |u|2dx � + ε � Ω |u|2dx (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='2) ≥ ε 2C � Ω |∇u|2dx + ε 2 � Ω |u|2dx ≥ min � ε 2C , ε 2 � ∥u ∥2 H1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' On the other hand, it is easy to check that the linear functional Λ : H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3) → R defined by Λ(v) = � Ω f · v dx + � ∂Ω g · v dx , v ∈ H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3) is continuous thanks to the H¨older inequality and the classical trace inequality for H1-functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Hence, we may write Λ ∈ (H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3))′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' With the notations introduced in this proof, the variational problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='7) may be written in the form a(u, v) = ⟨Λ, v⟩ for any v ∈ H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Introducing the linear continuous operator L : H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3) → (H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3))′ defined by ⟨Lu, v⟩ = a(u, v) for any u, v ∈ H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3) , we may write (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='7) in the form Lu = Λ (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='3) as an identity between elements of the dual space (H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3))′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' The next step is to introduce the following operator Rε : (H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3))′ → H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3) which maps each element h ∈ (H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3))′ into the unique solution w of the variational problem a(w, v) + ε(w, v)L2 = ⟨h, v⟩ for any v ∈ H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' 15 This problem admits a unique solution by the continuity and coercivity estimates (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='1), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='2) combined with the Lax-Milgram Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' In particular Rε is well defined and continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Moreover, Rε is invertible and by the Open Mapping Theorem its inverse is also continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' In the rest of the proof we denote by J : H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3) → (H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3))′ the linear operator defined by ⟨Ju, v⟩ = � Ω u · v dx for any u, v ∈ H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3) , which is compact as a consequence of the compact embedding H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3) ⊂ L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' We now introduce on H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3) the following scalar product (u, v)ε = a(u, v) + ε(u, v)L2 for any u, v ∈ H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3) , which is equivalent to the natural scalar product of H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3) thanks to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='1) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' In this way we may now define the compact self-adjoint linear operator Tε : H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3) → H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3) given by Rε ◦ J where by self-adjoint we mean (Tεu, v)ε = (u, Tεv)ε for any u, v ∈ H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Indeed, from the definition of Rε, J and (·, ·)ε we see that (Tεu, v)ε = (u, v)L2 = (u, Tεv)ε for any u, v ∈ H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' By definition of Rε and J we have that L = R−1 ε − εJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' In particular u ∈ H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3) is a solution of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='3) if and only if −ε−1 Rε(R−1 ε − εJ)u = −ε−1 RεΛ or equivalently Tεu − ε−1 u = w once we put w = −ε−1 RεΛ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Then, applying the Fredholm alternative to the operator Tε we deduce that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='3), or equivalently (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='7), admits a solution u ∈ H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3) if and only if w ∈ � Ker � T ∗ ε − ε−1 IH1 ��⊥ = � Ker � Tε − ε−1 IH1 ��⊥ (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='4) where T ∗ ε denotes the adjoint operator of Tε, IH1 denotes the identity map in H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3) and the orthogonal spaces are defined in the sense of the scalar product (·, ·)ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' We observe that a function v ∈ Ker � Tε − ε−1 IH1 � if and only if RεJv = ε−1 v and by the definition of Rε this is equivalent to a � ε−1 v, φ � + ε � ε−1 v, φ � L2 = ⟨Jv, φ⟩ for any φ ∈ H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3) and, in turn, recalling the definition of J this is equivalent to a(v, φ) = 0 for any φ ∈ H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' This shows that Ker � Tε − ε−1 IH1 � = V0 as we deduce by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Let us proceed by proving (i)-(iv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' The proof of (i) is complete once we show that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='4) is equivalent to condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Condition (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='4) is equivalent to a(w, v) + ε(w, v)L2 = 0 for any v ∈ V0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='5) being Ker � Tε − ε−1 IH1 � = V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' But w = −ε−1 RεΛ so that by definition of Rε, we infer a(w, v) + ε(w, v)L2 = −ε−1 [a(RεΛ, v) + ε(RεΛ, v)L2] = −ε−1 ⟨Λ, v⟩ for any v ∈ H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='6) Combining (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='5) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='6) we finally obtain ⟨Λ, v⟩ = 0 for any v ∈ V0, which is exactly (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='16) in view of the definition of the functional Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' For the proof of (ii) we observe that by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='2), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='7), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='13) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='14) we have for any v ∈ H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3) � Ω T(u + v0) : Dv dx = � Ω Tu : Dv dx + � Ω Tv0 : Dv dx = � Ω Tu : Dv dx = � Ω f · v dx + � ∂Ω g · v dS which shows that u + v0 is a solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' 16 For the proof of (iii) we consider two solutions u and w of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='7) and let v0 = w − u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='7) we obtain � Ω Tv0 : Dv dx = � Ω Tw : Dv dx − � Ω Tu : Dv dx = � Ω f · v dx + � ∂Ω g · v dS − � Ω f · v dx − � ∂Ω g · v dS = 0 for any v ∈ H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3) which immediately gives v0 ∈ V0 thanks to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Finally, let us proceed with the proof of (iv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' First we prove the existence of a solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='7) in V ⊥ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Let u be a generic solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='7) and consider its orthogonal decomposition u = u0 + u1 ∈ V0 ⊕ V ⊥ 0 with respect to the scalar product (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Then, u1 = u − u0 ∈ V ⊥ 0 and by part (ii) we deduce that u1 is still a solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Once we have proved existence, let us prove uniqueness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Let u, w ∈ V ⊥ 0 be two solutions of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Then, on one hand we have that u − w ∈ V ⊥ 0 and on the other hand u − v ∈ V0 thanks to part (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Therefore, u − w ∈ V0 ∩ V ⊥ 0 = {0} and this readily implies u = w thus completing the proof of (iv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='2 Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='1 Concerning part (i) of the Proposition we only give the proof of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='4) since the proof of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='5)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='6) can be obtained with a similar procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' For any function v ∈ H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3) we denote by ¯v = (¯v1, ¯v2, ¯v2) ∈ H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3) the function defined by ¯v1(x, y, z) = v1(x, y, −z) , ¯v2(x, y, z) = v2(x, y, −z) , ¯v3(x, y, z) = −v3(x, y, −z) , ∀ (x, y, z) ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='7) Let u be the unique solution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='3) in V ⊥ 0 and let ¯u be the corresponding function defined by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' We start by showing that ¯u solves problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' In doing this we show that it solves the variational problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='7) where in the present case f = 0 and g is the function defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' By direct computation one can see that for any test function v ∈ H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3) we have for any (x, y, z) ∈ Ω (D¯u : Dv)|(x,y,z) = (Du : D¯v)|(x,y,−z) , [(div ¯u)(div v]|(x,y,z) = [(div u)(div ¯v]|(x,y,−z) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='8) By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='7), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='2), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='8), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='1) and a change of variables, we obtain 2µ � Ω D¯u : Dv dx + λ � Ω (div ¯u)(div v) dx (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='9) = 2µ � Ω Du : D¯v dx + λ � Ω (div u)(div ¯v) dx = � ∂Ω g · ¯v dS = � ∂Ω g · v dS .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' By (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='9) we deduce that ¯u is a solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='7) and hence a weak solution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' We now prove that ¯u ∈ V ⊥ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Indeed, proceeding as in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='9) one can easily show that (¯u, v)T = (u, ¯v)T = 0 for any v ∈ V0 since u ∈ V ⊥ 0 and ¯v ∈ V0 whenever v ∈ V0, as one can deduce by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' This completes the proof of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Let us proceed with the proof of part (ii) and (iii) of the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' For any θ ∈ (−2π, 2π) we denote by Rθ : R2 → R2 the anticlockwise rotation of an angle θ and by Aθ the associate matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Clearly we have that the inverse map of Rθ is given by R−θ and A−1 θ = A−θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' We use the notation u = (u′, u3) ∈ R2 × R with u′ = (u1, u2) and we denote by ∇′u′ = � ∂u1 ∂x ∂u1 ∂y ∂u2 ∂x ∂u2 ∂y � 17 its Jacobian matrix in the x and y variables, and by D′u′ the corresponding symmetric gradient given by 1 2 � ∇′u′ + (∇′u′)T � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' more in general, throughout this proof we will use the symbol ∇′ for denoting the gradient with respect to the x and y variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' We now define uθ(x, y, z) = � R−θ � u1(Rθ(x, y), z), u2(Rθ(x, y), z) � , u3(Rθ(x, y), z) � for any (x, y, z) ∈ Ω .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Then, the Jacobian matrix ∇uθ ∈ R3×3 and in turn the matrix Duθ admit a representation in terms of four blocks of dimensions 2×2, 2×1, 1×2, 1×1 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' We proceed directly with the representation of Duθ: Duθ = � � � � A−θ D′u′� Rθ(x, y), z � Aθ A−θ ∂u′ ∂z � Rθ(x, y), z � + � ∇′u3 � Rθ(x, y), z � Aθ �T � A−θ ∂u′ ∂z � Rθ(x, y), z ��T +∇′u3 � Rθ(x, y), z � Aθ ∂u3 ∂z � Rθ(x, y), z � � � � � (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='10) In the same way, for any test function v ∈ H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3) and any θ ∈ (−2π, 2π) we may define the correspond- ing function vθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Looking at v as (v−θ)θ and applying (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='10) to v−θ we claim that for any (x, y, z) ∈ Ω Duθ(x, y, z) : Dv(x, y, z) = Du � Rθ(x, y), z � : Dv−θ � Rθ(x, y), z � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='11) This is a consequence of the fact that Aθ is orthogonal and the linear map Lθ : R2×2 → R2×2, Lθ(X) = A−θXAθ is an isometry in R2×2 as one can see by verifying the orthogonality of the associated matrix Mθ ∈ R4×4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' This implies (A−θ XAθ) : Y = Lθ(X) : Y = Lθ(X) : Lθ(L−1 θ (Y )) = X : L−1 θ (Y ) = X : (AθY A−θ) for any X, Y ∈ R2×2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' This arguments allow to treat the scalar products between the 2×2 block appearing in the representation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Even easier is to treat the scalar products between the 2 × 1 and 1 × 2 blocks thanks to the orthogonality of Aθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' This proves the claim (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' The invariance of the trace of a matrix X under maps of the form X �→ A−1XA combined with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='10) shows that div uθ(x, y, z) = div u � Rθ(x, y), z � and in particular for any (x, y, z) ∈ Ω we have (div uθ(x, y, z))(div v(x, y, z)) = � div u � Rθ(x, y), z �� � div v−θ � Rθ(x, y), z �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='12) By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='7), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='1), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='2), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='11), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='12), two changes of variables and the definitions of v−θ and g, we obtain 2µ � Ω Duθ : Dv dx + λ � Ω (div uθ)(div v) dx (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='13) = 2µ � Ω Du : Dv−θ dx + λ � Ω (div u) (div v−θ) dx = � ∂Ω g · v−θ dS = � ∂Ω g · v dS .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' We have just proved that uθ is still a weak solution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' We now show that uθ ∈ V ⊥ 0 as a consequence of the fact that u ∈ V ⊥ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Proceeding as in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='13), we infer (uθ, v)T = (u, v−θ)T for any v ∈ H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='14) We need to prove that if v ∈ V0 then v−θ ∈ V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' For any θ ∈ (−2π, 2π), let Bθ be the 3 × 3 matrix corresponding to an anticlockwise rotation of an angle θ around the z axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Clearly Bθ is orthogonal and B−1 θ = B−θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' With this notation we may write v−θ(x) = Bθ v(B−θ x) for any x ∈ R3 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='15) 18 where both x and v have to be considered vector columns in the right hand side of the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' If v ∈ V0, then by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='15) we have that v admits the following matrix representation v(x) = Mx + δ for any x ∈ R3 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='16) where M is an antisymmetric matrix and δ = (δ1 δ2 δ3)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Combining (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='15) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='16) we obtain v−θ(x) = BθMB−θ x + Bθδ where the matrix BθMB−θ is antisymmetric since (BθMB−θ)T = BT −θ MT BT θ = B−1 −θ(−M)B−1 θ = −BθMB−θ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' This proves that also v−θ ∈ V0 since it admits a representation like in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Now, if we choose v ∈ V0 in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='14), we readily see that (uθ, v)T = 0 being u ∈ V ⊥ 0 and v−θ ∈ V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' This proves that uθ ∈ V ⊥ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' By the uniqueness result stated in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='1 (iv) we infer that uθ = u for any θ ∈ (−2π, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Now the validity of (ii) and of the first part of (iii) follows immediately from the definition of uθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' It remains to observe that the vector field u′ is oriented radially in the xy-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' To do this, it is sufficient to combine the identity u = uθ with the identity u2(x, 0, z) = 0, valid for any a < x < b and z ∈ � − h 2, h 2 � , as a consequence of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='3 Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='2 Let us introduce the sequence of intervals Ik := � − h 2 + kh, h 2 + kh � , the corresponding sequence of domains Ωk := Ca,b × Ik and the sequence of functions gk : ∂Ωk → R3 gk(x, y, z) := � � � � � � � � � (0, 0, (−1)k χp(x, y)) if (x, y, z) ∈ Ca,b × � − h 2 + kh � , (0, 0, (−1)k+1 χp(x, y)) if (x, y, z) ∈ Ca,b × � h 2 + kh � , (0, 0, 0) if (x, y, z) ∈ ∂Ca,b × Ik .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='17) We know that the original function u is a weak solution of problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='3) in the sense that � Ω Tu : Dv dx = � ∂Ω g · v dS for any v ∈ H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='18) We need to find, starting from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='18), the equation solved, in the sense of distributions, by the periodic extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' First of all, we observe that by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='8), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='17), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='18) and some computations, we have � Ωk Tu : Dv dx = � ∂Ωk gk · v dS for any v ∈ H1(Ωk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='19) Now, letting φ = (φ1, φ2, φ3) ∈ D(Ca,b × R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3), by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='19) we infer � Ca,b×R Tu : Dφ dx = � k∈Z � Ωk Tu : Dφ dx = � k∈Z � ∂Ωk gk · φ dS = � k∈Z 2(−1)k+1 � Ca,b χp(x, y) φ3 � x, y, h 2 + kh � dxdy .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' This proves (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='9), where Λ is the distribution defined by ⟨Λ, φ⟩ := � Ca,b χp(x, y) � k∈Z 2(−1)k+1φ3 � x, y, h 2 + kh � dxdy (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='20) 19 for any φ = (φ1, φ2, φ3) ∈ D(Ca,b × R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' The distribution Λ admits a sort of factorization as a product of a function in the variables x and y and of a distribution acting on functions of the variable z: Λ = (Λ1, Λ2, Λ3) = � 0, 0, 2 χp � k∈Z (−1)k+1 δ h 2 +kh � where Λ1, Λ2, Λ3 ∈ D′(Ca,b × R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R) are the scalar distributions defined by ⟨Λi, φ⟩ := ⟨Λ, φ ei⟩ for any φ ∈ D(Ca,b × R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R) , with e1 = i, e2 = j, e3 = k, and δ h 2 +kh are Dirac delta distributions concentrated at z = h 2 + kh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Expanding in Fourier series the periodic distribution � k∈Z 2(−1)k+1 δ h 2 +kh we obtain (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='10), where the Fourier series converges in the sense of distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' For more details on this convergence see the arguments introduced in subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='4 Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='3 First of all we insert (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='11) into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='9);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' recalling the Hooke’s law (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='2) and exploiting (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='10), we obtain � � � � � � � � � � � � � � � � � � � −µ∆ϕ1 k + µπ2k2 h2 ϕ1 k − (λ + µ) �∂2ϕ1 k ∂x2 + ∂2ϕ2 k ∂x∂y + πk h ∂ϕ3 k ∂x � = 0 in Ca,b , −µ∆ϕ2 k + µπ2k2 h2 ϕ2 k − (λ + µ) � ∂2ϕ1 k ∂x∂y + ∂2ϕ2 k ∂y2 + πk h ∂ϕ3 k ∂y � = 0 in Ca,b , −µ∆ϕ3 k + µπ2k2 h2 ϕ3 k + πk h (λ + µ) �∂ϕ1 k ∂x + ∂ϕ2 k ∂y + πk h ϕ3 k � = Ψk in Ca,b , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='21) where the forcing term is defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' We observe that in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='21), the operator ∆ stands for the Laplace operator in the variables x and y, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' ∆ = ∂2/∂x2 + ∂2/∂y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Putting Φk := (ϕ1 k, ϕ2 k) and ¯n ∈ R2 the outward unit normal to ∂Ca,b, system (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='21) may be rewritten in the following form � � � −µ∆Φk + µ π2k2 h2 Φk − (λ + µ)∇(div Φk) − (λ + µ) πk h ∇ϕ3 k = 0 in Ca,b , −µ∆ϕ3 k + µ π2k2 h2 ϕ3 k + πk h (λ + µ) � div Φk + πk h ϕ3 k � = Ψk in Ca,b , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='22) or equivalently in the following form � � � −div(λ(div Φk)I + 2µDΦk) + µ π2k2 h2 Φk − (λ + µ) πk h ∇ϕ3 k = 0 in Ca,b , −µ∆ϕ3 k + (λ + 2µ) π2k2 h2 ϕ3 k + (λ + µ) πk h div Φk = Ψk in Ca,b , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='23) where D represents here the symmetric gradient in the two-dimensional case and I is the 2 × 2 identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' We also recall that by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='3), (Tu)n = 0 on ∂Ca,b × R so that by the Hooke’s law (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='2) we obtain � � � � � � � � � � � � � � � � � � � � � (λ + 2µ)x∂u1 ∂x + λx∂u2 ∂y + λx∂u3 ∂z + µy �∂u1 ∂y + ∂u2 ∂x � = 0 on ∂Ca,b × R , λy∂u1 ∂x + (λ + 2µ)y∂u2 ∂y + λy∂u3 ∂z + µx �∂u1 ∂y + ∂u2 ∂x � = 0 on ∂Ca,b × R , µx �∂u1 ∂z + ∂u3 ∂x � + µy �∂u2 ∂z + ∂u3 ∂y � = 0 on ∂Ca,b × R , 20 and by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='11) we obtain � � � λ(div Φk)¯n + 2µ(DΦk)¯n + λ πk h ϕ3 k ¯n = 0 on ∂Ca,b , µ∇ϕ3 k · ¯n − µ πk h Φk · ¯n = 0 on ∂Ca,b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='24) Let us derive the weak formulation of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='22)-(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Testing (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='23) with (w1, w2, w3), putting W = (w1, w2) and integrating by parts we obtain − � ∂Ca,b [(λ(div Φk)I + 2µDΦk) ¯n] · W ds + � Ca,b (λ(div Φk)I + 2µDΦk) : ∇W dxdy (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='25) + µπ2k2 h2 � Ca,b Φk · W dxdy − µπk h � Ca,b ∇ϕ3 k · W dxdy − λπk h � ∂Ca,b ϕ3 k ¯n · W ds + λπk h � Ca,b ϕ3 k div W dxdy − � ∂Ca,b µ∂ϕ3 k ∂¯n w3ds + µ � Ca,b ∇ϕ3 k · ∇w3 dxdy + (λ + 2µ)π2k2 h2 � Ca,b ϕ3 k w3 dxdy + µπk h � ∂Ca,b w3Φk · ¯n ds − µπk h � Ca,b Φk · ∇w3 dxdy + λπk h � Ca,b div Φk w3 dxdy = � Ca,b Ψk w3 dxdy .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' We observe that by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='24) the boundary integrals in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='25) disappear;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' on the other hand collecting the double integrals and recalling that DΦk : ∇W = DΦk : DW, we may write (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='25) in the form 2µ � Ca,b DΦk : DW dxdy + λ � Ca,b � div Φk + πk h ϕ3 k � � div W + πk h w3� dxdy (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='26) + µ � Ca,b (∇ϕ3 k − πk h Φk) · (∇w3 − πk h W)dxdy + 2µ π2k2 h2 � Ca,b ϕ3 kw3dxdy= � Ca,b Ψkw3dxdy for any w ∈ H1(Ca,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3), where W = (w1, w2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' This represents the weak form of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='22)-(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' For any k ≥ 2 even we observe that, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='13), ϕ1 k ≡ ϕ2 k ≡ ϕ3 k ≡ 0 in Ca,b, as one can deduce by testing (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='26) with (w1, w2, w2) = (ϕ1 k, ϕ2 k, ϕ3 k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' For any k ≥ 1 odd, we define the following bilinear form ¯ak(ϕ, w) := 2µ � Ca,b DΦ : DW dxdy + λ � Ca,b � div Φ + πk h ϕ3 k � � div W + πk h w3� dxdy (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='27) + µ � Ca,b (∇ϕ3 − πk h Φ) · (∇w3 − πk h W) dxdy + 2µ π2k2 h2 � Ca,b ϕ3 w3 dxdy for any ϕ, w ∈ H1(Ca,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3) where ϕ = (ϕ1, ϕ2, ϕ3), Φ := (ϕ1, ϕ2), w = (w1, w2, w3) and W = (w1, w2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' For the uniqueness issue we claim that for any ε > 0 there exists Cε > 0 such that ¯ak(ϕ, ϕ) + ε∥ϕ∥2 L2 ≥ Cε∥ϕ∥2 H1 for any ϕ ∈ H1(Ca,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='28) Suppose by contradiction that there exists ε > 0 such that for any m ≥ 1 there exists ϕm ∈ H1(Ca,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3) such that ¯ak(ϕm, ϕm) + ε∥ϕm∥2 L2 ≤ 1 m∥ϕm∥2 H1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='29) Up to normalization, it is not restrictive to assume that the sequence {ϕm} satisfies ∥ϕm∥H1 = 1 for any m ≥ 1, so that by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='27) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='29) we infer ϕm → 0 in L2(Ca,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3) , � Ca,b |DΦm|2dxdy → 0 , ∇ϕ3 m − kΦm → 0 in L2(Ca,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R2) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='30) 21 as m → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Applying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='11) in the two-dimensional case we obtain � Ca,b |∇Φm|2dxdy ≤ C �� Ca,b |DΦm|2dxdy + � Ca,b |Φm|2 dxdy � for some constant C > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' This, combined with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='30), proves that ∇Φm → 0 in L2(Ca,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R2×2) , ∇ϕ3 m → 0 in L2(Ca,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R2) and, in turn, that ϕm → 0 in H1(Ca,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' This contradicts the assumption ∥ϕm∥H1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' We have completed the proof of the claim (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Thanks to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='28), we may proceed as in the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='1 and apply the Fredholm alternative to show that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='23)-(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='24) admits a solution if and only if � Ca,b Ψk w3 dxdy = 0 for any w = (w1, w2, w3) ∈ ¯Vk (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='31) where ¯Vk := {w ∈ H1(Ca,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3) : ¯ak(w, v) = 0 for any v ∈ H1(Ca,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Testing the variational identity in the definition of ¯Vk with v = w, we readily see that for any k ≥ 1 we ¯Vk = {0} and hence, condition (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='31) is always satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' This completes the proof of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='5 Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='5 Before proceeding with the proof of the lemma, we devote the first part of this subsection to show that the functions Yk and Zk introduced in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='12) really satisfy (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' In order to simplify the notations we denote by Y and Z the unknown functions, omitting the index k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Testing (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='26) with a test function (w1, w2, w3) admitting in polar coordinates the following representation w1(ρ, θ) = H(ρ) cos θ , w2(ρ, θ) = H(ρ) sin θ , w3(ρ, θ) = K(ρ) , by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='12) we obtain 2µ � b a � ρY ′(ρ)H′(ρ) + Y (ρ)H(ρ) ρ � dρ + λ � b a ρ � Y ′(ρ) + Y (ρ) ρ + πk h Z(ρ) � � H′(ρ) + H(ρ) ρ + πk h K(ρ) � dρ (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='32) + µ � b a ρ � Z′(ρ) − πk h Y (ρ) � � K′(ρ) − πk h H(ρ) � dρ + 2µπ2k2 h2 � b a ρZ(ρ)K(ρ) dρ = � b a ρΨk(ρ)K(ρ) dρ with obvious meaning of the notation Ψk(ρ) being it a radial function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Collecting in a proper way the terms of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='32), we may rewrite it in the form � b a � (λ + 2µ)ρY ′(ρ) + λY (ρ) + λπk h ρZ(ρ) � H′(ρ) dρ (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='33) + � b a � λY ′(ρ) + (λ + 2µ)Y (ρ) ρ + µπ2k2 h2 ρY (ρ) − µπk h ρZ′(ρ) + λπk h Z(ρ) � H(ρ) dρ + � b a � µρZ′(ρ) − µπk h ρY (ρ) � K′(ρ) dρ + � b a � (λ + 2µ)π2k2 h2 ρZ(ρ) + λπk h ρY ′(ρ) + λπk h Y (ρ) � K(ρ) dρ = � b a ρΨk(ρ)K(ρ) dρ Integrating by parts the terms in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='33) containing H′(ρ) and K′(ρ), we see that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='33) is the variational formulation of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' 22 Let us proceed now with the proof of the lemma which is the main point of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Actually, we give here only a sketch of the proof since it essentially follows the ideas already introduced in the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' About the uniqueness issue, on the space H1(a, b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R2) it sufficient to define the bilinear form bk : H1(a, b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R2) × H1(a, b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R2) → R corresponding to the left hand side of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='32) and prove for it an estimate of the type (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Then, following again the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='3, one finds that the compatibility condition for Ψk is given by � b a ρΨk(ρ)K(ρ) dρ = 0 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='34) for any (H, K) ∈ H1(a, b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R2) satisfying bk � (H, K), (H, K) � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' A simple check shows that (H, K) ≡ (0, 0) so that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='34) is trivially satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' The Fredholm alternative then implies the existence of a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='6 Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='6 We omit for simplicity the dependence from the index k in the unknowns Yk and Zk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' For more clarity we divide the construction of this representation of Y and Z into different steps each of them is contained in the next subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='1 The solution of the homogeneous system We consider the homogeneous version of the system in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='14) � � � � � � � � � Y ′′(ρ) + Y ′(ρ) ρ − Y (ρ) ρ2 − α k2 Y (ρ) + β kZ′(ρ) = 0 ρ > 0 , Z′′(ρ) + Z′(ρ) ρ − γ k2Z(ρ) − δ k � Y ′(ρ) + Y (ρ) ρ � = 0 ρ > 0 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='35) where we put for simplicity α = π2µ h2(λ + 2µ) , β = π(λ + µ) h(λ + 2µ) , γ = π2(λ + 2µ) h2µ , δ = π(λ + µ) hµ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' We look for a solution admitting the following expansion � � � � � � � � � � � � � Y (ρ) = +∞ � n=−1 an ρn + (ln ρ) +∞ � n=0 bn ρn , Z(ρ) = +∞ � n=0 cn ρn + (ln ρ) +∞ � n=0 dn ρn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='36) Inserting the representation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='36) in the system (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='35), we obtain for each of the two equations the following identities: 23 +∞ � n=−1 n(n − 1)an ρn + +∞ � n=0 (n − 1)bn ρn + +∞ � n=0 nbn ρn + (ln ρ) +∞ � n=0 n(n − 1)bn ρn + +∞ � n=−1 nan ρn + +∞ � n=0 bn ρn + (ln ρ) +∞ � n=0 nbn ρn −αk2 +∞ � n=1 an−2 ρn − αk2(ln ρ) +∞ � n=2 bn−2 ρn − +∞ � n=−1 an ρn − (ln ρ) +∞ � n=0 bn ρn +βk +∞ � n=1 (n − 1)cn−1 ρn + βk +∞ � n=1 dn−1 ρn + βk(ln ρ) +∞ � n=1 (n − 1)dn−1 ρn = 0 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='37) +∞ � n=0 n(n − 1)cn ρn + +∞ � n=0 (n − 1)dn ρn + +∞ � n=0 ndn ρn + (ln ρ) +∞ � n=0 n(n − 1)dn ρn + +∞ � n=0 ncn ρn + +∞ � n=0 dn ρn + (ln ρ) +∞ � n=0 ndn ρn − γk2 +∞ � n=2 cn−2 ρn − γk2(ln ρ) +∞ � n=2 dn−2 ρn −δk +∞ � n=0 (n − 1)an−1 ρn − δk +∞ � n=1 bn−1 ρn − δk(ln ρ) +∞ � n=1 (n − 1)bn−1 ρn −δk +∞ � n=0 an−1 ρn − δk(ln ρ) +∞ � n=1 bn−1 ρn = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='38) To determine the values of the coefficients an, bn, cn, dn we need an iterative scheme starting from the values of the coefficients a−1, a0, a1, b0, b1, c0, c1, d0, d1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' The values of these nine parameters have to be determined collecting the coefficients of the terms ρ−1, ρ0, ρ0 ln ρ, ρ, ρ ln ρ appearing in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='37)-(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='38) and equating them to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' As a result of this procedure we obtain the following constraint: � a0 = b0 = c1 = d1 = 0 , 2b1 + βkd0 = αk2a−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='39) Among the left five parameters a−1, a1, b1, c0, d0 that may be possibly different from zero, a1, c0 and two among a−1, b1, d0 can be chosen arbitrarily, while the remaining one is determined by the equation in the second line of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='39);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' for example, we may choose arbitrarily a−1, a1, b1, c0 and put d0 = αk β a−1 − 2 βk b1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' In particular, we are interested in finding the general solution of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='35) as a linear combination of four linearly independent special solutions, denoted by Υj = (Y j, Zj) with j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' , 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' A possible choice for the independent solutions is given respectively by the assumption on the following combinations of coefficients: Υ1 : (a−1, a1, b1, c0) = (1, 0, 0, 0) , Υ2 : (a−1, a1, b1, c0) = (0, 1, 0, 0) , Υ3 : (a−1, a1, b1, c0) = (0, 0, 1, 0) , Υ4 : (a−1, a1, b1, c0) = (0, 0, 0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='40) By (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='37)-(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='38) we deduce the following linear system in the unknowns an, bn, cn−1, dn−1 with data 24 expressed in terms of an−2, bn−2, cn−3, dn−3: � � � � � � � � � � � (n2 − 1)an + 2nbn + βk(n − 1)cn−1 + βkdn−1 = αk2an−2 (n2 − 1)bn + βk(n − 1)dn−1 = αk2bn−2 (n − 1)2cn−1 + 2(n − 1)dn−1 = δk(n − 1)an−2 + δkbn−2 + γk2cn−3 (n − 1)2dn−1 = δk(n − 1)bn−2 + γk2dn−3 (n ≥ 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='41) We observe that the matrix of coefficients associated to system (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='41) is given by � � � � � � n2 − 1 2n β(n − 1)k βk 0 n2 − 1 0 β(n − 1)k 0 0 (n − 1)2 2(n − 1) 0 0 0 (n − 1)2 � � � � � � whose determinant is given by (n−1)6(n+1)2 ̸= 0, thus showing that the system is not singular for n ≥ 2 and hence admits a unique solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' With the restriction n ≥ 3 the coefficients a2, b2 remained excluded, but their calculation can be obtained from the first two equations of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='41) by choosing n = 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' this gives a2 = b2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' The linear independence of Υ1, Υ2, Υ3, Υ4 can be verified by looking at the asymptotic behavior of Y j(ρ), j = 1, 2, 3, 4 as ρ → 0+ in the four cases (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='40): case 1: Y 1(ρ) ∼ ρ−1 as ρ → 0+ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' case 2: Y 2(ρ) ∼ ρ as ρ → 0+ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' case 3: Y 3(ρ) ∼ ρ ln ρ as ρ → 0+ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' case 4: Y 4(ρ) = O(ρ2 ln ρ) as ρ → 0+ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' We observe that, after a suitable scaling, the dependence of system (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='35) from the parameter k can be dropped: given a solution (Y, Z) of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='35), we may define the functions �Y (t) = Y � h πk t � and �Z(t) = Z � h πk t � in such a way that the couple (�Y , �Z) solves system � � � � � � � � � �Y ′′(t) + �Y ′(t) t − �Y (t) t2 − �α �Y (t) + �β �Z′(t) = 0 t > 0 , �Z′′(t) + �Z′(t) t − �γ �Z(t) − �δ � �Y ′(t) + �Y (t) t � = 0 t > 0 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='42) where �α = µ/(λ + 2µ), �β = (λ + µ)/(λ + 2µ), �γ = (λ + 2µ)/µ and �δ = (λ + µ)/µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='2 The particular solution We write the nonhomogeneous system in the matrix form � � � � � � Y (ρ) Y ′(ρ) Z(ρ) Z′(ρ) � � � � � � ′ = � � � � � � 0 1 0 0 1 ρ2 + αk2 − 1 ρ 0 −βk 0 0 0 1 δk ρ δk γk2 − 1 ρ � � � � � � � � � � � � Y (ρ) Y ′(ρ) Z(ρ) Z′(ρ) � � � � � � + � � � � � � 0 0 0 − 1 µΨk(ρ) � � � � � � , ρ > 0 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='43) where the function Ψk = Ψk(ρ) is extended trivially outside the interval (a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' 25 Maintaining the order of the components, we may write the Wronskian matrix associated with Υ1, Υ2, Υ3, Υ4 in the form W(ρ) = � � � � � � Y 1(ρ) Y 2(ρ) Y 3(ρ) Y 4(ρ) (Y 1(ρ))′ (Y 2(ρ))′ (Y 3(ρ))′ (Y 4(ρ))′ Z1(ρ) Z2(ρ) Z3(ρ) Z4(ρ) (Z1(ρ))′ (Z2(ρ))′ (Z3(ρ))′ (Z4(ρ))′ � � � � � � , so that a particular solution Υ = (Y , Z) of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='43) is given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='3 The unique solution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='14) Applying the superposition principle we get (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' In order to obtain the unique solution (Y, Z) of the boundary value problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='14), it remains to determine the constants C1, C2, C3, C4 so that the boundary conditions at ρ = a and ρ = b are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' We check that the constants C1, C2, C3, C4 are uniquely determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' They solve the system A � � � � C1 C2 C3 C4 � � � � = � � � � � � � � � − � (λ + 2µ)Y ′(a) + λ a Y (a) + λ πk h Z(a) � − � (λ + 2µ)Y ′(b) + λ b Y (b) + λ πk h Z(b) � − � Z ′(a) − πk h Y (a) � − � Z ′(b) − πk h Y (b) � � � � � � � � � � where the matrix A = (aij),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' j ∈ {1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' 4},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' is given by a1j = (λ + 2µ)(Y j)′(a) + λ a Y j(a) + λ πk h Zj(a) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' a2j = (λ + 2µ)(Y j)′(b) + λ b Y j(b) + λ πk h Zj(b) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' a3j = (Zj)′(a) − πk h Y j(a) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' a4j = (Zj)′(b) − πk h Y j(b) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' We claim that the matrix A is not singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Consider the homogeneous linear system Ad = 0 with d = (D1, D2, D3, D4)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Then the function Γ = (G, H) given by Γ(ρ) = D1Υ1(ρ) + D2Υ2(ρ) + D3Υ3(ρ) + D4Υ4(ρ) solves system (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='35) coupled with the boundary conditions � � � � � � � � � � � � � � � (λ + 2µ)G′(a) + λ a G(a) + λ πk h H(a) = 0 , (λ + 2µ)G′(b) + λ b G(b) + λ πk h H(b) = 0 , H′(a) − πk h G(a) = 0 , H′(b) − πk h G(b) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='5 we then have that Γ ≡ (0, 0) in (a, b) but being Γ also a solution of system (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='35) for ρ ∈ (0, +∞), by local uniqueness for Cauchy problems, Γ ≡ (0, 0) in (0, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' The linear independence of the functions Υ1, Υ2, Υ3, Υ4, then implies D1 = D2 = D3 = D4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' We just proved that the linear system Ad = 0 admits only the trivial solution, thus completing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' □ 26 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='7 Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='7 The formal series contained in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='18) are a consequence of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='11), Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='3 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' It remains to show how those series converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' We start by proving the weak convergence in H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Let F be the linear functional defined by F(v) := � ∂Ω g · v dS for any v ∈ H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3) with g as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' We observe that thanks to the H¨older inequality and the trace inequality F ∈ (H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3))′: ��(H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='R3))′⟨F, v⟩H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='R3) �� ≤ 2p � π(b2 − a2) C(Ω) ∥v∥H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='R3) for any v ∈ H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3) , where C(Ω) is such that ∥trace(v)∥L2(∂Ω) ≤ C(Ω)∥v∥H1(Ω) for any v ∈ H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Writing F = (F1, F2, F3) we have that F1, F2, F3 ∈ (H1(Ω))′ with F1 = F2 are the null functionals and (H1(Ω))′⟨F3, v⟩H1(Ω) = − � Ca,b χp(x, y) � v � x, y, h 2 � − v � x, y, − h 2 �� dxdy for any v ∈ H1(Ω) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Let us define the sequence of partial sums corresponding to the Fourier expansion in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='10): SM(x, y, z) := χp(x, y) M � m=0 (−1)m+1 4 h sin �π h(2m + 1)z � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' We claim that SM ⇀ F3 weakly in (H1(Ω))′ as M → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' We first prove that the sequence {SM} is bounded in (H1(Ω))′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' In the next estimate we use the following notations: we put �Ω := Ca,b × � − h 2, 3h 2 � , we still denote by v the symmetric and 2h-periodic extension of a function v ∈ H1(Ω) (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='1) and by v � x, y, 3h 2 � and v � x, y, − h 2 � , the traces of a function v ∈ H1(�Ω) on the upper and lower faces of the hollow cylinder �Ω, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' ��(H1(Ω))′⟨SM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' v⟩H1(Ω) �� = ����� M � m=0 (−1)m+1 4 h � Ω χp(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' y) sin �π h(2m + 1)z � v(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' z) dxdydz ����� = ����� M � m=0 (−1)m+1 2 h � �Ω χp(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' y) sin �π h(2m + 1)z � v(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' z) dxdydz ����� (integration by parts) = ����� M � m=0 (−1)m+1 2 h �� Ca,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='b −h cos � 3π 2 (2m + 1) � χp(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' y) π(2m + 1) v � x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' 3h 2 � dxdy + � Ca,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='b h cos � − π 2 (2m + 1) � χp(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' y) π(2m + 1) v � x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' −h 2 � dxdy + � �Ω hχp(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' y) π(2m + 1) cos �π h(2m + 1)z � ∂v ∂z (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' z) dxdydz ����� (2h-periodicity of v) = ����� M � m=0 (−1)m+1 2 h �� �Ω hχp(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' y) π(2m + 1) cos �π h(2m + 1)z � ∂v ∂z (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' z) dxdydz ������ = ����� 2 π � Ca,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='b � M � m=0 (−1)m+1χp(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' y) 2m + 1 � 3h 2 − h 2 cos �π h(2m + 1)z � ∂v ∂z (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' z) dz � dxdy ����� (Cauchy-Schwarz inequality in Rn+1) ≤ 2p π � M � m=0 1 (2m + 1)2 � 1 2 � Ca,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='b � � M � m=0 �� 3h 2 − h 2 cos �π h(2m + 1)z � ∂v ∂z (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' z) dz �2� � 1 2 dxdy 27 (Bessel inequality) ≤ 2p π � +∞ � m=0 1 (2m + 1)2 � 1 2 � Ca,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='b � 1 h � 3h 2 − h 2 �∂v ∂z (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' z) �2 dz � 1 2 dxdy (H¨older inequality) ≤ 2p π � +∞ � m=0 1 (2m + 1)2 � 1 2� π(b2 − a2) �� Ca,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='b � 1 h � 3h 2 − h 2 �∂v ∂z (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' z) �2 dz � dxdy � 1 2 = 2p � b2 − a2 πh � +∞ � m=0 1 (2m + 1)2 � 1 2 ���� ∂v ∂z ���� L2(�Ω) ≤ 4p � b2 − a2 πh � +∞ � m=0 1 (2m + 1)2 � 1 2 ∥v∥H1(Ω) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' This readily implies ∥SM∥(H1(Ω))′ ≤ 4p � b2 − a2 πh � +∞ � m=0 1 (2m + 1)2 � 1 2 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='44) and boundedness of {SM} in (H1(Ω))′ is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Now we claim that � Ω SMφ dx → − � Ca,b χp(x, y) � φ � x, y, h 2 � − φ � x, y, − h 2 �� dxdy =(H1(Ω))′ ⟨F3, φ⟩H1(Ω) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='45) as M → +∞, for any φ ∈ C∞(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' First of all, by using the classical results about pointwise convergence of the Fourier Series applied to suitable 2h-periodic extensions of the functions z �→ φ(x, y, z) , z ∈ � − h 2, 0 � and z �→ φ(x, y, z) , z ∈ � 0, h 2 � one can show that for any (x, y) ∈ Ca,b � h 2 − h 2 SM(x, y, z)φ(x, y, z)dz → −χp(x, y) � φ � x, y, h 2 � − φ � x, y, − h 2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='46) Then applying to the test function φ the estimates used for proving boundedness of {SM} in (H1(Ω))′, one can show that for any M ����� � h 2 − h 2 SM(x, y, z)φ(x, y, z)dz ����� ≤ 2 √ 2p π � +∞ � m=0 1 (2m + 1)2 � 1 2 ���� ∂φ ∂z ���� L∞(Ω) for any (x, y) ∈ Ca,b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='47) By (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='46), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='47) and the Dominated Convergence Theorem the proof of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='45) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' With an essentially similar procedure one can prove that SM converges in the sense of distributions to Λ3 where Λ3 is the third component of the vector distribution Λ defined in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Since (H1(Ω))′ is a reflexive Banach space, by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='44) we infer that along suitable subsequences, the partial sums are weakly convergent in (H1(Ω))′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Thanks to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='45), we deduce that the weak limits of this subsequences coincide on the space C∞(Ω) and they equal F3 on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' By density of C∞(Ω) in H1(Ω), they actually coincide on the whole H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' This proves that all weakly convergent subsequences weakly converge to F3 and hence the sequence SM is itself weakly convergent to F3 in (H1(Ω))′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' We can now denote by SM = (0, 0, SM) the sequence of vector partial sums in such a way that SM ⇀ F weakly in (H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3))′ as M → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Now, let us consider the linear continuous operator L introduced in the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='1 and its restriction to V ⊥ 0 , where we recall that orthogonality is with respect to the scalar product (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Then, by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='1 (iv) we deduce that L|V ⊥ 0 : V ⊥ 0 → (H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3))′ is invertible and by the Open Mapping Theorem it follows that its inverse is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' 28 If we define UM := L−1 |V ⊥ 0 SM, then UM is the vector partial sum corresponding to the Fourier expansion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Since SM is weakly convergent in (H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3))′ to F, then the continuity of L|V ⊥ 0 implies that UM is weakly convergent in H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3) to the unique solution u of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='3) as M → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' The strong convergence UM → u in L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3) as M → +∞ is a consequence of the compactness of the embedding H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3) ⊂ L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' It remains to prove (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' In order to emphasize the dependence on k we reintroduce it for denoting the functions Yk and Zk appearing in the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Testing (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='32) with (H, K) = (Yk, Zk) we have 2µπ2 h2 k2 � b a ρ(Zk(ρ))2dρ ≤ � b a ρΨk(ρ)Zk(ρ) dρ from which we obtain �� b a (Zk(ρ))2dρ � 1 2 ≤ 2pbh √ b − a µaπ2 1 k2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='48) Testing again (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='32) with (H, K) = (Yk, Zk) we also have 2µ � b a (Yk(ρ))2 ρ dρ ≤ � b a ρΨk(ρ)Zk(ρ) dρ from which we obtain � b a (Yk(ρ))2 dρ ≤ 2pb2√ b − a µh �� b a (Zk(ρ))2dρ � 1 2 ≤ 4p2b3(b − a) µ2aπ2 1 k2 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='49) where in the last inequality we used (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='48).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Let us proceed by considering the difference between the partial sum for u1 and u1 itself: ∥U 1 M − u1∥2 L2(Ω) = � Ca,b ����� +∞ � m=M+1 Y2m+1(ρ) cos θ cos �(2m + 1)π h z ������ 2 L2(− h 2 , h 2) dxdy = h 2 +∞ � m=M+1 � Ca,b (cos2 θ)(Y2m+1(ρ))2dxdy = πh 2 +∞ � m=M+1 � b a (Y2m+1(ρ))2ρ dρ ≤ 2p2b4(b − a)h µ2aπ +∞ � m=M+1 1 (2m + 1)2 ≤ p2b4(b − a)h 2µ2aπ +∞ � m=M+1 1 m2 ≤ p2b4(b − a)h 2µ2aπ 1 M , where we also used (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='49).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' The estimate for ∥U 2 n − u2∥L2(Ω) gives the same result for obvious reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' With a completely similar procedure by exploiting this time (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='48), we obtain ∥U 3 n − u3∥2 L2(Ω) ≤ 2p2b3h3(b − a) µ2a2π4 +∞ � m=M+1 1 (2m + 1)4 ≤ p2b3h3(b − a) 8µ2a2π4 +∞ � m=M+1 1 m4 ≤ p2b3h3(b − a) 24µ2a2π4 1 M3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' The solution we found by means of the Fourier series expansion satisfies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='7) in the sense of traces of H1-functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' We conclude the proof of the theorem by observing that this solution coincides with the unique solution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='3) belonging to V ⊥ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' To see this, denote by u the solution found by means of the Fourier series expansion and by w the solution in V ⊥ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Both u and w possesses the symmetry properties stated in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='1 as it occurs to their difference u − w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' But from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='1 we have that u − w ∈ V0 and it is readily seen from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='15) that functions in V0 satisfying those symmetry properties are necessarily the null function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' This proves that u = w and completes the proof of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' □ 29 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='8 Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='1 We rewrite the homogeneous system (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='35) as in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='42) so that the corresponding series expansion can be written in the form � � � � � � � � � � � � � � � �Y (t) = +∞ � n=−1 �an tn + (ln t) +∞ � n=0 �bn tn , �Z(t) = +∞ � n=0 �cn tn + (ln t) +∞ � n=0 �dn tn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='50) The coefficients �an,�bn, �cn, �dn are related to the corresponding coefficients an, bn, cn, dn appearing in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='36), by the formulas �a−1 = πk h a−1 , �an = � h πk �n � an − ln �πk h � bn � , �bn = � h πk �n bn , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='51) �cn = � h πk �n � cn − ln �πk h � dn � , �dn = � h πk �n dn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Inserting (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='50) into (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='42) or alternatively combining (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='51) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='41), we see that �an,�bn, �cn, �dn solve the system � � � � � � � � � � � (n2 − 1)�an + 2n�bn + β(n − 1)�cn−1 + �β �dn−1 = �α�an−2 (n2 − 1)�bn + �β(n − 1) �dn−1 = �α�bn−2 (n − 1)2�cn−1 + 2(n − 1) �dn−1 = �δ(n − 1)�an−2 + �δ�bn−2 + �γ�cn−3 (n − 1)2 �dn−1 = �δ(n − 1)�bn−2 + �γ �dn−3 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='52) for n ≥ 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' moreover �a0 = �b0 = �c1 = �d1 = �a2 = �b2 = 0, the coefficients �a−1, �a1,�b1, �c0 may be chosen arbitrarily and �d0 = �α �β �a−1 − 2 �β �b1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' By direct computation one can verify that the unique solution of system (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='52) can be written in form � � � � �an �bn �cn−1 �dn−1 � � � � = � � � � � � � � � � − λ µ 1 (n+1)(n−1) 2λ µ n (n+1)2(n−1)2 − λ+µ µ 1 (n+1)(n−1)2 λ+µ µ 3n+1 (n+1)2(n−1)3 0 − λ µ 1 (n+1)(n−1) 0 − λ+µ µ 1 (n+1)(n−1)2 λ+µ µ 1 n−1 − λ+µ µ 1 (n−1)2 λ+2µ µ 1 (n−1)2 − 2(λ+2µ) µ 1 (n−1)3 0 λ+µ µ 1 n−1 0 λ+2µ µ 1 (n−1)2 � � � � � � � � � � � � � � �an−2 �bn−2 �cn−3 �dn−3 � � � � (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='53) for any n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' We are interested in the case n odd since when n is even, thanks to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='52), we know that �an = �bn = �cn−1 = �dn−1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Looking at (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='53), for any n ≥ 3 odd, we introduce the matrices Sn := � � − λ µ 1 (n+1)(n−1) − λ+µ µ 1 (n+1)(n−1)2 λ+µ µ 1 n−1 λ+2µ µ 1 (n−1)2 � � , Tn := � � 2λ µ n (n+1)2(n−1)2 λ+µ µ 3n+1 (n+1)2(n−1)3 − λ+µ µ 1 (n−1)2 − 2(λ+2µ) µ 1 (n−1)3 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' In this way, system (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='53) may written in the form � �bn �dn−1 � = Sn ��bn−2 �dn−3 � , � �an �cn−1 � = Sn � �an−2 �cn−3 � − Tn ��bn−2 �dn−3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' 30 After an iterative procedure we may write � � � � � � � � � � � � � � � � �bn �dn−1 � = � � (n−3)/2 � m=0 Sn−2m � � ��b1 �d0 � , � �an �cn−1 � = � � (n−3)/2 � m=0 Sn−2m � � � �a1 �c0 � − (n−3)/2 � j=0 � � � j� m=1 Sn−2m+2 � Tn−2j � � (n−3)/2 � m=j+1 Sn−2m � � ��b1 �d0 �� � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='54) for any n ≥ 3 odd, with the convention that for any sequence of matrices Am ∈ R2×2 m2 � m=m1 Am = �1 0 0 1 � and m2 � m=m1 Am = �0 0 0 0 � whenever m1 > m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' By induction one can verify that for any j ≤ n−3 2 j� m=0 Sn−2m = � � � − (j+1)λ+jµ µ(n+1)[n+1−2(j+1)] �j m=1(n+1−2m)2 − (j+1)(λ+µ) µ(n+1) �j+1 m=1(n+1−2m)2 (j+1)(λ+µ) µ[n+1−2(j+1)] �j m=1(n+1−2m)2 (j+1)λ+(j+2)µ µ �j+1 m=1(n+1−2m)2 � � � and, in turn, by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='2) we infer ����� j� m=0 Sn−2m ����� ∞ ≤ (λ + µ)(n − 2j)(j + 2) µ �j+1 m=1(n + 1 − 2m)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='55) In particular, with appropriate choices of the minimum and the maximum values of the index in the product (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='55) and with appropriate changes of index, for any n ≥ 3 odd, we obtain the estimates ������ (n−3)/2 � m=0 Sn−2m ������ ∞ ≤ 3(λ + µ)(n + 1) µ2n �� n−1 2 � !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' �2 , ����� j� m=1 Sn−2m+2 ����� ∞ ≤ (λ + µ)(n − 2j + 2)(j + 1) µ �j m=1(n + 1 − 2m)2 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='56) ������ (n−3)/2 � m=j+1 Sn−2m ������ ∞ ≤ 3(λ + µ)(n − 2j − 1) 2µ �(n−1)/2 m=j+2 (n + 1 − 2m)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' On the other hand, we observe that for the components of the matrices Sn and Tn the following inequalities hold true: |(Tn)ij| ≤ 3 n−1 |(Sn)ij| for any i, j ∈ {1, 2} and n ≥ 3, which, in turn, implies ∥Tn∥∞ ≤ 3 n−1 ∥Sn∥∞ = 3(λ + µ) 2n µ(n−1)3 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' the last inequality is obtained by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='55) with j = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' 31 Therefore, combining (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='55) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='56), for any n ≥ 3 odd, we obtain ������ (n−3)/2 � j=0 � � � j� m=1 Sn−2m+2 � Tn−2j � � (n−3)/2 � m=j+1 Sn−2m � � � � ������ ∞ (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='57) ≤ (n−3)/2 � j=0 ����� j� m=1 Sn−2m+2 ����� ∞ ∥Tn−2j∥∞ ������ (n−3)/2 � m=j+1 Sn−2m ������ ∞ ≤ (n−3)/2 � j=0 18(λ + µ)3(n − 2j + 2)(n − 2j)(j + 1) µ3 2n �� n−1 2 � !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' �2 ≤ 9(λ + µ)3 n(n + 2)(n2 − 1) 4µ3 2n �� n−1 2 � !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' �2 , where in the last inequality we used the estimate (n − 2j + 2)(n − 2j) ≤ n(n + 2) and the identity �(n−3)/2 j=0 (j + 1) = n2−1 8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Combining (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='1) with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='54), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='56) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='57), for any n ≥ 3 odd, we obtain ���� � �an �cn−1 ����� ∞ ≤ 3(λ + µ)(n + 1) µ2n �� n−1 2 � !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' �2 ���� ��a1 �c0 ����� ∞ + 9(λ + µ)3 n(n + 2)(n2 − 1) 4µ3 2n �� n−1 2 � !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' �2 ����� � �b1 �d0 ������ ∞ (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='58) ≤ 3(2λ + 5µ)(λ + µ)2 (n + 1)(3n3 + 3n2 − 6n + 4) 4µ3 2n �� n−1 2 � !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' �2 max{�a−1, �a1,�b1, �c0} and ����� � �bn �dn−1 ������ ∞ ≤ 3(λ + µ)(n + 1) µ2n �� n−1 2 � !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' �2 ����� � �b1 �d0 ������ ∞ ≤ 3(2λ + 5µ)(n + 1) µ2n �� n−1 2 � !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' �2 max{�a−1, �a1,�b1, �c0} (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='59) where we exploited the fact that �d0 = �α �β �a−1 − 2 �β �b1, accordingly with what already explained in the lines below (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='52), so that |�d0| ≤ �α + 2 �β max{�a−1,�b1} = 2λ + 5µ λ + µ max{�a−1,�b1} , from which it follows that max ����� ��a1 �c0 ����� ∞ , ����� � �b1 �d0 ������ ∞ � ≤ 2λ + 5µ λ + µ max{�a−1, �a1,�b1, �c0} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Since we are interested to the restrictions of the functions Y and Z to the interval [a, b], we have to evaluate the series expansion (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='50) of the functions �Y and �Z for t ∈ � πk h a, πk h b � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Let N odd be the number at which we want to truncate the series expansions in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Recalling that the coefficients �an,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='�bn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' �cn−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' �dn−1 vanish for n even,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' we may write � � � � � � � � � � � � � � � �Y (t) = � N � n=−1 �an tn + (ln t) N � n=0 �bn tn � + � +∞ � n=N+2 �an tn + (ln t) +∞ � n=N+2 �bn tn � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' �Z(t) = � N−1 � n=0 �cn tn + (ln t) N−1 � n=0 �dn tn � + � +∞ � n=N+1 �cn tn + (ln t) +∞ � n=N+1 �dn tn � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' and define the truncation error as Ek,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='N = max � max t∈[ πk h a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' πk h b] ����� +∞ � n=N+2 �an tn + (ln t) +∞ � n=N+2 �bn tn ����� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' max t∈[ πk h a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' πk h b] ����� +∞ � n=N+1 �cn tn + (ln t) +∞ � n=N+1 �dn tn ����� � 32 By (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='58) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='59), we see that for any t ∈ � πk h a, πk h b � we have 0 ≤ Ek,N ≤ �C(a, b, k) � +∞ � n=N+2 �πkb h �n ����� � �an �cn−1 ������ ∞ + +∞ � n=N+2 �πkb h �n ����� � �bn �dn−1 ������ ∞ � ≤ �C(a, b, k) max{�a−1, �a1,�b1, �c0} +∞ � n=N+2 n odd �πkb h �n 3(2λ + 5µ)(λ + µ)2 (n + 1)(3n3 + 3n2 − 6n + 8) 4µ3 2n �� n−1 2 � !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' �2 where we put �C(a, b, k) = max � 1, h πkb � max � 1, | ln � πka h � |, | ln � πkb h � | � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Since we are interested to truncation of the series expansion with a sufficiently large number of terms, letting P(n) := (n + 1)(3n3 + 3n2 − 6n + 8), it is not restrictive to assume N ≥ 3 in such a way that the sequence n �→ 2−nP(n) becomes decreasing for n ≥ N + 2 ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' In this way, for N ≥ 3 odd, we obtain for all t ∈ � πk h a, πk h b � 0 ≤ Ek,N ≤ �C(a, b, k) max{�a−1, �a1,�b1, �c0}3(2λ + 5µ)(λ + µ)2 P(N + 2) 4µ3 2N+2 � N+1 2 � !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' +∞ � m= N+1 2 � πkb h �2m+1 m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='60) ≤ �C(a, b, k) max{�a−1, �a1,�b1, �c0} �πkb h �N+2 e( πkb h ) 2 3(2λ + 5µ)(λ + µ)2 P(N + 2) 16µ3 2N �� N+1 2 � !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' �2 , where in the last estimate we used the Lagrange form of the reminder in the Taylor formula for the exponential function and P(N + 2) = (N + 3)(3N3 + 21N2 + 42N + 32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' According to the rescaling introduced in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='42) one may define the functions �Υj, whose series expan- sions are given by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='50) with coefficients in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='51) and with a−1, a1, b1, c0 given by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='40) in the cases corresponding to j ∈ {1, 2, 3, 4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' In these four cases, the quantity max{�a−1, �a1,�b1, �c0}, appearing in the right hand side of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='60), admits the following estimates: � � � � � � � � � � � � � � � � � max{�a−1, �a1,�b1, �c0} = πk h max � 1, µ λ+µ ln � πk h �� if j = 1, max{�a−1, �a1,�b1, �c0} = h πk if j = 2, max{�a−1, �a1,�b1, �c0} = h πk max � 1, 2(λ+2µ) λ+µ ln � πk h �� if j = 3, max{�a−1, �a1,�b1, �c0} = 1 if j = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='61) For k > 1 is easy to see that all the maximum in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='61) are less or equal than max � πk h , µ λ+µ πk h ln � πk h � , 2(λ+2µ) λ+µ h πk ln � πk h � � , so that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='2) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' □ 6 Conclusions In this work we started from an applicative problem, suggested by Studio De Miranda Associati, an engineering company expertized in building long span bridges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' They proposed to study the blister, a structural element in bridges where the steel forestay anchors to the deck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' The aim is to obtain an explicit formula to estimate the tensions in the blister, useful for the practical design of bridges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' 33 The problem can be solved through the resolution of the elasticity equation with a specific geometry and load configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Hence, the first step was to define the geometry of the element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Through some simplifications we end up with a hollow circular cylinder axially loaded at the end faces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' the volume of the cylinder represents the portion of the deck concrete where the stresses diffusion happens, while the applied load is given by the force that the stay has to transfer to the deck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Clearly this geometry and load configuration can be refined in order to model a real blister, but this is a first step in this way and we leave more sophisticated models to future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' As matter of fact, from literature we learn that the elasticity equation was explicitly solved only for very particular domains and load conditions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' in prisms [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' In this paper we provide the explicit solution for the hollow cylinder axially loaded, proceeding by steps: first of all we provide a periodic extension of the load in z direction, so that we expand the solution in Fourier series with respect to the variable z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Then we compute the Fourier coefficients in x and y passing to cylindrical coordinates and expanding such functions in power series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' In Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='7 we write the explicit solution for the problem, written in series expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' We point out that this solution may have an own interest in the construction science field, beyond the application to the blister.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' To employ directly the formula in real situations, such as the blister design, it is necessary to consider approximated solutions, giving some estimates on the errors due to the truncating of the series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' In Section 4 we proposed a case of study, where, fixing the parameters involved in the problem, we are able to find the distribution of the stresses in the cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' These plots can be obtained through a simple code, written in MATLAB® or GNU Octave®, running in brief time, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' 1-3 minutes, depending on the number where we truncate the series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' From these results it is possible to find the maximum and the minimum of the different stresses acting on the cylinder, their position on the element and an estimate on the error due to the truncation of the series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Knowing these values, the engineering designer can choice for instance the most appropriate strand anchorage from the commercial catalogue, see Figure 5, in order to not exceed specific limit stresses in the reinforced concrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Since the map of the tensions is given, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Figure 6, the engineer can design the steel reinforcements in the concrete, at least on a pre-dimensioning level, and can check the concrete cracking stresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' As we explained, to get more precise results on realistic blisters we should modify the geometry of the element and the configuration of the loads;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' this may be a future work, but we point out that, more the geometry and the distribution of the loads are complex more the expectations to find explicit solutions are few, so that the finite element analysis may be preferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Notations We give some notations that will be used throughout this paper about functional spaces and differential operators acting on scalar functions, vector valued functions, matrix valued functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' We denote by Ω a general domain in RN, N ≥ 1 where by domain we mean a connected open set in RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Given two vectors x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' , xN), y = (y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' , yN) ∈ RN we denote by x · y = �N i=1 xiyi their Euclidean scalar product and by |x| = √x · x the Euclidean modulus of x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' the ∞-norm of vectors is |x|∞ := max 1≤i≤N |xi|;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' RM×N: space of M × N matrices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' if A ∈ RM×N and x ∈ RN is a vector, Ax denotes the usual product of matrices where x has to be seen as a vector column;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' letting A = (aij), B = (bij) ∈ RN×N we denote by A : B = �N i,j=1 aijbij their Euclidean scalar product and by |A| = √ A : A its Euclidean modulus;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' given A ∈ RM×N we denote by AT ∈ RN×M its transpose;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' 34 given A ∈ RN×N we introduce the operator ∞-norm of matrices by ∥A∥∞ := sup x∈RN\\{0} |Ax|∞ |x|∞ so that we have in particular |Ax|∞ ≤ ∥A∥∞ |x|∞ for any x ∈ RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='1) Letting A = (aij), ∈ RN, the following characterization of ∥ · ∥∞ holds: ∥A∥∞ = max 1≤i≤N N � j=1 |aij|;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='2) being ∥ · ∥∞ an operator norm, it is sub-multiplicative in the sense that ∥AB∥∞ ≤ ∥A∥∞ ∥B∥∞ for any A, B ∈ RN×N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' some well known functional spaces of functions defined from on an open set Ω ⊂ RN to a vector space V which could be RM or a space of matrices: Ck(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' V ), Lp(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' V ), Hk(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' V ) with 0 ≤ k ≤ ∞ integer and 1 ≤ p ≤ ∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' for 0 ≤ k ≤ ∞ integer, Ck(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' V ) denotes the space of restrictions to Ω of functions in Ck(RN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' V );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' D(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' V ): space of C∞(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' V ) with compact support in Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' D′(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' V ): space of vector distributions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' the dual space of D(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' V );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' given a scalar function g ∈ C1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' R), we denote by ∇g ∈ C0(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Rn) its gradient;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' given a vector valued function u ∈ C1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' RM), we denote by ∇u ∈ C0(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' RM×N) its Jacobian matrix;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' given a vector valued function u ∈ C1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' RN), Ω ⊆ RN, we denote by Du ∈ C0(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' RN×N) its symmetric gradient defined by Du = ∇u + ∇uT 2 (linearized strain tensor when N = 3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' given U ∈ C1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' RM×N), Ω ⊆ RN, we denote by div U ∈ C0(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' RM) the vector field v = (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' , vM) such that vi = �N j=1 ∂Uij ∂xj , i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' , M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' given u = (u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' , uM) ∈ C2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' RM), we denote by ∆u ∈ C0(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' RM) the Laplacian of u defined component by component, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' ∆u = (∆u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' , ∆uM) where in the last identity ∆ denotes the usual Laplacian of a real valued function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Acknowledgments The two authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilit`a e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' The second author acknowledges partial financial support from the PRIN project 2017 “Direct and inverse problems for partial differential equations: theoretical aspects and applications”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' The authors acknowledge partial financial support from the INdAM - GNAMPA project 2022 “Modelli del 4° ordine per la dinamica di strutture ingegneristiche: aspetti analitici e applicazioni”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' The second author acknowledges partial financial support from the research project “Metodi e modelli per la matematica e le sue applicazioni alle scienze, alla tecnologia e alla formazione” Progetto di Ateneo 2019 of the University of Piemonte Orientale “Amedeo Avogadro”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' 35 References [1] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Berchio, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Falocchi, About symmetry in partially hinged composite plates, Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' 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545 (1966).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} +page_content=' 36' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE1T4oBgHgl3EQfgQQH/content/2301.03226v1.pdf'} diff --git a/kb_36/content/tmp_files/kb_36.pdf.txt b/kb_36/content/tmp_files/kb_36.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..85fad03adaab2cbbf8f6b2a85100004e8110d445 --- /dev/null +++ b/kb_36/content/tmp_files/kb_36.pdf.txt @@ -0,0 +1,1769 @@ +RESEARCH +Open Access +A metagenomics roadmap to the +uncultured genome diversity in hypersaline +soda lake sediments +Charlotte D. Vavourakis1 +, Adrian-Stefan Andrei2†, Maliheh Mehrshad2†, Rohit Ghai2, Dimitry Y. Sorokin3,4 +and Gerard Muyzer1* +Abstract +Background: Hypersaline soda lakes are characterized by extreme high soluble carbonate alkalinity. Despite the +high pH and salt content, highly diverse microbial communities are known to be present in soda lake brines but +the microbiome of soda lake sediments received much less attention of microbiologists. Here, we performed metagenomic +sequencing on soda lake sediments to give the first extensive overview of the taxonomic diversity found in these complex, +extreme environments and to gain novel physiological insights into the most abundant, uncultured prokaryote lineages. +Results: We sequenced five metagenomes obtained from four surface sediments of Siberian soda lakes with a pH 10 and +a salt content between 70 and 400 g L−1. The recovered 16S rRNA gene sequences were mostly from Bacteria, even in +the salt-saturated lakes. Most OTUs were assigned to uncultured families. We reconstructed 871 metagenome-assembled +genomes (MAGs) spanning more than 45 phyla and discovered the first extremophilic members of the Candidate Phyla +Radiation (CPR). Five new species of CPR were among the most dominant community members. Novel dominant +lineages were found within previously well-characterized functional groups involved in carbon, sulfur, and nitrogen +cycling. Moreover, key enzymes of the Wood-Ljungdahl pathway were encoded within at least four bacterial phyla +never previously associated with this ancient anaerobic pathway for carbon fixation and dissimilation, including the +Actinobacteria. +Conclusions: Our first sequencing effort of hypersaline soda lake sediment metagenomes led to two important +advances. First, we showed the existence and obtained the first genomes of haloalkaliphilic members of the CPR +and several hundred other novel prokaryote lineages. The soda lake CPR is a functionally diverse group, but the most +abundant organisms in this study are likely fermenters with a possible role in primary carbon degradation. Second, +we found evidence for the presence of the Wood-Ljungdahl pathway in many more taxonomic groups than those +encompassing known homo-acetogens, sulfate-reducers, and methanogens. Since only few environmental metagenomics +studies have targeted sediment microbial communities and never to this extent, we expect that our findings are relevant +not only for the understanding of haloalkaline environments but can also be used to set targets for future studies on marine +and freshwater sediments. +Keywords: Soda lake sediments, Metagenomics, Haloalkaliphilic extremophiles, Candidate Phyla Radiation, Wood-Ljungdahl +pathway +* Correspondence: G.Muijzer@uva.nl +†Adrian-Stefan Andrei and Maliheh Mehrshad contributed equally to this +work. +1Microbial Systems Ecology, Department of Freshwater and Marine Ecology, +Institute for Biodiversity and Ecosystem Dynamics, Faculty of Science, +University of Amsterdam, Postbus 94248, 1090, GE, Amsterdam, the +Netherlands +Full list of author information is available at the end of the article +© The Author(s). 2018 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 +International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and +reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to +the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver +(http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated. +Vavourakis et al. Microbiome (2018) 6:168 +https://doi.org/10.1186/s40168-018-0548-7 + +MicrobiomeBackground +Soda lakes are evaporative, athallasic salt lakes with low cal- +cium and magnesium concentrations and a high-alkaline +pH up to 11 buffered by dissolved (bi-) carbonate ions [1]. +They are constrained to arid regions across the globe, +mainly the tropical East African Rift Valley [2], the Libyan +Desert [3], the deserts in California and Nevada [4], and the +dry steppe belt of Central Asia that spans to southern Si- +beria, north-eastern Mongolia, and Inner Mongolia in +China [1]. On top of the extreme salinity and alkaline pH, +the Eurasian soda lakes experience extreme seasonal +temperature differences, causing highly unstable water re- +gimes and fluctuating salinities [5]. Yet, soda lakes harbor +diverse communities of haloalkaliphilic microbes, mostly +prokaryotes that are well adapted to survive and grow in +these extreme environments and consist of similar func- +tional groups in soda lakes around the world [1, 2, 6]. The +relative abundance of different groups is typically governed +by the salinity of the brine [1, 7, 8], and microbial-mediated +nutrient +cycles +become +partially +hampered +only +at +salt-saturating conditions [1]. +So far, all characterized prokaryotic lineages cultured +from soda lakes comprise over 70 different species within +more than 30 genera [1, 6, 9, 10]. From these, only a lim- +ited number of genomes have been sequenced today, +mostly from chemolithoautotrophic sulfur-oxidizing bac- +teria belonging to the genus Thioalkalivibrio (class Gam- +maproteobacteria) [1, 11, 12]. It is well established that +metagenomics enables the recovery of genomes and the +identification of novel genetic diversity where culturing ef- +forts fail [13, 14]. In recent years, next-generation sequen- +cing has recovered a massive number of genomes from +previously unknown groups of prokaryotes [15, 16], +including a strikingly large and diverse group called +“Candidate Phyla Radiation” (CPR), only distantly related +to other cultured bacterial lineages [17]. Previously, we +conducted a metagenomics study on soda lakes and re- +constructed novel genomes from uncultured Bacteroidetes +and “Candidatus Nanohaloarchaeaota” living in hypersa- +line Siberian soda brines [7]. Here, we turned our atten- +tion to the far more complex prokaryotic communities +living in the sediments of the hypersaline soda lakes from +the same region. We give a broad overview of all the +taxonomic groups sequenced and focus on the metabolic +diversity found in the reconstructed genomes of the most +abundant, uncultured organisms. +Results +Overall prokaryote community structure +The salinities from the studied soda lakes ranged from +moderately hypersaline (between 70 and 110 g L−1) to +salt-saturated (400 g L−1 salt). The soluble carbonate al- +kalinity was in the molar range, and the pH in all lakes +was around ten (see Additional file 1: Table S1). To give +an overview of the overall prokaryotic community com- +position in each of the samples, we looked at the taxo- +nomic classification of 16S rRNA genes recovered both +by amplicon sequencing and direct metagenomics se- +quencing (Fig. 1, see also Additional file 2: Figure S1; +Additional file 3). The prokaryotic communities of all +five sediment samples were highly diverse and consisted +mostly of uncultured taxonomic groups. Bacteria were +more abundant than Archaea, regardless of the salinity +of the overlaying brine [7] (Fig. 1). Euryarchaeota were +the second and third largest group in the sediments of +the two salt-saturated lakes comprising ~ 10 and ~ 20% +of the 16S rRNA genes in the metagenomes. Most +Euryarchaeota-related OTUs detected by amplicon se- +quencing belonged either to the uncultured Thermoplas- +mata group KTK 4A (SILVA classification) or the genera +Halohasta and Halorubrum (class Halobacteria). In ac- +cordance with cultivation-dependent studies [6], most +OTUs assigned to methanogens were from the class +Methanomicrobia, +especially +the +lithotrophic +genus +Methanocalculus (up to ~ 3%) and the methylotrophic +genus Methanosalsum (Additional file 3). +The varying ratio of the three dominant bacterial groups, +Firmicutes, Bacteroidetes (including the newly proposed +phyla +Rhodothermaeota +and +Balneolaeota +[18]), +and +Gammaproteobacteria, showed no clear trend in relation to +the salinity in the lakes, but when Firmicutes were domin- +ant, Bacteroidetes were less abundant and vice versa. Most +Firmicutes belonged to the order Clostridales. Uncultured +members from the family Syntrophomonadaceae had a +relative abundance of more than 5% in all five metagen- +omes and comprised in two lakes even ~ 11–20% of the +recovered amplicon sequences. The second most abundant +Firmicutes order was Halanaerobiales, particularly the +genus Halanaerobium (family Halanaerobiaceae) and un- +cultured members of the Halobacteroidaceae. The majority +of Bacteroidetes-related OTUs could not be assigned down +to the genus level. The uncultured ML635J-40 aquatic +group (order Bacteroidales) comprised at least 5% of all five +prokaryotic communities. This group has been previously +found to be abundant in Mono Lake [4] (a soda lake) and +in an anoxic bioreactor degrading cyanobacterial biomass +under haloalkaline conditions [19]. Two other highly abun- +dant (up to ~ 8%) uncultured groups from the class Balneo- +lia (proposed new phylum Balneolaeota [18]) were also +detected in other soda lakes before [3, 4]. Within the Gam- +maproteobacteria, the genus Thioalkalivibrio was abundant +(~ 3% of the total community), but also uncultured +members of HOC36 were prevailing at moderate salinities. +Members of the Deltaproteobacteria, Alphaproteobacteria, +and Chloroflexi comprised up to ~ 10% of the detected 16S +rRNA gene in some of the metagenomes. The GIF9 family +of the class Dehalococcoidia was among the top three most +abundant OTUs in two lakes. The extremely salt-tolerant +Vavourakis et al. Microbiome (2018) 6:168 +Page 2 of 18 + +and alkaliphilic genera Desulfonatronobacter (order Desulfo- +bacterales) and Desulfonatronospira (order Desulfovibrio- +nales) +were +the +dominant +Deltaproteobacteria. +Highly +abundant OTUs, within the Actinobacteria belonged to the +class Nitriliruptoria and within the Alphaproteobacteria to +the family Rhodobacteraceae and the genus Roseibaca. The +important nitrifying genus Nitrobacter (Alphaproteobacteria) +was present in only one of the lakes with moderate salinity +(Additional file 3). +Some bacterial top-level taxa appeared less dominant +(< 5%) from the 16S rRNA genes recovered from the +metagenomes but were represented mainly by a single +highly abundant OTU in the amplicon sequences, in- +cluding the haloalkaliphilic genus Truepera within the +phylum Deinococcus-Thermus, the genus Spirochaeata +within the phylum Spirochaetes, the family BSN166 +within the phylum Ignavibacteriae, the BD2–11 terres- +trial group within the Gemmatimonadetes, and the +WCHB1–41 +order +within +the +Verrucomicrobia. +All +OTUs +within +the +Thermotogae +and +Lentisphaerae +belonged to uncultured genera from the family Kosmoto- +gaceae and Oligosphaeraceae, respectively. All Tenericu- +tes-related OTUs belonged to the class Mollicutes, and +especially the order NB1-n was dominant. In contrast, +the phylum Planctomycetes was relatively diverse, with +at least 11 different genus-level OTUs spread over four +class-level groups. +High-throughput genome recovery +We obtained 717 medium-quality (≥ 50% complete, +< 10% contamination) and 154 near-complete (≥ 90% +complete, < 5% contamination) metagenome-assembled +genomes (MAGs) across three major prokaryote groups: +Archaea, Bacteria, and CPR (see Additional file 4 and +Additional file 2: Figure S2). Figures 2 and 3 show the +top-level phylogeny of all MAGs based on 16 ribosomal +proteins. The reference database used contains a repre- +sentative for each major prokaryote lineage [17]. We +a +b +Fig. 1 Abundant prokaryotic groups in five hypersaline soda lake sediments. a Relative abundance of the top-level taxa (those with ≥ 1% abundance +in at least one dataset) based on 16S rRNA reads in unassembled metagenomic datasets. b Relative abundance of the 16S rRNA OTUs (those with sum +of abundance in all datasets ≥ 3%) recovered by amplicon sequencing assigned where possible down to the genus-level. Three of the assessed soda +lakes have a moderate salinity (70–110 g L−1), two are salt-saturated (400 g L− 1) +Vavourakis et al. Microbiome (2018) 6:168 +Page 3 of 18 + +colored the different phyla from which we obtained a +MAG +in +alternate +blue +and +orange +colors, +and +highlighted the MAGs obtained here in a darker shade. +Many MAGs belonged to uncultured groups commonly +detected in soda lake 16S rRNA gene surveys, over 100 +MAGs still belonged to candidate prokaryote phyla and +divisions that to our knowledge were never detected be- +fore in soda lakes, including CPR. Although only few +MAGs had near-complete 16S rRNA genes, in most +cases we were able to link available taxonomic gene an- +notations and ribosomal protein phylogeny to the SILVA +taxonomy of the OTUs assigned to the amplicon se- +quences, while cross-checking the abundance profiles of +both MAGs (Additional file 5) and OTUs. +The soda lake CPR recovered from the metagenomes was +restricted to a few distinct phyla within the Parcubacteria +group, mostly affiliating with “Candidatus Nealsonbacteria” +and “Ca. Zambryskibacteria” [15] (Fig. 2). The first group of +MAGs encompassed four different branches in our riboso- +mal protein tree, suggesting a high-phylogenetic diversity, +with 33 putative new species sampled here (ANI and con- +DNA matrices given in Additional file 6). The “Ca. Zambrys- +kibacteria-”related MAGs consisted of at least five new +species. Few MAGs were recovered from CPR groups also +detected by amplicon sequencing (see Additional file 2: +Figure S1), namely the “Ca. Dojkabacteria” (former WS6), +“Ca. Saccharibacteria” (former TM7), CPR2, and “Ca. +Katanobacteria” (former WWE3). +Fig. 2 Maximum-likelihood phylogeny of the CPR and archaeal MAGs based on 16 ribosomal proteins. The archaeal tree is unrooted. The CPR tree is rooted +to the Wirthbacteria. Alternate orange and blue colors show phyla/classes from which we obtained MAGs (labeled as “Phyla present”). Reconstructed MAGs of +this study are highlighted by darker shades (labeled as “MAG present”). Phyla/classes for which there was no representative in the reconstructed MAGs of this +study are shown as gray cartoons (labeled as “Phyla not present”), and the numerical labels are represented at the bottom. Colored circles at the nodes show +confidence percentage of the bootstraps analysis (100×) +Vavourakis et al. Microbiome (2018) 6:168 +Page 4 of 18 + +Most archaeal MAGs belonged to the phylum Euryarch- +aeota and the abundant classes Halobacteria, Methanomi- +crobia, and Thermoplasmata (related to OTU KTK 4A) +within. In addition, three Thermoplasmata-related MAGs +that encoded for the key enzyme for methanogenesis +(methyl-coenzyme M reductase, mcr) affiliated with refer- +ence genomes from Methanomassilicoccales, the seventh +order of methanogens have been recovered [20, 21]. +Another MCR-encoding MAG was closely related to the +latest +discovered +group +of +poly-extremophilic, +methyl-reducing methanogens from hypersaline lakes +from the class Methanonatronarchaeia [9] (related to +OTU ST-12K10A). We recovered also one MAG from the +class Methanobacteria and a high-quality MAG from the +WCHA1–57 +group +(“Candidatus +Methanofastidiosa” +[22]) in the candidate division WSA2 (Arc I). Several +MAGs were recovered from the DPANN archaeal +groups “Ca. Diapherotrites,” “Ca. Aenigmarchaeota,” +(see Additional file 2: Figure S3) and “Ca. Woesearch- +aeota” (former Deep Sea Hydrothermal Vent Group 6, +DHVEG-6). Although we did not reconstruct any +reasonable-sized MAGs from the TACK superphylum, +we found several 16S rRNA genes on the assembled +contigs that affiliated to the Thaumarchaeota (see +Additional file 1: Table S2). +Nearly every known bacterial phylum had an extremo- +philic lineage sampled from our hypersaline soda lake +sediments (Fig. 3). In most cases, the soda lake lineages +clearly formed separate branches appearing as sister +groups to known reference lineages. The highest genome +recovery was from the same top-level taxonomic groups +that were also abundant in our 16S rRNA gene analysis. +From the Verrucomicrobia, most MAGs belonged to the +order WCHB1-41 (16S rRNA gene identity 92–100%). +However, in our ribosomal protein tree, they branched +within the phylum Lentisphaerae. Sixteen Tenericutes +MAGs from at least 12 different species (Additional file 6) +were closely related to the NB1-n group of Mollicutes. +Based on the recovered genome size and encoded meta- +bolic potential, these organisms are free-living anaerobic +fermenters of simple sugars, similar to what has recently +been +proposed +for +“Candidatus +Izimaplasma” +[23]. +Fig. 3 Maximum-likelihood phylogeny of the bacterial MAGs (CPR excluded) based on 16 ribosomal proteins. Alternate orange and blue colors show phyla/ +classes from which we obtained MAGs (labeled as “Phyla present”). Reconstructed MAGs of this study are highlighted by darker shades (labeled as “MAG +present”). Phyla/classes for which there was no representative in the reconstructed MAGs of this study are shown as gray cartoons (labeled as “Phyla not +present”), and the numerical labels are represented at the bottom. Colored circles at the nodes show confidence percentage of the bootstraps analysis (100×) +Vavourakis et al. Microbiome (2018) 6:168 +Page 5 of 18 + +Several MAGs belonged to the candidate phyla “Ca. +Omnitrophica,” “Ca. Atribacteria,” and “Ca. Acetother- +mia” (former OP1), which were moderately abundant +also in some sediment (see Additional file 2: Figure S1). +For the latter phylum, we suspect that four MAGs were +more closely related to ca. div. WS1 and “Ca. Lindow- +bacteria” for which only few reference genomes are +currently available in NCBI (see Additional file 2: +Figure S4). Due to a high-sequencing coverage, we also +managed to reconstruct several MAGs from rare Bacteria +(< 100 amplicon sequences detected, see Additional file 2: +Figure S1), including the phyla “Ca. Hydrogenedentes,” +“Ca. Cloacimonetes,” ca. div. BRC1, Elusimicrobia, Caldi- +serica, and “Ca. Latescibacteria.” The MAGs from the +latter phylum were more closely related to the recently +proposed phylum “Ca. Handelsmanbacteria” [15]. Two +additional MAGs with 16S rRNA gene fragments with +94–95% identity to the class MD2898-B26 (Nitrospinae) +were more likely members of ca. div. KSB3 (proposed +“Ca. Moduliflexus” [24], see Additional file 2: Figure S5). +Draft genomes of haloalkaliphilic CPR +Strikingly, members of the CPR related to “Ca. Nealson- +bacteria” and “Ca. Vogelbacteria” were among the top +5% of abundant organisms in the surface sediments of +the soda lakes, especially those with moderate salinity +(Fig. 4). Like most members of the CPR, the MAGs of +the four most abundant “Ca. Nealsonbacteria” seem to +be anaerobic fermenters [25]. They lacked a complete +TCA cycle and most complexes from the oxidative elec- +tron transfer chain, except for the subunit F of a +NADH-quinone oxidoreductase (complex I, nuoF, nuoG, +nuoA) and coxB genes (complex II). All CPR MAGs had +a near-complete glycolysis pathway (Embden-Meyerhof- +Parnas) encoded, but pentose phosphate pathways were +severely truncated. The commonly encoded F- and +V-type ATPase can establish a membrane potential for +symporter-antiporters by utilizing the ATP formed by +substrate-level phosphorylation during fermentation. All +CPR have V-type ATPases that can translocate Na+ in +addition to H+ (see Additional file 2: Figure S6), while +only two members of the “Ca. Falkowbacteria” had puta- +tive Na+-coupled F-type ATPases (see Additional file 2: +Figure S7). The coupling of ATP hydrolysis to sodium +translocation is advantageous to maintain pH homeosta- +sis in alkaline environments. Interestingly, with only two +exceptions [26, 27], all CPR genomes recovered from +other environments with neutral pH were reported to +encode only F-type ATPases [28–32]. One low-abundant +MAG affiliated to “Ca. Peregrinibacteria” contained both +the +large +subunit +of +a +RuBisCO +(type +II/III, +see +Additional file 2: Figure S8) and a putative phosphoribu- +lokinase (PRK, K00855) encoded in the same contig. +This is remarkable because PRK homologs were not +previously identified among CPR, and RuBisCo form II/ +III was inferred to function in a nucleoside salvage path- +way [33]. One “Ca. Saccharibacteria” MAG encoded for +a putative channelrhodopsin (see Additional file 2: +Figure S9). This is the first rhodopsin found among the +CPR and suggests that this enigmatic group of organ- +isms may have acquired evolutionary adaptations to a +life in sunlit surface environments. +A previous study showed that most CPR has coccoid +cell morphotypes with a monoderm cell envelope resem- +bling those from Gram-positives and Archaea but with a +distinct S-layer [34]. Thick peptidoglycans coated with +acidic surface polymers such as teichoic acids help pro- +tect the cells of Gram-positives against reactive hydroxyl +ions in highly alkaline environments [35] (Fig. 5a). All +soda lake CPR had indeed the capability for peptidogly- +can biosynthesis, but we found proteins typical for +Gram-negatives for the biosynthesis of lipopolysaccha- +rides (see Additional file 1: Table S3), homologous to the +inner membrane proteins of type II secretion systems +and +to +several +proteins +associated +to +the +outer +membrane and peptidoglycan, including OmpA. +It remains to be determined whether the soda lake +CPR also lacks an outer membrane and perhaps anchor +lipopolysaccharides, S-layer proteins, and lipoproteins to +the inner cell membrane or peptidoglycan. We also +found gene encoding cardiolipin and squalene synthases. +Increased levels of cardiolipin and the presence of squa- +lene make the cytoplasmic membrane less leaky for +protons [36]. In addition, cation/proton exchangers are +known to play a crucial role for pH homeostasis in alka- +liphilic prokaryotes as they help acidify the cytoplasm +during the extrusion of cations [35]. Putative Na+/H+ +exchangers of the Nha-type and multi-subunit Mnh-type +were found only within a few soda lake CPR. Secondary +active transport of K+ might be mediated in most soda +lake CPR by KefB (COG0475)/kch Kef-type, glutathione- +dependent K+ transport systems, with or without H+ +antiport (67,68). +Various soda lake CPR had an acidic proteome, with +pI curves resembling those found in extremely halophilic +Bacteria. Intracellular proteins enriched in acidic amino +acids might be an adaptation to a “salt-in” strategy, i.e., +maintaining high intracellular potassium (K+) concentra- +tions to keep osmotic balance [7, 37] (Fig. 5b; see +Additional file 2: Figure S10). Such a strategy is energet- +ically favorable over de novo synthesis or import of +osmolytes such as ectoine and glycine betaine. We did +not find genes for the synthesis of organic osmolytes and +homologs of ABC-type transporters for primary active +uptake of proline/glycine betaine which were encoded +only in one MAG (Fig. 5a). For the “Ca. Nealsonbac- +teria” and “Ca. Vogelbacteria,” the salt-in strategy might +be a unique feature for the soda lake species explaining +Vavourakis et al. Microbiome (2018) 6:168 +Page 6 of 18 + +their high abundance in the hypersaline soda lake sedi- +ments, as we did not found an acidic proteome pre- +dicted from genomes obtained from other non-saline +environments (See Additional file 2: Figure S11). The +uptake of K+ ions remains enigmatic for most soda lake +CPR. Low-affinity Trk-type K+ uptake transporters (gen- +erally with symport of H+) (67,68) were encoded only by +a limited number of MAGs. We found three MAGs +Fig. 4 Relative abundance and metabolic potential of the dominant species. Abundance values, expressed as reads per kilobase of MAG per gigabase +of mapped reads (RPKG), were averaged for the top ten abundant MAGs from each dataset that were (likely) the same species (Additional file 5, +Additional file 6). Population genomes were ranked by their “salinity preference scores”: those recruiting relatively more from moderate salinity datasets +(cold colors) are drawn to the top, from high salinity datasets (warm colors) to the bottom. The metabolic potential derived from functional marker +genes (Additional file 7) is depicted by the colored symbols. CBB = Calvin-Benson-Bassham cycle, DNRA = dissimilatory nitrite reduction to ammonia, +fix. = fixation, red. = reduction, ox. = oxidation, dis. = disproportionation +Vavourakis et al. Microbiome (2018) 6:168 +Page 7 of 18 + +a +b +Fig. 5 (See legend on next page.) +Vavourakis et al. Microbiome (2018) 6:168 +Page 8 of 18 + +encoding for Kdp-type sensor kinases (kdpD) but no +corresponding genes for the response regulator (kdpE) +or for Kdp-ATPases that function as the inducible, high- +affinity K+ transporters in other Bacteria (67,68). Finally, +mechanosensitive ion channels (mscS, mscL) and ABC- +type multidrug transport systems (AcrAB, ccmA, EmrA, +MdlB, NorM) and sodium efflux permeases (NatB) were +encoded in almost every MAG. The first might rapidly +restore the turgor pressure under fluctuating salinity +conditions by releasing cytoplasmic ions [38]. +Novel abundant groups involved in sulfur, nitrogen, and +carbon cycles +A new species of Thioalkalivibrio (family Ectothiorhodospir- +aceae) was by far the most abundant in the sediments of +the two salt-saturated lakes (Fig. 4). In the sediment of +Bitter-1, also a purple sulfur bacterium from the same fam- +ily was highly abundant. It was closely related to Halorho- +dospira, a genus also frequently cultured from hypersaline +soda lakes [1]. None of the abundant Ectothiorhodospira- +ceae spp. had already a species-representative genome +sequenced (Additional file 6). The potential of the Thioalk- +alivibrio spp. for chemolithotrophic sulfur oxidation was +evident (Additional file 7; see Additional file 8: Information +S1). Interestingly, the encoded nitrogen metabolisms were +quite versatile. While Thioalkalivibrio sp. 1 had the poten- +tial for nitrate reduction to nitrite, Thioalkalivibrio sp. 2 +might perform dissimilatory nitrite reduction to ammonia +(DNRA; see Additional file 2: Figure S12). +Two +deltaproteobacterial +lineages +of +dissimilatory +sulfate-reducing bacteria (SRB) were highly abundant in +the soda lake sediment of Bitter-1. One MAG from the +family Desulfobacteraceae (order Desulfobacterales) is +the first genome from the genus Desulfonatronobacter. It +encodes the genes for complete sulfate reduction to sul- +fide using various electron donors, as well as for the +complete oxidation of volatile fatty acids and alcohols, a +unique +feature +for +the +genus +Desulfonatronobacter +among haloalkaliphilic SRB [10] (see Additional file 8: +Information S2). Fumarate and nitrite (DNRA, NrfAH) +could be used as alternative electron acceptors. The sec- +ond dominant lineage was a new species from the genus +Desulfonatronospira (family Desulfohalobiaceae, order +Desulfovibrionales). Like other members of this genus, it +had the potential to reduce or disproportionate partially +reduced sulfur compounds. In addition, it could also use +nitrite as an alternative electron acceptor (NrfAH) [6]. +A novel lineage of gammaproteobacterial SOB was +highly abundant in the sediments of the moderately hy- +persaline Cock Soda Lake. It appeared as a sister group of +the family Xanthomonadaceae in the ribosomal protein +tree. This heterotrophic organism could conserve energy +through aerobic respiration. It might detoxify sulfide by +oxidizing it to elemental sulfur (sqr) with subsequent re- +duction or disproportionation of the polysulfides (psrA) +chemically formed from the sulfur. It also encoded the po- +tential for DNRA (nrfA and napC). Genes likely involved +in sulfide detoxification (sqr and psrA) were found also in +several other abundant MAGs of heterotrophs, including +one new abundant species from the family of Nitrilirup- +toraceae (class Nitriliruptoria, phylum Actinobacteria). +We found a wide variety of carbohydrate-active enzymes +in these MAGs, such as cellulases (GH1 family) in +addition to genes for glycolysis and TCA cycle and a +chlorophyll/bacteriochlorophyll a/b synthase (bchG gene). +The latter was also found in other Actinobacteria from the +genus Rubrobacter [39]. No evidence was found for +nitrile-degrading potential. +A second novel, uncultured lineage of Gammaproteo- +bacteria that was highly abundant at moderate salinities +branched in our ribosomal protein tree as a sister group +to the family Halothiobacillaceae. The MAGs encoded +for a versatile metabolism typical for purple non-sulfur +bacteria. The MAGs contained puf genes, bch genes, +genes for carotenoid biosynthesis (not shown), and a +Calvin cycle for photoautotrophic growth. Alternatively, +energy may be conserved through aerobic respiration, +while acetate and proprionate could be taken up via an +acetate permease (actP) and further used for acetyl-CoA +biosynthesis and carbon assimilation. Since the sqr gene +was present, but no dsr or sox genes, the organism +might oxidize sulfide only to elemental sulfur. One bin +contained also nifDKH genes suggesting putative diazo- +trophy, as well as a coenzyme F420 hydrogenase suggest- +ing photoproduction of hydrogen [40]. +The abundant Euryarchaeota organism showed a clear +preference for higher salinities. We obtained one highly +abundant MAG from the class Thermoplasmata that +encoded a full-length 16S rRNA gene only distantly re- +lated (91,2% identity, e value 0) to that of a member of +the KTK 4A group found in a hypersaline endoevaporitic +microbial mat [8]. The abundant soda lake organism is +likely a new genus and species. All KTK 4A-related +MAGs found here encoded for similar heterotrophic, +fermentative +metabolisms, +with +the +potential +for +(See figure on previous page.) +Fig. 5 Potential mechanisms for regulating the intracellular pH and cytoplasmic ion content in different CPR phyla. a Membrane transporters, +channels, and lipids. Peptidoglycan is depicted in gray and S-layer proteins in cyan. b Predicted isoelectric points (bin width 0.2) for the coding +sequences of MAGs. A representative proteome is depicted for each phylum for which several members had a pronounced acidic peak (see also +Additional file 2: Figure S11) +Vavourakis et al. Microbiome (2018) 6:168 +Page 9 of 18 + +anaerobic formate and CO oxidation. The KTK 4A +might be also primary degraders since they encoded pu- +tative cellulases (CAZY-families GH1, GH5) and chiti- +nases (GH18). Interestingly, half of the MAGs encoded a +putative +chlorophyll/bacteriochlorophyll +a/b +synthase +(bchG), which is highly unusual for Archaea. Although +little can be inferred from the presence of only one +marker gene, a functional bchG was previously also +found in Crenarchaeota [41]. The remaining two highly +abundant Euryarchaeota-related MAGs belonged to a +new species of Halorubrum (Additional file 6). +Key genes of the Wood-Ljungdahl pathway found in +novel phylogenetic groups +More than 50 MAGs were related to the family Syntro- +phomonadaceae (class Clostridia, phylum Firmicutes) +based on ribosomal protein phylogeny. All 16S rRNA +gene sequences found in the MAGS had 86–95% iden- +tity to sequences obtained from uncultured organisms +related to the genus Dethiobacter. While an isolated +strain of Dethiobacter alkaliphilus is a facultative auto- +troph +that +respires +thiosulfate, +elemental +sulfur +or +polysulfides with hydrogen as an electron donor [42] or +disproportionates +sulfur +[43], +other +haloalkaliphilic +members +of +the +Syntrophomonadaceae +are +reverse +acetogens, oxidizing acetate in syntrophy with a hydro- +genotrophic partner [44]. Two populations (different +species, Additional file 6) were especially abundant in +Cock Soda Lake (Fig. 4). They encoded for a full +CODH/ACS complex, the key enzyme for the reductive +acetyl-CoA or Wood-Ljungdahl pathway (WL) and a +complete +Eastern +branch +for +CO2 +conversion +to +5-methyl-tetrahydrofolate (Additional file 9) [45, 46]. +Acetogens use the WL to reduce CO2 to acetyl-CoA, +which can be fixed into the cell or used to conserve en- +ergy via acetogenesis. Syntrophic acetate oxidizers, some +sulfate reducing bacteria and aceticlastic methanogens +run the WL in reverse. Syntrophomonadaceae sp. 2 +encoded for a putative thiosulfate/polysulfide reductase +as well as a phosphotransacetylase (pta) and an acetate +kinase (ack) for the ATP-dependent conversion of acet- +ate to acetyl-CoA. Although alternative pathways for the +latter interconversion can exist, this second species has +the complete potential for (reversed) acetogenesis. +Highly remarkable was the presence of a bacterial-type +CODH/ACS +complex +and +a +near-complete +eastern +branch of the WL in a highly abundant species in Cock +Soda Lake from the family Coriobacteriaceae (phylum +Actinobacteria). This prompted us to scan all 871 MAGs +for the presence of acsB encoding for the beta-subunit +of the oxido-reductase module of CODH/ACS. We con- +firmed an encoded +(near)-complete +WL in several +additional organisms belonging to phylogenetic groups +not +previously +associated +with +this +pathway +[46] +(Additional file 9). We removed the Coriobacteriaceae +acsB genes from the final dataset to construct a phylo- +genetic tree since they were < 500 aa (Fig. 6) but found +seven MAGs from the OPB41 class within the Actino- +bacteria (16S rRNA gene fragment identity 94–96%). +The eastern branch of WL can function independently +in folate-dependent C1 metabolism [45], but the pres- +ence of the Western-branch in a phylum that comprises +mostly aerobic isolates is very surprising. The WL in +combination with the potential for acetate to acetyl-CoA +interconversion (pta/ack) and a glycolysis pathway were +also present in the soda lake MAGs from the phyla “Ca. +Handelsmanbacteria,” “Ca. Atribacteria” (latter branched +within the “Ca. Acetothermia”), and the class LD1-PA32 +(Chlamydiae), suggesting all these uncultured organisms +might be heterotrophic acetogens. However, it should be +noted that a PFOR typically connecting glycolysis to the +WL was only encoded in the LD1-PA32 MAGs. More- +over, from the genetic make-up alone, it cannot be +excluded that acetate is activated, and the WL run in +reverse for syntrophic acetate oxidation. Finally, the +novel acsB genes from soda lake Halanaerobiaceae, +Natranaerobiaceae, and Halobacteroidaceae (Firmicutes) +and from Brocadiaceae and Planctomycetaceae (Plancto- +mycetes) disrupt the previously proposed dichotomy +between Terrabacteria and Gracilicutes bacterial groups +unifying 16S rRNA and acsB gene phylogenies [46] and +suggest a far more complex evolutionary history of the +WL pathway than previously anticipated. +Discussion +Extensive +classical +microbiology +efforts +have +been +already undertaken to explore the unique extremophilic +microbial communities inhabiting soda lakes. These un- +covered the presence of most of the functional groups +participating in carbon, nitrogen, sulfur, and minor +element cycling at haloalkaline conditions. The main re- +sults of this work are summarized in several recent re- +views [1, 6, 47, 48]. Since most microbes, including +those living in soda lakes, still evade all cultivation ef- +forts, a very effective way to discover new microbes and +assess their physiology based on their genetic repertoire +is either through single cell genomics or by directly se- +quenced environmental DNA. This exploratory metage- +nomics +study +performed +on +soda +lake +sediments +effectively overcame the existing cultivation bottleneck. +First, we expanded the known diversity of CPR consider- +ably with the first genomes of poly-extremophiles sam- +pled from soda lake sediments. Although the presence of +16S rRNA genes from CPR in marine sediments and hy- +persaline microbial mats was previously shown [34], +until now, CPR MAGs were mainly obtained from deep, +subsurface environments [15, 26, 29, 32, 49–52], and hu- +man microbiota [30]. Despite being highly abundant +Vavourakis et al. Microbiome (2018) 6:168 +Page 10 of 18 + +100 % +90-100 % +70-90 % +50-70 % +some MAGs +all MAGs +Bootstraps +Genes present +Glycolysis (EMP) +PFOR +WL-Eastern branch +H4MPT +TH4 +WL-Western branch +CODH/ACS +Acetogenesis/ +acetate activation +(pta/ack) +0.4 +PVC group (Chlamydiae LD1-PA32) +Syntrophorhabdus aromaticivorans +PVC group bacterium CSSed11_184 +Aerophobetes bacterium SCGC_AAA255-F10 +Ca. Acetothermia +Ca. Handelsmanbacteria +Planctomycetaceae +Anaerolineae +Firmicutes +Brocadiaceae +Planctomycetes +Methanomassiliicoccales +Halobacteroidaceae +Natranaerobiaceae +Methanomicrobiales +Desulfonatronospira +Firmicutes +Dehalococcoidia +Armatimonadetes bacterium CSP1-3 +Deltaproteobacteria +Thermodesulfobacteria +Desulfobulbaceae +Halanaerobiaceae +Nitrospirae +Actinobacteria (OPB41) +Fig. 6 Maximum likelihood phylogeny of the bacterial-type acetyl-coA synthases (acsB) found in the MAGs. Only sequences ≥ 500 aa +were included. Lineages for which we discovered the Wood-Ljungdahl (WL) in this study are highlighted in orange, and the presence +of genes in the respective MAGs related to WL, glycolysis, pyruvate, and acetate conversion is indicated by the colored symbols (see +also Additional file 9: Dataset S6). Additional lineages found in this study are marked in blue. The three was rooted according to [46]. +Circles at the nodes show confidence percentage of the bootstraps analysis (100×). EMP = Embden-Meyerhof-Parnas, PFOR = pyruvate:ferredoxin +oxidoreductase complex, pta = phosphotransacetylase gene, ack = acetate kinase gene, H4MPT = tetrahydromethanopterin-linked pathway, TH4 = +tetrahydrofolate pathway, CODH/ACS = carbon monoxide dehydrogenase/acetyl-CoA synthase. PVC group bacterium CSSed11_184 is likely a member +of the WCHB1-41 class within the Verrucomicrobia +Vavourakis et al. Microbiome (2018) 6:168 +Page 11 of 18 + +here, CPR went unnoticed in previous amplicon sequen- +cing studies. This might be due to the fact that many +CPR representatives have random inserts of various +length in their 16S rRNA genes or due to primer mis- +matches [29, 34]. This illustrates also that direct metage- +nomics should not only be preferred over amplicon +sequencing to infer functional potential, but the former +is far more effective for the discovery of novel organ- +isms. Second, we obtained many more genomes from +“traditional” bacterial phyla such as the Planctomycetes +and Chloroflexi, as well as candidate phyla, for which no +soda lake isolates, hence no genomes were previously +obtained. Third, even within the sulfur cycle, the most +active and frequently studied element cycle in soda lakes +[1], we found considerable metabolic novelty. Finally, we +found the Wood-Ljungdahl pathway in several novel +phyla, not closely related to any known acetogens, +methanogens, or sulfate-reducing bacteria [46]. The lat- +ter shows that our sequencing recovery effort has also +significantly contributed to the discovery of metabolic +novelty within various prokaryote phylogenetic groups. +Salinity is often considered to be the major factor +shaping prokaryote community composition in diverse +habitats [53, 54]. Extreme halophilic Euryarchaeota +seem to be always the dominant group in salt-saturated +hypersaline brines, both those with neutral or alkaline +pH [1, 7, 37]. Here, we found that although these +haloarchaea are still relatively more abundant in the sed- +iments exposed to brines with salt-saturating conditions, +the clear majority of microbes in all investigated hyper- +saline soda lake sediments are Bacteria. It could be +hypothesized that the sediment is a hide-out for the +extreme alkalinity and salinity governing the water +column, and that sediment stratification, especially in +the anoxic part, offers plenty of opportunities for niche +diversification. On the other hand, it should no longer +be a surprise that soda lakes are such productive and +biodiverse +systems. +First, +it +has +been +previously +elaborated that soda lake organisms are exposed to +approximately half the osmotic pressure in sodium +carbonate-dominated +brines +compared +to +sodium +chloride-dominated brines with the same Na+ molarity +[47]. Second, nitrogen limitation in the community can +be overcome when many members contribute to the +fixation of atmospheric N2, and various forms of organic +nitrogen are efficiently recycled. The soda lakes exam- +ined in this study were also eutrophic, and sulfur com- +pounds were abundant. Sulfide is also far less toxic at +high pH as it mostly occurs in the form of bisulfide +(HS−). Besides the evident high metabolic and taxo- +nomic diversity of dissimilatory sulfur-cycling bacteria, a +diverse heterotrophic community can be sustained com- +prising both generalist and very specialized carbon de- +graders. Less eutrophic soda lakes might not suffer from +carbon +limitation +either, +due +to +a +presence +of +high-bicarbonate concentrations. These effectively elim- +inate the inorganic carbon limitation for primary pro- +ducers who are highly active in soda lakes, especially +Cyanobacteria [55, 56]. Third, light that penetrates the +surface of the sediment seems to stimulate oxygenic and +anoxygenic phototrophic growth. Moreover, various het- +erotrophs, such as the rhodopsin-containing haloarchaea +and Bacteroidetes, have the option to tap into this un- +limited energy source for example to help sustain the +costly maintenance of osmotic balance. Unexpectedly, +we even found the first rhodopsin encoded by a member +of the CPR. Fourth, tight syntrophic relations, as pro- +posed for CPR members and Syntrophomonadaceae +spp., might be the solution to successful growth in an +energetically challenging environment. +Since our metagenomes are snapshots in time and space, +the failure to reconstruct specific MAGs gives no conclu- +sive evidence for the absence of certain microbial-mediated +element transformation in hypersaline soda lake sediments. +Additionally, technical limitations of the assembly and bin- +ning of highly micro-diverse genome populations might +hamper genome recovery [57]. More importantly, the +abundance of a specific microbe is not necessarily corre- +lated to the importance of its performance in an ecosystem. +Many metabolic capacities are redundant, and often key +transformations are reserved for a few rare organisms that +might proliferate for a short time-span when specific condi- +tions allow for it. For example, although no MAGs were re- +covered from chemolithoautotrophic nitrifiers [58], we did +detect a Nitrobacter-related OTU by amplicon sequencing +and assembled 16S rRNA genes from Thaumarchaeota, +suggesting bacterial and archaeal nitrifiers are present in +the surface sediments of soda lakes at very low abundance. +Finally, the method of DNA isolation might impact the +community composition apparent in the final metagenome +sequenced. Environmental samples contain complex mix- +tures of different organisms, and it is impossible to find a +protocol where the DNA from every single organism is ex- +tracted as efficiently without compromising the final quality +of the extracted DNA. However, since we find all the im- +portant taxonomic and functional groups known from pre- +vious cultivation-dependent studies back in either our +amplicon sequencing datasets or our directly sequenced +metagenomes, we are confident that the community com- +position and the MAGs presented here are representative +for the microbiomes of the soda lake sediments in the +Kulunda Steppe. +Conclusion +Years of intensive microbiological research on soda lakes +seem to have paid off, since many of the described gen- +era we could detect here have a cultured representative +from soda lakes. However, as many of the abundant +Vavourakis et al. Microbiome (2018) 6:168 +Page 12 of 18 + +lineages and groups found in soda lake sediments are +still uncultured, metagenomics proved to be a helpful +tool to gain primary insights in the potential physiology +and ecology of these poly-extremophilic prokaryotes. +We reconstructed the first genomes for many of such +organisms and proposed new functional roles for the +most abundant ones. Future studies should provide +more in depth analyses of these genomes, especially +from the less abundant organisms that might perform +key ecological processes, such as methanogens and nitri- +fiers. In addition, they should focus on gaining physio- +logical culture-based evidence or proof for in situ +activity for the abundant organisms described here. The +key metabolic insights provided by this metagenomics +study can lead to the design of new cultivation strategies. +In general, sediment communities are far more complex +than those found in the corresponding water column +[53, 59] and are therefore often considered too complex +for efficient metagenomic analysis. Many of the novel +lineages found here may therefore have related neutro- +philic lineages in marine and freshwater sediments that +await discovery. We demonstrate here that, by providing +sufficient sequencing depth, the “state of the art metage- +nomics toolbox” can effectively be used on sediments as +well. +Methods +Site description and sample collection +The top 10 cm sediments from four hypersaline, eutrophic +soda lakes located in the Kulunda Steppe (south-western +Siberia, Altai, Russia) were sampled in July of 2010 and +2011. General features and exact location of the sampled +soda lakes are summarized in Additional file 1: Table S1; a +map of the area was published previously [5]. Cock Soda +Lake (a stand-alone lake, sampled both in 2010 and 2011) +and Tanatar-3 (Tanatar system) were moderately hypersa- +line (~ 100 g L−1) with sandy sediment, while Tanatar-1 +and Bitter-1 (Bitter system) were salt-saturated (400 g L−1) +with sulfide-rich sapropel sediments underlined by rock +trona deposits [7, 60]. Especially, Bitter-1 harbors a very +active microbial community, probably due to its high- +organic and -mineral content. Surface sediments were col- +lected by a plastic corer into sterile glass containers and +transported to the laboratory in a cooler. +DNA isolation, 16S rRNA amplicon, and metagenomic +sequencing +The colloidal fraction of each sediment sample (~ 10% +of 50 g) was separated from the course sandy fraction by +several short (30–60 s) low-speed (1–2,000 rpm in +50 mL Falcon tubes) centrifugation steps and washed +with 1–2 M NaCl solution. The pelleted colloidal sedi- +ment fraction was first subjected to 3 cycles of freezing +in liquid nitrogen/thawing, then re-suspended in 0.1 M +Tris (pH 8)/10 mM EDTA, and then subjected to harsh +bead beating treatment. Next, the samples were incu- +bated with lysozyme (15 mg/mL) for 2 h at 37 °C +followed by a SDS (10% w/v) and proteinase K (10 μg/ +mL) treatment for 30 min. at 45 °C. High molecular +weight DNA was isolated using phenol/chloroform ex- +traction, quality-checked, and sequenced as described +previously [7]. Direct high-throughput sequencing of the +DNA was performed on an Illumina HiSeq 2000 plat- +form to generate 150 b paired-end reads. Amplification +of the V4-V6 region of prokaryote 16S rRNA genes +using barcoded 926F-1392R primers, amplicon purifica- +tion, quantification, and Roche (454)-sequencing was +performed together in a batch with brine samples from +the same sampling campaigns. Barcodes and adapter se- +quences were removed from de-multiplexed amplicon +sequence reads and analyzed with the automated NGS +analysis pipeline of the SILVA rRNA gene database pro- +ject [61] (SILVAngs 1.3, database release version 128) +using default parameters. The OTUs (97% identity) +assigned down to the genus level were only considered +when they had a relative abundance ≥ 0.1% in at least +one of the five datasets. +Processing metagenomics reads, assembly, binning, and +post-binning +Metagenomic raw reads were quality trimmed using +Sickle [62] (version 1.33), and only reads ≥ 21 b were +retained. The prokaryotic community structure at taxo- +nomic top levels was extrapolated from ten million ran- +domly sampled singletons from each dataset. Candidate +16S rRNA fragments > 90 b were identified [63] and +compared against the SILVA SSU database 128 (blastn, +min. length 90, min. identity 80%, e value 1e-5). To ver- +ify that the microbial community composition was in- +deed +mostly +prokaryotic, +we +did +a +more +general +screening of the metagenomics reads that identified also +candidate 18S rRNA fragments > 90 b (see Additional +file 1: Tables S4-S5). The complete trimmed read sets +were assembled into contigs ≥ 1 kb with MEGAHIT [64] +(v1.0.3–6-gc3983f9) using paired-end mode, k min = 21, +k max = 131, k step = 10. Genes were predicted using +Prodigal [65] (v.2.6.2) and RNAs with rna_hmm3 [66] +and tRNAscan-SE [67]. Assembled 16S rRNA sequences +were compared to a manually curated version from the +SILVA SSU database (e value ≥ 1e-5). Predicted protein +sequences +were +annotated +against +KEGG +with +GhostKOALA (genus_prokaryotes + family_eukaryotes ++ viruses) [68]. Marker genes for central metabolic +pathways and key environmental element transforma- +tions were identified based on K number assignments +[15, 69–71]. +Contigs ≥ 2.5 kb were binned with METABAT [72] +(superspecific mode) based on differential coverage +Vavourakis et al. Microbiome (2018) 6:168 +Page 13 of 18 + +values obtained by mapping all five trimmed readsets to +all five contig sets with Bowtie2 [73]. The bins were sub- +jected to post-binning (an overview of the workflow is +given in Additional file 2: Figure S13). Bins were +assessed with lineage-specific single copy genes using +CheckM [74] and further processed with the metage- +nomics workflow in Anvi’o [75] (v2.3.2). Since Candidate +Phyla Radiant (CPR) is not included in the CheckM ref- +erence trees and are likely to have low-genome com- +pleteness, we used an existing training file of 797 CPR +genomes to identify putative CPR bins [76]. Bins with +CheckM-completeness ≥ 50% (884 out of 1778) and an +additional four CPR bins were further processed. Coding +sequences +were +annotated +for +taxonomy +against +NCBI-nr (July, 2017) with USEARCH [77] (5.2.32) to +verify that most hits in each bin were to prokaryotic ref- +erences. Phage or viral contigs were manually removed. +Genome +contamination (redundancy) +was estimated +based on marker sets of universal single copy genes +identified for Bacteria [30] and Archaea [78] as imple- +mented in Anvi’o. Genome coverage was obtained by +mapping trimmed reads with BBMap [79] v36.x (kfilter +31, subfilter 15, maxindel 80). Bins with ≥ 5% redun- +dancy were further refined with Anvi’o using circle phy- +lograms +(guide +trees +tnf-cov: +euclidian +ward) +and +scanned again for CPR. Post-binning resulted in a total +of 2499 metagenome-assembled genomes (MAGs), of +which 871 were either medium-quality genome drafts +(CheckM estimated completeness ≥ 50% and contamin- +ation ≤ 10% [80], Additional file 4) or lower quality draft +genomes from CPR. +Phylogeny of the MAGs was assessed based on 16 +single-copy ribosomal proteins and representative refer- +ence genomes of major prokaryote lineages across the +tree of life [17]. Individual ribosomal proteins in our +MAGs were identified by K number assignments. Only +ribosomal proteins ≥ 80 aa were considered. Initial +maximum-likelihood (ML) trees were constructed to de- +termine which organisms belonged to the Archaea, Bac- +teria, or CPR with FastTree 2 [81] (WAG + CAT). Final +separate trees for the three distant evolutionary groups +were constructed in the same manner. Each ribosomal +protein set was aligned separately with MAFFT [82] +(v7.055b, − auto) and concatenated only if a MAG +encoded at least 8 out of 16 proteins. For all trees, a +100× posterior bootstraps +analysis +was +performed. +Phylogenetic trees were visualized together with gen- +ome statistics and abundance information using iTOL +[83]. We cross-checked the taxonomic assignments +based on the phylogeny of the ribosomal protein cas- +sette +with +the +top +hit +contig annotations +against +NCBI-nr and with the reference lineage obtained with +CheckM. Lastly, we manually corrected the MAGs for +misplaced 16S rRNA genes. The final trees presented +in the manuscript were redrawn using FigTree v1.4.3 +[84]. +Detailed genome analyses +CPR +MAGs +were +re-annotated +more +thoroughly: +genes were predicted with Prokka [85], and functional +predictions were performed by running InterProScan +5 locally on the supplied COG, CDD, TIGRFAMs, +HAMAP, Pfam, and SMART databases [86]. BLAST +Koala was used for KEGG pathway predictions [68]. +To find putative carbohydrate-active enzymes in all +final MAGs, we used the web-resource dbCAN [87] +to annotate all predicted proteins ≥ 80 aa against +CAZy [88]. +To identify the top ten abundant MAGs from each re- +spective dataset, ten million randomly sampled single- +tons were mapped onto each MAG with a cut-off of 95% +identity in minimum of 50 bases. Coverage values were +additionally normalized for genome size and expressed +as reads per kilobase of sequence per gigabase of +mapped reads (RPKG) [89]. A positive score (from 871 +to 1) was assigned to each MAG according to the rank- +ing of the summed RPKG of MAGs in the high-salinity +datasets (B1Sed10 and T1Sed) and a negative score ac- +cording to the ranking of the summed RPKGs in the +moderate salinity datasets (CSSed10, CSSed11, T3Se +d10). Both scores were summed to get a “salinity prefer- +ence score” with MAGs recruiting preferably from high +salinity datasets on the positive end, moderate salinity +datasets in the negative end, and those without prefer- +ence in the middle. +We determined species delineation for the most +abundant MAGs and their closest reference genomes +(NCBI-nr) by Average Nucleotide Identity (ANI) and +conserved DNA-matrices, as follows [90]: ANI ≥ 95%, +conDNA ≥ 69% = same species, ANI ≥ 95%, condDNA +< 69% = might be same species, ANI < 95%, condDNA +< 69% = different species. Single gene trees based on +maximum +likelihood +were +constructed +with +un- +trimmed alignments (MAFFT, L-INS-i model) and +FastTree 2 (WAG + CAT, increased accuracy, -spr4 +-mlacc 2 -slownni) using 100× bootstraps. References +were pulled from eggNOG (v4.5.1) [91] and supple- +mented +with +sequences +from +NCBI-nr +or +refined +according to [7, 33, 46, 92–94]. The curated MAGs +were +scanned +for +the +presence +of +rhodopsin +sequences with the hmmsearch software [95] and a +profile +hidden +Markov +model +(HMM) +of +the +bacteriorhodopsin-like protein family (Pfam accession +number +PF01036). +The +identified +sequences +with +significant similarity were aligned together with a +curated database composed of a collection of type-1 +rhodopsins, using MAFFT (L-INS-i accuracy model) +[82]. This protein alignment was further utilized to +Vavourakis et al. Microbiome (2018) 6:168 +Page 14 of 18 + +construct a maximum likelihood tree with 100× boot- +strap with FastTree 2 [81]. All other genes were +identified using the KEGG annotation. +Additional files +Additional file 1: Table S1. General features of the four sampled soda +lakes at time of sampling. Table S2. SILVA classification of the 16S rRNA +gene sequences found in all ≥1 kb contigs of five soda sediment +metagenomic datasets. Table S3. Enzymes involved in lipopolysaccharide +biosynthesis found among different members of the CPR. Table S4. +Sub-kingdom classification of candidate SSU rRNA gene fragments +found in subsamples of 10 million random forward reads from the +five soda sediment metagenomes. Table S5. Top-level taxonomic +classification of the 18S rRNA gene fragments found in subsamples +of 10 million random forward reads from the five soda sediment +metagenomes. Table S6. Description of the metagenomic datasets, +NCBI Sequence Read Archive (SRA) accession numbers and general +statistics of the assembled contigs. (PDF 740 kb) +Additional file 2: Figure S1. Taxonomic fingerprints determined by 16S +rRNA gene amplicon sequencing. Figure S2. Genome statistics of the +871 MAGs. Figure S3. Phylogeny of MAGs belonging to “Candidatus +Aenigmarchaeota” and “Ca. Nanohaloarchaeota”. Figure S4. Phylogeny of +MAGs related to “Candidatus Acetothermia”, candidate division WS1 and +“Candidatus Lindowbacteria”. Figure S5. Phylogeny of MAGs related to +candidate division KSB3 and “Candidatus Schekmanbacteria”. Figure S6. +Multiple sequence alignment of the V-type ATPase subunits K. Figure S7. +Multiple sequence alignment of the F-type ATPase subunits c. Figure S8. +Maximum likelihood tree of the large subunits of RuBisCo and RubisCo- +like proteins. Figure S9. Maximum likelihood tree of the putative +rhodopsins. Figure S10. Predicted isoelectric points (pI) profiles of all +MAGs from CPR members. Figure S11. Predicted isoelectric points +profiles for members of the “Ca. Nealsonbacteria” and “Ca. Vogelbacteria”. +Figure S12. Multiple sequence alignment of the dissimilatory +cytochrome c nitrite reductases (nrfA/TvNiR, K03385). Figure S13. +Overview of the post-binning workflow used for genome recovery. +(PDF 6548 kb) +Additional file 3: Dataset S1. Relative abundance of the OTUs assigned +to the genus-level within the Archaea, Bacteria and organelles from +Eukaryota detected by 16S rRNA gene amplicon sequencing. The OTUs +with less than 0.1% abundance accross all five datasets are not shown. +The names of highly abundant genera (≥1% in at least one of the data- +sets) are shown in bold. (XLSX 24 kb) +Additional file 4: Dataset S2. Organism names, statistics and general +description incl. Completeness and contamination estimates, phylogeny +and DDBJ/EMBL/Genbank accession numbers of the metagenome +assembled genomes (MAGs) described in this paper. All submitted +versions described in this paper are version XXXX01000000. Size = +recovered genome size, Completeness (Compl1), contamination (Cont), +strain heterogenity (Str het) and Taxon CheckM were inferred from +lineage-specific marker sets and a reference tree build with CheckM [74]. +Additional completeness (compl2) and redundancy (red) estimates were +inferred based on the presence of universal single copy genes for Bacteria +and Archaea [75]. Decision and confidence intervals from the Candidate +Phyla Radiation (CPR) scan [75] are given, as well as the taxonomy of the +besthit in SILVA when 16S rRNA genes were present. Phylum/class 16 +ribosomal proteins is the taxonomy derived from our ribosomal protein +trees (see main text: Figs. 2 and 3). OTU gives the inferred link of a +population genome with our 16S rRNA gene amplicon dataset +(Additional file 3). (XLSX 253 kb) +Additional file 5: Dataset S3. Estimated abundance and derived +salinity preference from each MAG in each metagenomic dataset +expressed as Reads per Kilobase of MAG per Gigabase of mapped reads +(RPKG) and “salinity preference score” (see Methods section), basis for +Fig. 4. (XLSX 143 kb) +Additional file 6: Dataset S4. Average Nucleotide Identity (ANI) and +conserved DNA (condna) matrices to determine species delineation +between the most abundant MAGs shown in Fig. 4, closely related +(less abundant) MAGs and NCBI reference genomes. Decision matrix +shows: 1 = same species, − 1 = might be same species, 0 = different +species (see Methods section). (XLSX 1161 kb) +Additional file 7: Dataset S5. Sheet 1 Presence and absence of marker +genes and putative carbohydrate-active enzymes in the MAGs to infer putative +roles in C, N and S element cycles based on K-number assignments and CAZy +annotations. Sheet 2 Summary basis for Fig. 4. (XLSX 41 kb) +Additional file 8: Information S1. More detailed description of the +main metabolisms encoded by Thioalkalivibrio-related MAGs. +Information S2 More detailed description of the main metabolisms +encoded by Deltaproteobacterial-related MAGs. (PDF 219 kb) +Additional file 9: Dataset 6. Sheet 1 shows the MAGs positive for the +marker gene acsB (K14138) encoding an acetyl-CoA synthase (ACS). The +basis for Fig. 6, namely presence and absence of key genes involved in +the Wood-Ljungdahly pathway, acetogenesis, methanogenesis, glycolysis +and pyruvate to CO2 conversion is shown for each MAG. Sheet 2 shows +the MAGs positive for the marker gene cdhC (K00193) encoding for the +beta subunit of an acetyl-CoA decarboxylase synthase complex. While +acsB and cdhC correspond roughly to the Bacterial-type and Archaeal- +type (methanogens) enzymes with the same function, we found few +discrepancies between marker gene and genome phylogeny within the +Methanomassiliicoccaceae and Chloroflexi. (XLSX 52 kb) +Acknowledgments +We thank Dr. Nikolai Chernych for his technical assistance during the +isolation and purification of metagenomics DNA. We also thank the +Department of Energy Joint Genome Institute for sequencing the +metagenomes. +Funding +CDV and GM were supported by the ERC Advanced Grant PARASOL (no. 322551). +A-SA and RG were supported by the research grant 17-04828S from the Grant +Agency of the Czech Republic. MM was supported by the Czech Academy of +Sciences (Postdoc program PPPLZ application number L200961651). DYS was +supported by the SIAM/Gravitation Program (Dutch Ministry of Education and +Science, grant 24002002) and by the Russian Science Foundation (grant 16–14- +00121). Sequencing was performed by the U.S. Department of Energy Joint +Genome Institute, a DOE Office of Science User Facility, as part of the Community +Sequencing Program (contract no. DE-AC02- 05CH11231). +Availability of data and materials +The raw sequence reads of the five metagenomes have been deposited to +the NCBI Sequence Read Archive (see Additional file 1: Table S6 for accession +numbers and read and contig statistics). The final 871 MAGs described in this +paper have been deposited as Whole Genome Shotgun projects at DDBJ/ +EMBL/GenBank, and accession numbers are listed in Additional file 4 +(BioProject ID PRJNA434545). All versions described in this paper are version +XXXX01000000. The cleaned and dereplicated amplicon sequence datasets +are available in FigShare (https://figshare.com/s/7684627445e3621aba24). +Maximum likelihood trees based on the concatenated alignment of 16 +ribosomal proteins, basis for Figs. 2 and 3, in newick format (.tre file) and +complementary datasets (used to plot completeness, contamination, +genome recovery size, G + C mol% and RPKG in iTOL), as well as K number +assignments for the predicted proteins of all MAGs (KEGG-orthologues, +Ghost Koala) and the fully annotated CPR MAGs supporting the conclusions +of this article are also available in FigShare (https://figshare.com/s/ +7684627445e3621aba24). +Authors’ contributions +GM and DYS initiated this study and were responsible for the fieldwork, +sample preparation, and sequencing effort. CDV conceptualized the research +goals under supervision of DYS and GM, and performed the bioinformatics +analysis under close guidance of A-SA and RG. CDV is the primary author of +this manuscript. MM, RG, and CDV prepared the main figures. All authors +read and approved the final manuscript. +Ethics approval and consent to participate +Not applicable. +Vavourakis et al. Microbiome (2018) 6:168 +Page 15 of 18 + +Consent for publication +Not applicable. +Competing interests +The authors declare that they have no competing interests. +Publisher’s Note +Springer Nature remains neutral with regard to jurisdictional claims in +published maps and institutional affiliations. +Author details +1Microbial Systems Ecology, Department of Freshwater and Marine Ecology, +Institute for Biodiversity and Ecosystem Dynamics, Faculty of Science, +University of Amsterdam, Postbus 94248, 1090, GE, Amsterdam, the +Netherlands. 2Department of Aquatic Microbial Ecology, Institute of +Hydrobiology, Biology Centre CAS, Na Sadkach 7, 370 05 Ceske Budejovice, +Czech Republic. 3Winogradsky Institute of Microbiology, Research Centre of +Biotechnology, Russian Academy of Sciences, 60 let Oktyabrya pr-t, 7, bld. 2, +Moscow, Russian Federation117312. 4Environmental Biotechnology, +Department of Biotechnology, Delft University of Technology, Van der +Maasweg 9, 2629, HZ, Delft, the Netherlands. +Received: 23 June 2018 Accepted: 3 September 2018 +References +1. +Sorokin DY, Berben T, Melton ED, Overmars L, Vavourakis CD, Muyzer G. +Microbial diversity and biogeochemical cycling in soda lakes. 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Microbiome (2018) 6:168 +Page 18 of 18 + diff --git a/kb_36/content/tmp_files/load_file.txt b/kb_36/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0fa759a85ccdce1e77a559ac05dec6fd86aee726 --- /dev/null +++ b/kb_36/content/tmp_files/load_file.txt @@ -0,0 +1,1147 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf,len=1146 +page_content='RESEARCH Open Access A metagenomics roadmap to the uncultured genome diversity in hypersaline soda lake sediments Charlotte D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Vavourakis1 , Adrian-Stefan Andrei2†, Maliheh Mehrshad2†, Rohit Ghai2, Dimitry Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Sorokin3,4 and Gerard Muyzer1* Abstract Background: Hypersaline soda lakes are characterized by extreme high soluble carbonate alkalinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Despite the high pH and salt content, highly diverse microbial communities are known to be present in soda lake brines but the microbiome of soda lake sediments received much less attention of microbiologists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Here, we performed metagenomic sequencing on soda lake sediments to give the first extensive overview of the taxonomic diversity found in these complex, extreme environments and to gain novel physiological insights into the most abundant, uncultured prokaryote lineages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Results: We sequenced five metagenomes obtained from four surface sediments of Siberian soda lakes with a pH 10 and a salt content between 70 and 400 g L−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' The recovered 16S rRNA gene sequences were mostly from Bacteria, even in the salt-saturated lakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Most OTUs were assigned to uncultured families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' We reconstructed 871 metagenome-assembled genomes (MAGs) spanning more than 45 phyla and discovered the first extremophilic members of the Candidate Phyla Radiation (CPR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Five new species of CPR were among the most dominant community members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Novel dominant lineages were found within previously well-characterized functional groups involved in carbon, sulfur, and nitrogen cycling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Moreover, key enzymes of the Wood-Ljungdahl pathway were encoded within at least four bacterial phyla never previously associated with this ancient anaerobic pathway for carbon fixation and dissimilation, including the Actinobacteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Conclusions: Our first sequencing effort of hypersaline soda lake sediment metagenomes led to two important advances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' First, we showed the existence and obtained the first genomes of haloalkaliphilic members of the CPR and several hundred other novel prokaryote lineages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' The soda lake CPR is a functionally diverse group, but the most abundant organisms in this study are likely fermenters with a possible role in primary carbon degradation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Second, we found evidence for the presence of the Wood-Ljungdahl pathway in many more taxonomic groups than those encompassing known homo-acetogens, sulfate-reducers, and methanogens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Since only few environmental metagenomics studies have targeted sediment microbial communities and never to this extent, we expect that our findings are relevant not only for the understanding of haloalkaline environments but can also be used to set targets for future studies on marine and freshwater sediments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Keywords: Soda lake sediments, Metagenomics, Haloalkaliphilic extremophiles, Candidate Phyla Radiation, Wood-Ljungdahl pathway Correspondence: G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content='Muijzer@uva.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content='nl †Adrian-Stefan Andrei and Maliheh Mehrshad contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' 1Microbial Systems Ecology, Department of Freshwater and Marine Ecology, Institute for Biodiversity and Ecosystem Dynamics, Faculty of Science, University of Amsterdam, Postbus 94248, 1090, GE, Amsterdam, the Netherlands Full list of author information is available at the end of the article © The Author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' 2018 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content='0 International License (http://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content='org/licenses/by/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content='0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' The Creative Commons Public Domain Dedication waiver (http://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content='org/publicdomain/zero/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content='0/) applies to the data made available in this article, unless otherwise stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Vavourakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Microbiome (2018) 6:168 https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content='1186/s40168-018-0548-7 MicrobiomeBackground Soda lakes are evaporative, athallasic salt lakes with low cal- cium and magnesium concentrations and a high-alkaline pH up to 11 buffered by dissolved (bi-) carbonate ions [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' They are constrained to arid regions across the globe, mainly the tropical East African Rift Valley [2], the Libyan Desert [3], the deserts in California and Nevada [4], and the dry steppe belt of Central Asia that spans to southern Si- beria, north-eastern Mongolia, and Inner Mongolia in China [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' On top of the extreme salinity and alkaline pH, the Eurasian soda lakes experience extreme seasonal temperature differences, causing highly unstable water re- gimes and fluctuating salinities [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Yet, soda lakes harbor diverse communities of haloalkaliphilic microbes, mostly prokaryotes that are well adapted to survive and grow in these extreme environments and consist of similar func- tional groups in soda lakes around the world [1, 2, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' The relative abundance of different groups is typically governed by the salinity of the brine [1, 7, 8], and microbial-mediated nutrient cycles become partially hampered only at salt-saturating conditions [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' So far, all characterized prokaryotic lineages cultured from soda lakes comprise over 70 different species within more than 30 genera [1, 6, 9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' From these, only a lim- ited number of genomes have been sequenced today, mostly from chemolithoautotrophic sulfur-oxidizing bac- teria belonging to the genus Thioalkalivibrio (class Gam- maproteobacteria) [1, 11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' It is well established that metagenomics enables the recovery of genomes and the identification of novel genetic diversity where culturing ef- forts fail [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' In recent years, next-generation sequen- cing has recovered a massive number of genomes from previously unknown groups of prokaryotes [15, 16], including a strikingly large and diverse group called “Candidate Phyla Radiation” (CPR), only distantly related to other cultured bacterial lineages [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Previously, we conducted a metagenomics study on soda lakes and re- constructed novel genomes from uncultured Bacteroidetes and “Candidatus Nanohaloarchaeaota” living in hypersa- line Siberian soda brines [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Here, we turned our atten- tion to the far more complex prokaryotic communities living in the sediments of the hypersaline soda lakes from the same region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' We give a broad overview of all the taxonomic groups sequenced and focus on the metabolic diversity found in the reconstructed genomes of the most abundant, uncultured organisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Results Overall prokaryote community structure The salinities from the studied soda lakes ranged from moderately hypersaline (between 70 and 110 g L−1) to salt-saturated (400 g L−1 salt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' The soluble carbonate al- kalinity was in the molar range, and the pH in all lakes was around ten (see Additional file 1: Table S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' To give an overview of the overall prokaryotic community com- position in each of the samples, we looked at the taxo- nomic classification of 16S rRNA genes recovered both by amplicon sequencing and direct metagenomics se- quencing (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' 1, see also Additional file 2: Figure S1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Additional file 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' The prokaryotic communities of all five sediment samples were highly diverse and consisted mostly of uncultured taxonomic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Bacteria were more abundant than Archaea, regardless of the salinity of the overlaying brine [7] (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Euryarchaeota were the second and third largest group in the sediments of the two salt-saturated lakes comprising ~ 10 and ~ 20% of the 16S rRNA genes in the metagenomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Most Euryarchaeota-related OTUs detected by amplicon se- quencing belonged either to the uncultured Thermoplas- mata group KTK 4A (SILVA classification) or the genera Halohasta and Halorubrum (class Halobacteria).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' In ac- cordance with cultivation-dependent studies [6], most OTUs assigned to methanogens were from the class Methanomicrobia, especially the lithotrophic genus Methanocalculus (up to ~ 3%) and the methylotrophic genus Methanosalsum (Additional file 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' The varying ratio of the three dominant bacterial groups, Firmicutes, Bacteroidetes (including the newly proposed phyla Rhodothermaeota and Balneolaeota [18]), and Gammaproteobacteria, showed no clear trend in relation to the salinity in the lakes, but when Firmicutes were domin- ant, Bacteroidetes were less abundant and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Most Firmicutes belonged to the order Clostridales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Uncultured members from the family Syntrophomonadaceae had a relative abundance of more than 5% in all five metagen- omes and comprised in two lakes even ~ 11–20% of the recovered amplicon sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' The second most abundant Firmicutes order was Halanaerobiales, particularly the genus Halanaerobium (family Halanaerobiaceae) and un- cultured members of the Halobacteroidaceae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' The majority of Bacteroidetes-related OTUs could not be assigned down to the genus level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' The uncultured ML635J-40 aquatic group (order Bacteroidales) comprised at least 5% of all five prokaryotic communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' This group has been previously found to be abundant in Mono Lake [4] (a soda lake) and in an anoxic bioreactor degrading cyanobacterial biomass under haloalkaline conditions [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Two other highly abun- dant (up to ~ 8%) uncultured groups from the class Balneo- lia (proposed new phylum Balneolaeota [18]) were also detected in other soda lakes before [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Within the Gam- maproteobacteria, the genus Thioalkalivibrio was abundant (~ 3% of the total community), but also uncultured members of HOC36 were prevailing at moderate salinities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Members of the Deltaproteobacteria, Alphaproteobacteria, and Chloroflexi comprised up to ~ 10% of the detected 16S rRNA gene in some of the metagenomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' The GIF9 family of the class Dehalococcoidia was among the top three most abundant OTUs in two lakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' The extremely salt-tolerant Vavourakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Microbiome (2018) 6:168 Page 2 of 18 and alkaliphilic genera Desulfonatronobacter (order Desulfo- bacterales) and Desulfonatronospira (order Desulfovibrio- nales) were the dominant Deltaproteobacteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Highly abundant OTUs, within the Actinobacteria belonged to the class Nitriliruptoria and within the Alphaproteobacteria to the family Rhodobacteraceae and the genus Roseibaca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' The important nitrifying genus Nitrobacter (Alphaproteobacteria) was present in only one of the lakes with moderate salinity (Additional file 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Some bacterial top-level taxa appeared less dominant (< 5%) from the 16S rRNA genes recovered from the metagenomes but were represented mainly by a single highly abundant OTU in the amplicon sequences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' in- cluding the haloalkaliphilic genus Truepera within the phylum Deinococcus-Thermus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' the genus Spirochaeata within the phylum Spirochaetes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' the family BSN166 within the phylum Ignavibacteriae,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' the BD2–11 terres- trial group within the Gemmatimonadetes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' and the WCHB1–41 order within the Verrucomicrobia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' All OTUs within the Thermotogae and Lentisphaerae belonged to uncultured genera from the family Kosmoto- gaceae and Oligosphaeraceae, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' All Tenericu- tes-related OTUs belonged to the class Mollicutes, and especially the order NB1-n was dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' In contrast, the phylum Planctomycetes was relatively diverse, with at least 11 different genus-level OTUs spread over four class-level groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' High-throughput genome recovery We obtained 717 medium-quality (≥ 50% complete, < 10% contamination) and 154 near-complete (≥ 90% complete, < 5% contamination) metagenome-assembled genomes (MAGs) across three major prokaryote groups: Archaea, Bacteria, and CPR (see Additional file 4 and Additional file 2: Figure S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Figures 2 and 3 show the top-level phylogeny of all MAGs based on 16 ribosomal proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' The reference database used contains a repre- sentative for each major prokaryote lineage [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' We a b Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' 1 Abundant prokaryotic groups in five hypersaline soda lake sediments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' a Relative abundance of the top-level taxa (those with ≥ 1% abundance in at least one dataset) based on 16S rRNA reads in unassembled metagenomic datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' b Relative abundance of the 16S rRNA OTUs (those with sum of abundance in all datasets ≥ 3%) recovered by amplicon sequencing assigned where possible down to the genus-level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Three of the assessed soda lakes have a moderate salinity (70–110 g L−1), two are salt-saturated (400 g L− 1) Vavourakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Microbiome (2018) 6:168 Page 3 of 18 colored the different phyla from which we obtained a MAG in alternate blue and orange colors, and highlighted the MAGs obtained here in a darker shade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Many MAGs belonged to uncultured groups commonly detected in soda lake 16S rRNA gene surveys, over 100 MAGs still belonged to candidate prokaryote phyla and divisions that to our knowledge were never detected be- fore in soda lakes, including CPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Although only few MAGs had near-complete 16S rRNA genes, in most cases we were able to link available taxonomic gene an- notations and ribosomal protein phylogeny to the SILVA taxonomy of the OTUs assigned to the amplicon se- quences, while cross-checking the abundance profiles of both MAGs (Additional file 5) and OTUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' The soda lake CPR recovered from the metagenomes was restricted to a few distinct phyla within the Parcubacteria group, mostly affiliating with “Candidatus Nealsonbacteria” and “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Zambryskibacteria” [15] (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' The first group of MAGs encompassed four different branches in our riboso- mal protein tree, suggesting a high-phylogenetic diversity, with 33 putative new species sampled here (ANI and con- DNA matrices given in Additional file 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' The “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Zambrys- kibacteria-”related MAGs consisted of at least five new species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Few MAGs were recovered from CPR groups also detected by amplicon sequencing (see Additional file 2: Figure S1), namely the “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Dojkabacteria” (former WS6), “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Saccharibacteria” (former TM7), CPR2, and “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Katanobacteria” (former WWE3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' 2 Maximum-likelihood phylogeny of the CPR and archaeal MAGs based on 16 ribosomal proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' The archaeal tree is unrooted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' The CPR tree is rooted to the Wirthbacteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Alternate orange and blue colors show phyla/classes from which we obtained MAGs (labeled as “Phyla present”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Reconstructed MAGs of this study are highlighted by darker shades (labeled as “MAG present”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Phyla/classes for which there was no representative in the reconstructed MAGs of this study are shown as gray cartoons (labeled as “Phyla not present”), and the numerical labels are represented at the bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Colored circles at the nodes show confidence percentage of the bootstraps analysis (100×) Vavourakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Microbiome (2018) 6:168 Page 4 of 18 Most archaeal MAGs belonged to the phylum Euryarch- aeota and the abundant classes Halobacteria, Methanomi- crobia, and Thermoplasmata (related to OTU KTK 4A) within.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' In addition, three Thermoplasmata-related MAGs that encoded for the key enzyme for methanogenesis (methyl-coenzyme M reductase, mcr) affiliated with refer- ence genomes from Methanomassilicoccales, the seventh order of methanogens have been recovered [20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Another MCR-encoding MAG was closely related to the latest discovered group of poly-extremophilic, methyl-reducing methanogens from hypersaline lakes from the class Methanonatronarchaeia [9] (related to OTU ST-12K10A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' We recovered also one MAG from the class Methanobacteria and a high-quality MAG from the WCHA1–57 group (“Candidatus Methanofastidiosa” [22]) in the candidate division WSA2 (Arc I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Several MAGs were recovered from the DPANN archaeal groups “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Diapherotrites,” “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Aenigmarchaeota,” (see Additional file 2: Figure S3) and “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Woesearch- aeota” (former Deep Sea Hydrothermal Vent Group 6, DHVEG-6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Although we did not reconstruct any reasonable-sized MAGs from the TACK superphylum, we found several 16S rRNA genes on the assembled contigs that affiliated to the Thaumarchaeota (see Additional file 1: Table S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Nearly every known bacterial phylum had an extremo- philic lineage sampled from our hypersaline soda lake sediments (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' In most cases, the soda lake lineages clearly formed separate branches appearing as sister groups to known reference lineages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' The highest genome recovery was from the same top-level taxonomic groups that were also abundant in our 16S rRNA gene analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' From the Verrucomicrobia, most MAGs belonged to the order WCHB1-41 (16S rRNA gene identity 92–100%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' However, in our ribosomal protein tree, they branched within the phylum Lentisphaerae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Sixteen Tenericutes MAGs from at least 12 different species (Additional file 6) were closely related to the NB1-n group of Mollicutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Based on the recovered genome size and encoded meta- bolic potential, these organisms are free-living anaerobic fermenters of simple sugars, similar to what has recently been proposed for “Candidatus Izimaplasma” [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' 3 Maximum-likelihood phylogeny of the bacterial MAGs (CPR excluded) based on 16 ribosomal proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Alternate orange and blue colors show phyla/ classes from which we obtained MAGs (labeled as “Phyla present”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Reconstructed MAGs of this study are highlighted by darker shades (labeled as “MAG present”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Phyla/classes for which there was no representative in the reconstructed MAGs of this study are shown as gray cartoons (labeled as “Phyla not present”), and the numerical labels are represented at the bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Colored circles at the nodes show confidence percentage of the bootstraps analysis (100×) Vavourakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Microbiome (2018) 6:168 Page 5 of 18 Several MAGs belonged to the candidate phyla “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Omnitrophica,” “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Atribacteria,” and “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Acetother- mia” (former OP1), which were moderately abundant also in some sediment (see Additional file 2: Figure S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' For the latter phylum, we suspect that four MAGs were more closely related to ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' div.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' WS1 and “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Lindow- bacteria” for which only few reference genomes are currently available in NCBI (see Additional file 2: Figure S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Due to a high-sequencing coverage, we also managed to reconstruct several MAGs from rare Bacteria (< 100 amplicon sequences detected, see Additional file 2: Figure S1), including the phyla “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Hydrogenedentes,” “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Cloacimonetes,” ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' div.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' BRC1, Elusimicrobia, Caldi- serica, and “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Latescibacteria.” The MAGs from the latter phylum were more closely related to the recently proposed phylum “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Handelsmanbacteria” [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Two additional MAGs with 16S rRNA gene fragments with 94–95% identity to the class MD2898-B26 (Nitrospinae) were more likely members of ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' div.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' KSB3 (proposed “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Moduliflexus” [24], see Additional file 2: Figure S5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Draft genomes of haloalkaliphilic CPR Strikingly, members of the CPR related to “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Nealson- bacteria” and “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Vogelbacteria” were among the top 5% of abundant organisms in the surface sediments of the soda lakes, especially those with moderate salinity (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Like most members of the CPR, the MAGs of the four most abundant “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Nealsonbacteria” seem to be anaerobic fermenters [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' They lacked a complete TCA cycle and most complexes from the oxidative elec- tron transfer chain, except for the subunit F of a NADH-quinone oxidoreductase (complex I, nuoF, nuoG, nuoA) and coxB genes (complex II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' All CPR MAGs had a near-complete glycolysis pathway (Embden-Meyerhof- Parnas) encoded, but pentose phosphate pathways were severely truncated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' The commonly encoded F- and V-type ATPase can establish a membrane potential for symporter-antiporters by utilizing the ATP formed by substrate-level phosphorylation during fermentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' All CPR have V-type ATPases that can translocate Na+ in addition to H+ (see Additional file 2: Figure S6), while only two members of the “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Falkowbacteria” had puta- tive Na+-coupled F-type ATPases (see Additional file 2: Figure S7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' The coupling of ATP hydrolysis to sodium translocation is advantageous to maintain pH homeosta- sis in alkaline environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Interestingly, with only two exceptions [26, 27], all CPR genomes recovered from other environments with neutral pH were reported to encode only F-type ATPases [28–32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' One low-abundant MAG affiliated to “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Peregrinibacteria” contained both the large subunit of a RuBisCO (type II/III, see Additional file 2: Figure S8) and a putative phosphoribu- lokinase (PRK, K00855) encoded in the same contig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' This is remarkable because PRK homologs were not previously identified among CPR, and RuBisCo form II/ III was inferred to function in a nucleoside salvage path- way [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' One “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Saccharibacteria” MAG encoded for a putative channelrhodopsin (see Additional file 2: Figure S9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' This is the first rhodopsin found among the CPR and suggests that this enigmatic group of organ- isms may have acquired evolutionary adaptations to a life in sunlit surface environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' A previous study showed that most CPR has coccoid cell morphotypes with a monoderm cell envelope resem- bling those from Gram-positives and Archaea but with a distinct S-layer [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Thick peptidoglycans coated with acidic surface polymers such as teichoic acids help pro- tect the cells of Gram-positives against reactive hydroxyl ions in highly alkaline environments [35] (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' 5a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' All soda lake CPR had indeed the capability for peptidogly- can biosynthesis, but we found proteins typical for Gram-negatives for the biosynthesis of lipopolysaccha- rides (see Additional file 1: Table S3), homologous to the inner membrane proteins of type II secretion systems and to several proteins associated to the outer membrane and peptidoglycan, including OmpA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' It remains to be determined whether the soda lake CPR also lacks an outer membrane and perhaps anchor lipopolysaccharides, S-layer proteins, and lipoproteins to the inner cell membrane or peptidoglycan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' We also found gene encoding cardiolipin and squalene synthases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Increased levels of cardiolipin and the presence of squa- lene make the cytoplasmic membrane less leaky for protons [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' In addition, cation/proton exchangers are known to play a crucial role for pH homeostasis in alka- liphilic prokaryotes as they help acidify the cytoplasm during the extrusion of cations [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Putative Na+/H+ exchangers of the Nha-type and multi-subunit Mnh-type were found only within a few soda lake CPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Secondary active transport of K+ might be mediated in most soda lake CPR by KefB (COG0475)/kch Kef-type, glutathione- dependent K+ transport systems, with or without H+ antiport (67,68).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Various soda lake CPR had an acidic proteome, with pI curves resembling those found in extremely halophilic Bacteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Intracellular proteins enriched in acidic amino acids might be an adaptation to a “salt-in” strategy, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=', maintaining high intracellular potassium (K+) concentra- tions to keep osmotic balance [7, 37] (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' 5b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' see Additional file 2: Figure S10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Such a strategy is energet- ically favorable over de novo synthesis or import of osmolytes such as ectoine and glycine betaine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' We did not find genes for the synthesis of organic osmolytes and homologs of ABC-type transporters for primary active uptake of proline/glycine betaine which were encoded only in one MAG (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' 5a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' For the “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Nealsonbac- teria” and “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Vogelbacteria,” the salt-in strategy might be a unique feature for the soda lake species explaining Vavourakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Microbiome (2018) 6:168 Page 6 of 18 their high abundance in the hypersaline soda lake sedi- ments, as we did not found an acidic proteome pre- dicted from genomes obtained from other non-saline environments (See Additional file 2: Figure S11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' The uptake of K+ ions remains enigmatic for most soda lake CPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Low-affinity Trk-type K+ uptake transporters (gen- erally with symport of H+) (67,68) were encoded only by a limited number of MAGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' We found three MAGs Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' 4 Relative abundance and metabolic potential of the dominant species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Abundance values, expressed as reads per kilobase of MAG per gigabase of mapped reads (RPKG), were averaged for the top ten abundant MAGs from each dataset that were (likely) the same species (Additional file 5, Additional file 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Population genomes were ranked by their “salinity preference scores”: those recruiting relatively more from moderate salinity datasets (cold colors) are drawn to the top, from high salinity datasets (warm colors) to the bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' The metabolic potential derived from functional marker genes (Additional file 7) is depicted by the colored symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' CBB = Calvin-Benson-Bassham cycle, DNRA = dissimilatory nitrite reduction to ammonia, fix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' = fixation, red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' = reduction, ox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' = oxidation, dis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' = disproportionation Vavourakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Microbiome (2018) 6:168 Page 7 of 18 a b Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' 5 (See legend on next page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=') Vavourakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Microbiome (2018) 6:168 Page 8 of 18 encoding for Kdp-type sensor kinases (kdpD) but no corresponding genes for the response regulator (kdpE) or for Kdp-ATPases that function as the inducible, high- affinity K+ transporters in other Bacteria (67,68).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Finally, mechanosensitive ion channels (mscS, mscL) and ABC- type multidrug transport systems (AcrAB, ccmA, EmrA, MdlB, NorM) and sodium efflux permeases (NatB) were encoded in almost every MAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' The first might rapidly restore the turgor pressure under fluctuating salinity conditions by releasing cytoplasmic ions [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Novel abundant groups involved in sulfur, nitrogen, and carbon cycles A new species of Thioalkalivibrio (family Ectothiorhodospir- aceae) was by far the most abundant in the sediments of the two salt-saturated lakes (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' In the sediment of Bitter-1, also a purple sulfur bacterium from the same fam- ily was highly abundant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' It was closely related to Halorho- dospira, a genus also frequently cultured from hypersaline soda lakes [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' None of the abundant Ectothiorhodospira- ceae spp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' had already a species-representative genome sequenced (Additional file 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' The potential of the Thioalk- alivibrio spp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' for chemolithotrophic sulfur oxidation was evident (Additional file 7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' see Additional file 8: Information S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Interestingly, the encoded nitrogen metabolisms were quite versatile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' While Thioalkalivibrio sp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' 1 had the poten- tial for nitrate reduction to nitrite, Thioalkalivibrio sp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' 2 might perform dissimilatory nitrite reduction to ammonia (DNRA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' see Additional file 2: Figure S12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Two deltaproteobacterial lineages of dissimilatory sulfate-reducing bacteria (SRB) were highly abundant in the soda lake sediment of Bitter-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' One MAG from the family Desulfobacteraceae (order Desulfobacterales) is the first genome from the genus Desulfonatronobacter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' It encodes the genes for complete sulfate reduction to sul- fide using various electron donors, as well as for the complete oxidation of volatile fatty acids and alcohols, a unique feature for the genus Desulfonatronobacter among haloalkaliphilic SRB [10] (see Additional file 8: Information S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Fumarate and nitrite (DNRA, NrfAH) could be used as alternative electron acceptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' The sec- ond dominant lineage was a new species from the genus Desulfonatronospira (family Desulfohalobiaceae, order Desulfovibrionales).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Like other members of this genus, it had the potential to reduce or disproportionate partially reduced sulfur compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' In addition, it could also use nitrite as an alternative electron acceptor (NrfAH) [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' A novel lineage of gammaproteobacterial SOB was highly abundant in the sediments of the moderately hy- persaline Cock Soda Lake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' It appeared as a sister group of the family Xanthomonadaceae in the ribosomal protein tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' This heterotrophic organism could conserve energy through aerobic respiration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' It might detoxify sulfide by oxidizing it to elemental sulfur (sqr) with subsequent re- duction or disproportionation of the polysulfides (psrA) chemically formed from the sulfur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' It also encoded the po- tential for DNRA (nrfA and napC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Genes likely involved in sulfide detoxification (sqr and psrA) were found also in several other abundant MAGs of heterotrophs, including one new abundant species from the family of Nitrilirup- toraceae (class Nitriliruptoria, phylum Actinobacteria).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' We found a wide variety of carbohydrate-active enzymes in these MAGs, such as cellulases (GH1 family) in addition to genes for glycolysis and TCA cycle and a chlorophyll/bacteriochlorophyll a/b synthase (bchG gene).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' The latter was also found in other Actinobacteria from the genus Rubrobacter [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' No evidence was found for nitrile-degrading potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' A second novel, uncultured lineage of Gammaproteo- bacteria that was highly abundant at moderate salinities branched in our ribosomal protein tree as a sister group to the family Halothiobacillaceae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' The MAGs encoded for a versatile metabolism typical for purple non-sulfur bacteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' The MAGs contained puf genes, bch genes, genes for carotenoid biosynthesis (not shown), and a Calvin cycle for photoautotrophic growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Alternatively, energy may be conserved through aerobic respiration, while acetate and proprionate could be taken up via an acetate permease (actP) and further used for acetyl-CoA biosynthesis and carbon assimilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Since the sqr gene was present, but no dsr or sox genes, the organism might oxidize sulfide only to elemental sulfur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' One bin contained also nifDKH genes suggesting putative diazo- trophy, as well as a coenzyme F420 hydrogenase suggest- ing photoproduction of hydrogen [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' The abundant Euryarchaeota organism showed a clear preference for higher salinities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' We obtained one highly abundant MAG from the class Thermoplasmata that encoded a full-length 16S rRNA gene only distantly re- lated (91,2% identity, e value 0) to that of a member of the KTK 4A group found in a hypersaline endoevaporitic microbial mat [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' The abundant soda lake organism is likely a new genus and species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' All KTK 4A-related MAGs found here encoded for similar heterotrophic, fermentative metabolisms, with the potential for (See figure on previous page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=') Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' 5 Potential mechanisms for regulating the intracellular pH and cytoplasmic ion content in different CPR phyla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' a Membrane transporters, channels, and lipids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Peptidoglycan is depicted in gray and S-layer proteins in cyan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' b Predicted isoelectric points (bin width 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content='2) for the coding sequences of MAGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' A representative proteome is depicted for each phylum for which several members had a pronounced acidic peak (see also Additional file 2: Figure S11) Vavourakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Microbiome (2018) 6:168 Page 9 of 18 anaerobic formate and CO oxidation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' The KTK 4A might be also primary degraders since they encoded pu- tative cellulases (CAZY-families GH1, GH5) and chiti- nases (GH18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Interestingly, half of the MAGs encoded a putative chlorophyll/bacteriochlorophyll a/b synthase (bchG), which is highly unusual for Archaea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Although little can be inferred from the presence of only one marker gene, a functional bchG was previously also found in Crenarchaeota [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' The remaining two highly abundant Euryarchaeota-related MAGs belonged to a new species of Halorubrum (Additional file 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Key genes of the Wood-Ljungdahl pathway found in novel phylogenetic groups More than 50 MAGs were related to the family Syntro- phomonadaceae (class Clostridia, phylum Firmicutes) based on ribosomal protein phylogeny.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' All 16S rRNA gene sequences found in the MAGS had 86–95% iden- tity to sequences obtained from uncultured organisms related to the genus Dethiobacter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' While an isolated strain of Dethiobacter alkaliphilus is a facultative auto- troph that respires thiosulfate, elemental sulfur or polysulfides with hydrogen as an electron donor [42] or disproportionates sulfur [43], other haloalkaliphilic members of the Syntrophomonadaceae are reverse acetogens, oxidizing acetate in syntrophy with a hydro- genotrophic partner [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Two populations (different species, Additional file 6) were especially abundant in Cock Soda Lake (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' They encoded for a full CODH/ACS complex, the key enzyme for the reductive acetyl-CoA or Wood-Ljungdahl pathway (WL) and a complete Eastern branch for CO2 conversion to 5-methyl-tetrahydrofolate (Additional file 9) [45, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Acetogens use the WL to reduce CO2 to acetyl-CoA, which can be fixed into the cell or used to conserve en- ergy via acetogenesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Syntrophic acetate oxidizers, some sulfate reducing bacteria and aceticlastic methanogens run the WL in reverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Syntrophomonadaceae sp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' 2 encoded for a putative thiosulfate/polysulfide reductase as well as a phosphotransacetylase (pta) and an acetate kinase (ack) for the ATP-dependent conversion of acet- ate to acetyl-CoA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Although alternative pathways for the latter interconversion can exist, this second species has the complete potential for (reversed) acetogenesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Highly remarkable was the presence of a bacterial-type CODH/ACS complex and a near-complete eastern branch of the WL in a highly abundant species in Cock Soda Lake from the family Coriobacteriaceae (phylum Actinobacteria).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' This prompted us to scan all 871 MAGs for the presence of acsB encoding for the beta-subunit of the oxido-reductase module of CODH/ACS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' We con- firmed an encoded (near)-complete WL in several additional organisms belonging to phylogenetic groups not previously associated with this pathway [46] (Additional file 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' We removed the Coriobacteriaceae acsB genes from the final dataset to construct a phylo- genetic tree since they were < 500 aa (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' 6) but found seven MAGs from the OPB41 class within the Actino- bacteria (16S rRNA gene fragment identity 94–96%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' The eastern branch of WL can function independently in folate-dependent C1 metabolism [45], but the pres- ence of the Western-branch in a phylum that comprises mostly aerobic isolates is very surprising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' The WL in combination with the potential for acetate to acetyl-CoA interconversion (pta/ack) and a glycolysis pathway were also present in the soda lake MAGs from the phyla “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Handelsmanbacteria,” “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Atribacteria” (latter branched within the “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Acetothermia”), and the class LD1-PA32 (Chlamydiae), suggesting all these uncultured organisms might be heterotrophic acetogens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' However, it should be noted that a PFOR typically connecting glycolysis to the WL was only encoded in the LD1-PA32 MAGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' More- over, from the genetic make-up alone, it cannot be excluded that acetate is activated, and the WL run in reverse for syntrophic acetate oxidation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Finally, the novel acsB genes from soda lake Halanaerobiaceae, Natranaerobiaceae, and Halobacteroidaceae (Firmicutes) and from Brocadiaceae and Planctomycetaceae (Plancto- mycetes) disrupt the previously proposed dichotomy between Terrabacteria and Gracilicutes bacterial groups unifying 16S rRNA and acsB gene phylogenies [46] and suggest a far more complex evolutionary history of the WL pathway than previously anticipated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Discussion Extensive classical microbiology efforts have been already undertaken to explore the unique extremophilic microbial communities inhabiting soda lakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' These un- covered the presence of most of the functional groups participating in carbon, nitrogen, sulfur, and minor element cycling at haloalkaline conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' The main re- sults of this work are summarized in several recent re- views [1, 6, 47, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Since most microbes, including those living in soda lakes, still evade all cultivation ef- forts, a very effective way to discover new microbes and assess their physiology based on their genetic repertoire is either through single cell genomics or by directly se- quenced environmental DNA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' This exploratory metage- nomics study performed on soda lake sediments effectively overcame the existing cultivation bottleneck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' First, we expanded the known diversity of CPR consider- ably with the first genomes of poly-extremophiles sam- pled from soda lake sediments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Although the presence of 16S rRNA genes from CPR in marine sediments and hy- persaline microbial mats was previously shown [34], until now, CPR MAGs were mainly obtained from deep, subsurface environments [15, 26, 29, 32, 49–52], and hu- man microbiota [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Despite being highly abundant Vavourakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Microbiome (2018) 6:168 Page 10 of 18 100 % 90-100 % 70-90 % 50-70 % some MAGs all MAGs Bootstraps Genes present Glycolysis (EMP) PFOR WL-Eastern branch H4MPT TH4 WL-Western branch CODH/ACS Acetogenesis/ acetate activation (pta/ack) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content='4 PVC group (Chlamydiae LD1-PA32) Syntrophorhabdus aromaticivorans PVC group bacterium CSSed11_184 Aerophobetes bacterium SCGC_AAA255-F10 Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Acetothermia Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Handelsmanbacteria Planctomycetaceae Anaerolineae Firmicutes Brocadiaceae Planctomycetes Methanomassiliicoccales Halobacteroidaceae Natranaerobiaceae Methanomicrobiales Desulfonatronospira Firmicutes Dehalococcoidia Armatimonadetes bacterium CSP1-3 Deltaproteobacteria Thermodesulfobacteria Desulfobulbaceae Halanaerobiaceae Nitrospirae Actinobacteria (OPB41) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' 6 Maximum likelihood phylogeny of the bacterial-type acetyl-coA synthases (acsB) found in the MAGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Only sequences ≥ 500 aa were included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Lineages for which we discovered the Wood-Ljungdahl (WL) in this study are highlighted in orange, and the presence of genes in the respective MAGs related to WL, glycolysis, pyruvate, and acetate conversion is indicated by the colored symbols (see also Additional file 9: Dataset S6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Additional lineages found in this study are marked in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' The three was rooted according to [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Circles at the nodes show confidence percentage of the bootstraps analysis (100×).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' EMP = Embden-Meyerhof-Parnas, PFOR = pyruvate:ferredoxin oxidoreductase complex, pta = phosphotransacetylase gene, ack = acetate kinase gene, H4MPT = tetrahydromethanopterin-linked pathway, TH4 = tetrahydrofolate pathway, CODH/ACS = carbon monoxide dehydrogenase/acetyl-CoA synthase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' PVC group bacterium CSSed11_184 is likely a member of the WCHB1-41 class within the Verrucomicrobia Vavourakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Microbiome (2018) 6:168 Page 11 of 18 here, CPR went unnoticed in previous amplicon sequen- cing studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' This might be due to the fact that many CPR representatives have random inserts of various length in their 16S rRNA genes or due to primer mis- matches [29, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' This illustrates also that direct metage- nomics should not only be preferred over amplicon sequencing to infer functional potential, but the former is far more effective for the discovery of novel organ- isms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Second, we obtained many more genomes from “traditional” bacterial phyla such as the Planctomycetes and Chloroflexi, as well as candidate phyla, for which no soda lake isolates, hence no genomes were previously obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Third, even within the sulfur cycle, the most active and frequently studied element cycle in soda lakes [1], we found considerable metabolic novelty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Finally, we found the Wood-Ljungdahl pathway in several novel phyla, not closely related to any known acetogens, methanogens, or sulfate-reducing bacteria [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' The lat- ter shows that our sequencing recovery effort has also significantly contributed to the discovery of metabolic novelty within various prokaryote phylogenetic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Salinity is often considered to be the major factor shaping prokaryote community composition in diverse habitats [53, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Extreme halophilic Euryarchaeota seem to be always the dominant group in salt-saturated hypersaline brines, both those with neutral or alkaline pH [1, 7, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Here, we found that although these haloarchaea are still relatively more abundant in the sed- iments exposed to brines with salt-saturating conditions, the clear majority of microbes in all investigated hyper- saline soda lake sediments are Bacteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' It could be hypothesized that the sediment is a hide-out for the extreme alkalinity and salinity governing the water column, and that sediment stratification, especially in the anoxic part, offers plenty of opportunities for niche diversification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' On the other hand, it should no longer be a surprise that soda lakes are such productive and biodiverse systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' First, it has been previously elaborated that soda lake organisms are exposed to approximately half the osmotic pressure in sodium carbonate-dominated brines compared to sodium chloride-dominated brines with the same Na+ molarity [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Second, nitrogen limitation in the community can be overcome when many members contribute to the fixation of atmospheric N2, and various forms of organic nitrogen are efficiently recycled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' The soda lakes exam- ined in this study were also eutrophic, and sulfur com- pounds were abundant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Sulfide is also far less toxic at high pH as it mostly occurs in the form of bisulfide (HS−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Besides the evident high metabolic and taxo- nomic diversity of dissimilatory sulfur-cycling bacteria, a diverse heterotrophic community can be sustained com- prising both generalist and very specialized carbon de- graders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Less eutrophic soda lakes might not suffer from carbon limitation either, due to a presence of high-bicarbonate concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' These effectively elim- inate the inorganic carbon limitation for primary pro- ducers who are highly active in soda lakes, especially Cyanobacteria [55, 56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Third, light that penetrates the surface of the sediment seems to stimulate oxygenic and anoxygenic phototrophic growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Moreover, various het- erotrophs, such as the rhodopsin-containing haloarchaea and Bacteroidetes, have the option to tap into this un- limited energy source for example to help sustain the costly maintenance of osmotic balance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Unexpectedly, we even found the first rhodopsin encoded by a member of the CPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Fourth, tight syntrophic relations, as pro- posed for CPR members and Syntrophomonadaceae spp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=', might be the solution to successful growth in an energetically challenging environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Since our metagenomes are snapshots in time and space, the failure to reconstruct specific MAGs gives no conclu- sive evidence for the absence of certain microbial-mediated element transformation in hypersaline soda lake sediments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Additionally, technical limitations of the assembly and bin- ning of highly micro-diverse genome populations might hamper genome recovery [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' More importantly, the abundance of a specific microbe is not necessarily corre- lated to the importance of its performance in an ecosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Many metabolic capacities are redundant, and often key transformations are reserved for a few rare organisms that might proliferate for a short time-span when specific condi- tions allow for it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' For example, although no MAGs were re- covered from chemolithoautotrophic nitrifiers [58], we did detect a Nitrobacter-related OTU by amplicon sequencing and assembled 16S rRNA genes from Thaumarchaeota, suggesting bacterial and archaeal nitrifiers are present in the surface sediments of soda lakes at very low abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Finally, the method of DNA isolation might impact the community composition apparent in the final metagenome sequenced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Environmental samples contain complex mix- tures of different organisms, and it is impossible to find a protocol where the DNA from every single organism is ex- tracted as efficiently without compromising the final quality of the extracted DNA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' However, since we find all the im- portant taxonomic and functional groups known from pre- vious cultivation-dependent studies back in either our amplicon sequencing datasets or our directly sequenced metagenomes, we are confident that the community com- position and the MAGs presented here are representative for the microbiomes of the soda lake sediments in the Kulunda Steppe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Conclusion Years of intensive microbiological research on soda lakes seem to have paid off, since many of the described gen- era we could detect here have a cultured representative from soda lakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' However, as many of the abundant Vavourakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Microbiome (2018) 6:168 Page 12 of 18 lineages and groups found in soda lake sediments are still uncultured, metagenomics proved to be a helpful tool to gain primary insights in the potential physiology and ecology of these poly-extremophilic prokaryotes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' We reconstructed the first genomes for many of such organisms and proposed new functional roles for the most abundant ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Future studies should provide more in depth analyses of these genomes, especially from the less abundant organisms that might perform key ecological processes, such as methanogens and nitri- fiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' In addition, they should focus on gaining physio- logical culture-based evidence or proof for in situ activity for the abundant organisms described here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' The key metabolic insights provided by this metagenomics study can lead to the design of new cultivation strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' In general, sediment communities are far more complex than those found in the corresponding water column [53, 59] and are therefore often considered too complex for efficient metagenomic analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Many of the novel lineages found here may therefore have related neutro- philic lineages in marine and freshwater sediments that await discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' We demonstrate here that, by providing sufficient sequencing depth, the “state of the art metage- nomics toolbox” can effectively be used on sediments as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Methods Site description and sample collection The top 10 cm sediments from four hypersaline, eutrophic soda lakes located in the Kulunda Steppe (south-western Siberia, Altai, Russia) were sampled in July of 2010 and 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' General features and exact location of the sampled soda lakes are summarized in Additional file 1: Table S1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' a map of the area was published previously [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Cock Soda Lake (a stand-alone lake, sampled both in 2010 and 2011) and Tanatar-3 (Tanatar system) were moderately hypersa- line (~ 100 g L−1) with sandy sediment, while Tanatar-1 and Bitter-1 (Bitter system) were salt-saturated (400 g L−1) with sulfide-rich sapropel sediments underlined by rock trona deposits [7, 60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Especially, Bitter-1 harbors a very active microbial community, probably due to its high- organic and -mineral content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Surface sediments were col- lected by a plastic corer into sterile glass containers and transported to the laboratory in a cooler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' DNA isolation, 16S rRNA amplicon, and metagenomic sequencing The colloidal fraction of each sediment sample (~ 10% of 50 g) was separated from the course sandy fraction by several short (30–60 s) low-speed (1–2,000 rpm in 50 mL Falcon tubes) centrifugation steps and washed with 1–2 M NaCl solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' The pelleted colloidal sedi- ment fraction was first subjected to 3 cycles of freezing in liquid nitrogen/thawing, then re-suspended in 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content='1 M Tris (pH 8)/10 mM EDTA, and then subjected to harsh bead beating treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Next, the samples were incu- bated with lysozyme (15 mg/mL) for 2 h at 37 °C followed by a SDS (10% w/v) and proteinase K (10 μg/ mL) treatment for 30 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' at 45 °C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' High molecular weight DNA was isolated using phenol/chloroform ex- traction, quality-checked, and sequenced as described previously [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Direct high-throughput sequencing of the DNA was performed on an Illumina HiSeq 2000 plat- form to generate 150 b paired-end reads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Amplification of the V4-V6 region of prokaryote 16S rRNA genes using barcoded 926F-1392R primers, amplicon purifica- tion, quantification, and Roche (454)-sequencing was performed together in a batch with brine samples from the same sampling campaigns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Barcodes and adapter se- quences were removed from de-multiplexed amplicon sequence reads and analyzed with the automated NGS analysis pipeline of the SILVA rRNA gene database pro- ject [61] (SILVAngs 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content='3, database release version 128) using default parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' The OTUs (97% identity) assigned down to the genus level were only considered when they had a relative abundance ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content='1% in at least one of the five datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Processing metagenomics reads, assembly, binning, and post-binning Metagenomic raw reads were quality trimmed using Sickle [62] (version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content='33), and only reads ≥ 21 b were retained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' The prokaryotic community structure at taxo- nomic top levels was extrapolated from ten million ran- domly sampled singletons from each dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Candidate 16S rRNA fragments > 90 b were identified [63] and compared against the SILVA SSU database 128 (blastn, min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' length 90, min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' identity 80%, e value 1e-5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' To ver- ify that the microbial community composition was in- deed mostly prokaryotic, we did a more general screening of the metagenomics reads that identified also candidate 18S rRNA fragments > 90 b (see Additional file 1: Tables S4-S5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' The complete trimmed read sets were assembled into contigs ≥ 1 kb with MEGAHIT [64] (v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content='3–6-gc3983f9) using paired-end mode, k min = 21, k max = 131, k step = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Genes were predicted using Prodigal [65] (v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content='2) and RNAs with rna_hmm3 [66] and tRNAscan-SE [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Assembled 16S rRNA sequences were compared to a manually curated version from the SILVA SSU database (e value ≥ 1e-5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Predicted protein sequences were annotated against KEGG with GhostKOALA (genus_prokaryotes + family_eukaryotes + viruses) [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Marker genes for central metabolic pathways and key environmental element transforma- tions were identified based on K number assignments [15, 69–71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Contigs ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content='5 kb were binned with METABAT [72] (superspecific mode) based on differential coverage Vavourakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Microbiome (2018) 6:168 Page 13 of 18 values obtained by mapping all five trimmed readsets to all five contig sets with Bowtie2 [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' The bins were sub- jected to post-binning (an overview of the workflow is given in Additional file 2: Figure S13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Bins were assessed with lineage-specific single copy genes using CheckM [74] and further processed with the metage- nomics workflow in Anvi’o [75] (v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Since Candidate Phyla Radiant (CPR) is not included in the CheckM ref- erence trees and are likely to have low-genome com- pleteness, we used an existing training file of 797 CPR genomes to identify putative CPR bins [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Bins with CheckM-completeness ≥ 50% (884 out of 1778) and an additional four CPR bins were further processed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Coding sequences were annotated for taxonomy against NCBI-nr (July, 2017) with USEARCH [77] (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content='32) to verify that most hits in each bin were to prokaryotic ref- erences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Phage or viral contigs were manually removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Genome contamination (redundancy) was estimated based on marker sets of universal single copy genes identified for Bacteria [30] and Archaea [78] as imple- mented in Anvi’o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Genome coverage was obtained by mapping trimmed reads with BBMap [79] v36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content='x (kfilter 31, subfilter 15, maxindel 80).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Bins with ≥ 5% redun- dancy were further refined with Anvi’o using circle phy- lograms (guide trees tnf-cov: euclidian ward) and scanned again for CPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Post-binning resulted in a total of 2499 metagenome-assembled genomes (MAGs), of which 871 were either medium-quality genome drafts (CheckM estimated completeness ≥ 50% and contamin- ation ≤ 10% [80], Additional file 4) or lower quality draft genomes from CPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Phylogeny of the MAGs was assessed based on 16 single-copy ribosomal proteins and representative refer- ence genomes of major prokaryote lineages across the tree of life [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Individual ribosomal proteins in our MAGs were identified by K number assignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Only ribosomal proteins ≥ 80 aa were considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Initial maximum-likelihood (ML) trees were constructed to de- termine which organisms belonged to the Archaea, Bac- teria, or CPR with FastTree 2 [81] (WAG + CAT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Final separate trees for the three distant evolutionary groups were constructed in the same manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Each ribosomal protein set was aligned separately with MAFFT [82] (v7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content='055b, − auto) and concatenated only if a MAG encoded at least 8 out of 16 proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' For all trees, a 100× posterior bootstraps analysis was performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Phylogenetic trees were visualized together with gen- ome statistics and abundance information using iTOL [83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' We cross-checked the taxonomic assignments based on the phylogeny of the ribosomal protein cas- sette with the top hit contig annotations against NCBI-nr and with the reference lineage obtained with CheckM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Lastly, we manually corrected the MAGs for misplaced 16S rRNA genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' The final trees presented in the manuscript were redrawn using FigTree v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content='3 [84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Detailed genome analyses CPR MAGs were re-annotated more thoroughly: genes were predicted with Prokka [85], and functional predictions were performed by running InterProScan 5 locally on the supplied COG, CDD, TIGRFAMs, HAMAP, Pfam, and SMART databases [86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' BLAST Koala was used for KEGG pathway predictions [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' To find putative carbohydrate-active enzymes in all final MAGs, we used the web-resource dbCAN [87] to annotate all predicted proteins ≥ 80 aa against CAZy [88].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' To identify the top ten abundant MAGs from each re- spective dataset, ten million randomly sampled single- tons were mapped onto each MAG with a cut-off of 95% identity in minimum of 50 bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Coverage values were additionally normalized for genome size and expressed as reads per kilobase of sequence per gigabase of mapped reads (RPKG) [89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' A positive score (from 871 to 1) was assigned to each MAG according to the rank- ing of the summed RPKG of MAGs in the high-salinity datasets (B1Sed10 and T1Sed) and a negative score ac- cording to the ranking of the summed RPKGs in the moderate salinity datasets (CSSed10, CSSed11, T3Se d10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Both scores were summed to get a “salinity prefer- ence score” with MAGs recruiting preferably from high salinity datasets on the positive end, moderate salinity datasets in the negative end, and those without prefer- ence in the middle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' We determined species delineation for the most abundant MAGs and their closest reference genomes (NCBI-nr) by Average Nucleotide Identity (ANI) and conserved DNA-matrices, as follows [90]: ANI ≥ 95%, conDNA ≥ 69% = same species, ANI ≥ 95%, condDNA < 69% = might be same species, ANI < 95%, condDNA < 69% = different species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Single gene trees based on maximum likelihood were constructed with un- trimmed alignments (MAFFT, L-INS-i model) and FastTree 2 (WAG + CAT, increased accuracy, -spr4 mlacc 2 -slownni) using 100× bootstraps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' References were pulled from eggNOG (v4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content='1) [91] and supple- mented with sequences from NCBI-nr or refined according to [7, 33, 46, 92–94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' The curated MAGs were scanned for the presence of rhodopsin sequences with the hmmsearch software [95] and a profile hidden Markov model (HMM) of the bacteriorhodopsin-like protein family (Pfam accession number PF01036).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' The identified sequences with significant similarity were aligned together with a curated database composed of a collection of type-1 rhodopsins, using MAFFT (L-INS-i accuracy model) [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' This protein alignment was further utilized to Vavourakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Microbiome (2018) 6:168 Page 14 of 18 construct a maximum likelihood tree with 100× boot- strap with FastTree 2 [81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' All other genes were identified using the KEGG annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Additional files Additional file 1: Table S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' General features of the four sampled soda lakes at time of sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Table S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' SILVA classification of the 16S rRNA gene sequences found in all ≥1 kb contigs of five soda sediment metagenomic datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Table S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Enzymes involved in lipopolysaccharide biosynthesis found among different members of the CPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Table S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Sub-kingdom classification of candidate SSU rRNA gene fragments found in subsamples of 10 million random forward reads from the five soda sediment metagenomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Table S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Top-level taxonomic classification of the 18S rRNA gene fragments found in subsamples of 10 million random forward reads from the five soda sediment metagenomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Table S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Description of the metagenomic datasets, NCBI Sequence Read Archive (SRA) accession numbers and general statistics of the assembled contigs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' (PDF 740 kb) Additional file 2: Figure S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Taxonomic fingerprints determined by 16S rRNA gene amplicon sequencing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Figure S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Genome statistics of the 871 MAGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Figure S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Phylogeny of MAGs belonging to “Candidatus Aenigmarchaeota” and “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Nanohaloarchaeota”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Figure S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Phylogeny of MAGs related to “Candidatus Acetothermia”, candidate division WS1 and “Candidatus Lindowbacteria”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Figure S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Phylogeny of MAGs related to candidate division KSB3 and “Candidatus Schekmanbacteria”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Figure S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Multiple sequence alignment of the V-type ATPase subunits K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Figure S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Multiple sequence alignment of the F-type ATPase subunits c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Figure S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Maximum likelihood tree of the large subunits of RuBisCo and RubisCo- like proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Figure S9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Maximum likelihood tree of the putative rhodopsins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Figure S10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Predicted isoelectric points (pI) profiles of all MAGs from CPR members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Figure S11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Predicted isoelectric points profiles for members of the “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Nealsonbacteria” and “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Vogelbacteria”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Figure S12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Multiple sequence alignment of the dissimilatory cytochrome c nitrite reductases (nrfA/TvNiR, K03385).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Figure S13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Overview of the post-binning workflow used for genome recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' (PDF 6548 kb) Additional file 3: Dataset S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Relative abundance of the OTUs assigned to the genus-level within the Archaea, Bacteria and organelles from Eukaryota detected by 16S rRNA gene amplicon sequencing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' The OTUs with less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content='1% abundance accross all five datasets are not shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' The names of highly abundant genera (≥1% in at least one of the data- sets) are shown in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' (XLSX 24 kb) Additional file 4: Dataset S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Organism names, statistics and general description incl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Completeness and contamination estimates, phylogeny and DDBJ/EMBL/Genbank accession numbers of the metagenome assembled genomes (MAGs) described in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' All submitted versions described in this paper are version XXXX01000000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Size = recovered genome size, Completeness (Compl1), contamination (Cont), strain heterogenity (Str het) and Taxon CheckM were inferred from lineage-specific marker sets and a reference tree build with CheckM [74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Additional completeness (compl2) and redundancy (red) estimates were inferred based on the presence of universal single copy genes for Bacteria and Archaea [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Decision and confidence intervals from the Candidate Phyla Radiation (CPR) scan [75] are given, as well as the taxonomy of the besthit in SILVA when 16S rRNA genes were present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Phylum/class 16 ribosomal proteins is the taxonomy derived from our ribosomal protein trees (see main text: Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' 2 and 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' OTU gives the inferred link of a population genome with our 16S rRNA gene amplicon dataset (Additional file 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' (XLSX 253 kb) Additional file 5: Dataset S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Estimated abundance and derived salinity preference from each MAG in each metagenomic dataset expressed as Reads per Kilobase of MAG per Gigabase of mapped reads (RPKG) and “salinity preference score” (see Methods section), basis for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' (XLSX 143 kb) Additional file 6: Dataset S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Average Nucleotide Identity (ANI) and conserved DNA (condna) matrices to determine species delineation between the most abundant MAGs shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' 4, closely related (less abundant) MAGs and NCBI reference genomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Decision matrix shows: 1 = same species, − 1 = might be same species, 0 = different species (see Methods section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' (XLSX 1161 kb) Additional file 7: Dataset S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Sheet 1 Presence and absence of marker genes and putative carbohydrate-active enzymes in the MAGs to infer putative roles in C, N and S element cycles based on K-number assignments and CAZy annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Sheet 2 Summary basis for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' (XLSX 41 kb) Additional file 8: Information S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' More detailed description of the main metabolisms encoded by Thioalkalivibrio-related MAGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Information S2 More detailed description of the main metabolisms encoded by Deltaproteobacterial-related MAGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' (PDF 219 kb) Additional file 9: Dataset 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Sheet 1 shows the MAGs positive for the marker gene acsB (K14138) encoding an acetyl-CoA synthase (ACS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' The basis for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' 6, namely presence and absence of key genes involved in the Wood-Ljungdahly pathway, acetogenesis, methanogenesis, glycolysis and pyruvate to CO2 conversion is shown for each MAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Sheet 2 shows the MAGs positive for the marker gene cdhC (K00193) encoding for the beta subunit of an acetyl-CoA decarboxylase synthase complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' While acsB and cdhC correspond roughly to the Bacterial-type and Archaeal- type (methanogens) enzymes with the same function, we found few discrepancies between marker gene and genome phylogeny within the Methanomassiliicoccaceae and Chloroflexi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' (XLSX 52 kb) Acknowledgments We thank Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Nikolai Chernych for his technical assistance during the isolation and purification of metagenomics DNA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' We also thank the Department of Energy Joint Genome Institute for sequencing the metagenomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Funding CDV and GM were supported by the ERC Advanced Grant PARASOL (no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' 322551).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' A-SA and RG were supported by the research grant 17-04828S from the Grant Agency of the Czech Republic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' MM was supported by the Czech Academy of Sciences (Postdoc program PPPLZ application number L200961651).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' DYS was supported by the SIAM/Gravitation Program (Dutch Ministry of Education and Science, grant 24002002) and by the Russian Science Foundation (grant 16–14- 00121).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Sequencing was performed by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Department of Energy Joint Genome Institute, a DOE Office of Science User Facility, as part of the Community Sequencing Program (contract no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' DE-AC02- 05CH11231).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Availability of data and materials The raw sequence reads of the five metagenomes have been deposited to the NCBI Sequence Read Archive (see Additional file 1: Table S6 for accession numbers and read and contig statistics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' The final 871 MAGs described in this paper have been deposited as Whole Genome Shotgun projects at DDBJ/ EMBL/GenBank, and accession numbers are listed in Additional file 4 (BioProject ID PRJNA434545).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' All versions described in this paper are version XXXX01000000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' The cleaned and dereplicated amplicon sequence datasets are available in FigShare (https://figshare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content='com/s/7684627445e3621aba24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Maximum likelihood trees based on the concatenated alignment of 16 ribosomal proteins, basis for Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' 2 and 3, in newick format (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content='tre file) and complementary datasets (used to plot completeness, contamination, genome recovery size, G + C mol% and RPKG in iTOL), as well as K number assignments for the predicted proteins of all MAGs (KEGG-orthologues, Ghost Koala) and the fully annotated CPR MAGs supporting the conclusions of this article are also available in FigShare (https://figshare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content='com/s/ 7684627445e3621aba24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Authors’ contributions GM and DYS initiated this study and were responsible for the fieldwork, sample preparation, and sequencing effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' CDV conceptualized the research goals under supervision of DYS and GM, and performed the bioinformatics analysis under close guidance of A-SA and RG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' CDV is the primary author of this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' MM, RG, and CDV prepared the main figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' All authors read and approved the final manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Ethics approval and consent to participate Not applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Vavourakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Microbiome (2018) 6:168 Page 15 of 18 Consent for publication Not applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Competing interests The authors declare that they have no competing interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Author details 1Microbial Systems Ecology, Department of Freshwater and Marine Ecology, Institute for Biodiversity and Ecosystem Dynamics, Faculty of Science, University of Amsterdam, Postbus 94248, 1090, GE, Amsterdam, the Netherlands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' 2Department of Aquatic Microbial Ecology, Institute of Hydrobiology, Biology Centre CAS, Na Sadkach 7, 370 05 Ceske Budejovice, Czech Republic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' 3Winogradsky Institute of Microbiology, Research Centre of Biotechnology, Russian Academy of Sciences, 60 let Oktyabrya pr-t, 7, bld.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' 2, Moscow, Russian Federation117312.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' 4Environmental Biotechnology, Department of Biotechnology, Delft University of Technology, Van der Maasweg 9, 2629, HZ, Delft, the Netherlands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Received: 23 June 2018 Accepted: 3 September 2018 References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Sorokin DY, Berben T, Melton ED, Overmars L, Vavourakis CD, Muyzer G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_36/content/kb_36.pdf'} +page_content=' Microbial diversity and biogeochemical cycling in soda lakes.' metadata={'source': 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0000000000000000000000000000000000000000..64193c9aa7226c165805f81430e39b944cd53080 --- /dev/null +++ b/kdE_T4oBgHgl3EQf5hys/content/tmp_files/2301.08359v1.pdf.txt @@ -0,0 +1,1367 @@ +Deep Reinforcement Learning for Gas Trading +Yuanrong Wang +University College London +London, UK +Shell Global Solutions International +UK +yuanrong.wang@cs.ucl.ac.uk +raymond.wang2@shell.com +Yinsen Miao +Shell International Exploration and +Production +USA +yinsenm@gmail.com +Alexander CY Wong +Shell Global Solutions International +The Netherlands +alexwong_92@hotmail.com +Nikita P Granger +Shell Energy North America +USA +Nikita.Granger@shell.com +Christian Michler∗ +Shell Global Solutions International +The Netherlands +C.Michler@shell.com +ABSTRACT +Deep Reinforcement Learning (Deep RL) has been explored for a +number of applications in finance and stock trading. In this pa- +per, we present a practical implementation of Deep RL for trading +natural gas futures contracts. The Sharpe Ratio obtained exceeds +benchmarks given by trend following and mean reversion strategies +as well as results reported in literature. Moreover, we propose a +simple but effective ensemble learning scheme for trading, which +significantly improves performance through enhanced model sta- +bility and robustness as well as lower turnover and hence lower +transaction cost. We discuss the resulting Deep RL strategy in terms +of model explainability, trading frequency and risk measures. +CCS CONCEPTS +• Computing methodologies → Reinforcement learning. +KEYWORDS +Deep Reinforcement Learning, Gas Trading, Systematic Trading +ACM Reference Format: +Yuanrong Wang, Yinsen Miao, Alexander CY Wong, Nikita P Granger, +and Christian Michler. 2022. Deep Reinforcement Learning for Gas Trading. +In Proceedings of (Conference’17). ACM, New York, NY, USA, 8 pages. https: +//doi.org/XXXXXXX.XXXXXXX +1 +INTRODUCTION +The profitability of a trading strategy hinges on a good timing of +entering and exiting a position. Researchers, traders and quants +have been exploiting fundamental and technical analysis to analyze +the market with the aim of predicting future market movements +ever since. However, among the ever increasing complexity and the +∗Corresponding author. +Permission to make digital or hard copies of all or part of this work for personal or +classroom use is granted without fee provided that copies are not made or distributed +for profit or commercial advantage and that copies bear this notice and the full citation +on the first page. Copyrights for components of this work owned by others than ACM +must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, +to post on servers or to redistribute to lists, requires prior specific permission and/or a +fee. Request permissions from permissions@acm.org. +Conference’17, , +© 2022 Association for Computing Machinery. +ACM ISBN 978-x-xxxx-xxxx-x/YY/MM...$15.00 +https://doi.org/XXXXXXX.XXXXXXX +notoriously low signal-to-noise ratio of financial markets, many +expert-designed rule-based strategies fail to cope with changing +market conditions. Recent advancements in Artificial Intelligence +and data science start to bring a competitive edge to trading by +learning the market dynamics. With the vast amount of historical +data, models’ non-linear learning and inference ability extract pat- +terns in time series and exploit volatility and directional signals for +position taking. +Natural gas is one of the most liquid commodity markets. En- +ergy futures and options contracts are usually traded on deriva- +tive marketplaces such as the Chicago Mercantile Exchange (CME) +or over-the-counter. Given the high volatility and similar market +dynamics between gas and equity markets, the classic technical +indicators are considered to still provide analytical signals for the +underlying gas price. However, as commodities are influenced more +by supply and demand, the fundamental analysis is very different +from stocks that are mostly correlated to the valuation of underly- +ing firms. Key contributing factors to the gas price movements are +often macro economic, including regional storage and production, +global demand, alternative power production as well as weather. +These differences prevent many stock traders to enter the energy +market. Nevertheless, some trading algorithms have shown to be +transferable, see e.g. [16, 20, 23]. +Broadly speaking, quantitative models in commodity trading +focus on the following aspects: Event-driven traders react to sys- +tem events and failures. Similar to how corporate news influences +the price of stocks, a failure in a transmission system or a genera- +tor [13, 25] and sometimes in a policy [36] can greatly impact the +price of energy. Then, technical analysis and fundamental analysis +are two main methodologies in the market, where strategies learn +from past data for future price forecasting. An early success story +of Moshiri [32] employed a non-linear Artificial Neural Network +model to predict crude oil futures which outperformed the tradi- +tional econometric-based models. Since then, ensemble learning +methods by Yu and Jammazi [17, 53] combined Machine Learn- +ing (ML) and econometric models by Godarzi and Zhang [9, 54], +and also deep learning models by Zhao, Tang, Zhao and Safari +[37, 45, 56, 57] have been proposed and designed. More recently +with the advancement in hardware and execution speed, both ag- +gressive and passive high frequency trading algorithms have been +arXiv:2301.08359v1 [q-fin.TR] 19 Jan 2023 + +Conference’17, , +Y.Wang et al. +applied in commodity trading, see [28]. As the order book structure +in natural gas futures on CME and equities on stock exchanges +are similar, momentum ignition, order anticipation and arbitrage +trading have been explored, see, e.g., Fishe [8]. +Despite the abundance of data and learning ability of ML mod- +els, many issues arise from the noisy time-series data, especially +in finance, and biased supervised labelling in the trading environ- +ment. An extremely low signal-to-noise ratio in financial time- +series is an inherent impediment despite considerable efforts in +time-series filtering by means of, e.g., information filtering net- +works [26, 27, 46, 50]. Moreover, the general two-step approach in +constructing a trading strategy is ill-posed, see [30]: Firstly, a super- +vised model forecasts asset price changes with a defined investment +horizon. Then, the forecasts are fed to a strategy module to generate +actual trading strategies. In the first step, the supervised models +are normally labelled by future prices with a defined investment +horizon. This approach limits generalization of the model. In addi- +tion, besides the predictions, other features, e.g., liquidity, market +micro-structure, are usually not included in the second step. This +ideal simplification of market dynamics fails to explicitly address +the interaction between a trading strategy and the real market in +the form of market impact during execution. +Reinforcement Learning (RL) agents learn policies (strategies) +directly by optimising a numerical reward signal by interacting +with a virtual market environment that is usually derived from +historical market data. This has been shown to prevent many of +the aforementioned issues in supervised ML by Sutton [42]. Instead +of back-propagating the loss between labelled ground truth and +the prediction, a simulation environment is built based on market +data for Deep RL agents to explore and exploit. Next, actions are +evaluated based on the reward from the interaction with the market +environment, and a policy (trading strategy) is learned through it- +erative interactions. In 2013, Mnih[29] published the seminal work +of Deep Q-learning Network (DQN), which marks the transition +from Reinforcement Learning to Deep Reinforcement Learning. +Since then, more algorithms have been proposed, e.g, PPO [38], +and DDPG [22]. Early literature has already attempted to apply +traditional RL in the financial market, like stock pricing and selec- +tion [7, 21, 44], optimal trade execution [2, 19, 34], high frequency +trading [39, 48] and portfolio trading [11, 18, 33]. In the last few +years, Spooner [41] has used several Time-Difference methods for +market making. [51] applied DDPG for stock trading. RoaVicens +[35] has embarked on simulating an order book environment in +the presence of competitive agents. Yang [52] researched ensemble +methods between different Deep RL agents. Other recent Deep RL +research in financial applications has been summarized by Hambly +[14]. Besides the academic community, JP Morgan in 2018 has re- +leased a working paper [1] introducing their RL based limit order +engine highlighting the potential for applications in trading. +In 2019, Zhang [55] compared DQN, PG, A2C algorithms for +equity, FX, fixed income and commodity trading as the current +state-of-the-art benchmark. Our work builds on the structure of the +DQN applied to commodity trading [55] and combines it with the +implementation from Udacity [47]. For performance improvement +and model explainability, we further incorporate a game-theoretical +SHAP (SHapley Additive exPlanations) plot from Lundberg[24] +for feature interpretation and selection. Furthermore, to address +the high compute cost of training Deep RL models discussed in +Gudimella [11], we leverage the Microsoft Bons.ai platform [10] +with its containerized simulation environments, high parallelism +and automated hyper-parameter tuning which accelerates model +training and testing. Yang [52] and Carta[5] emphasized that the +same model with different initializations can lead to different out- +comes due to the long path-dependency especially for problems in +trading. To prevent bias from back-testing and enhance robustness +of the model, Borokova [4] present an online ensemble learning +method for LSTM in high-frequency equity price prediction, where +models are weighted by their recent performance. Inspired by their +framework, we propose and design an ensemble learning scheme, +where agents are trained in parallel before filtering by training per- +formance, and then averaged by threshold voting. The implementa- +tion details are presented in Section 3, and results are discussed in +Section 4. We will discuss the resulting ensemble learning scheme +under several practical aspects in Section 6. +2 +PROBLEM DESCRIPTION +2.1 +Data +We trade the front month NBP UK Natural Gas Futures contract. +The future contracts are for physical delivery through the trans- +fer of rights in respect of Natural Gas at the National Balancing +Point (NBP) Virtual Trading Point, operated by National Grid, the +transmission system operator in the UK. Delivery is made equally +each day throughout the delivery period. The NBP virtual trading +point acts as a central exchange. The price during a trading day can +be characterized by a so-called candle of low, high, open and close +price [31]. Alongside the trading volume of the day, we shall refer +to these five features as price features. Other than the price features, +technical analysis is employed to construct technical features. In +this experiment, we compute MACD, Price RSI, Volume RSI, PCA +1st component and volatility adjusted returns of 1, 2 and 3 months +as additional features [6]. Furthermore, to investigate the effect +of natural gas fundamental features, we also include demand data +(industrial, gas to power and residential demand), production data +from UK production fields, LNG data in all 3 UK terminals, data of +pipeline imports from Norway, Netherlands, Belgium and Ireland +as well as storage data in all active facilities. A full list of features +is shown in Table 1. +3 +DEEP RL METHODOLOGY +3.1 +Set up of virtual market simulation +environment +In Reinforcement Learning, agents learn policies from interacting +with a simulation environment. Hence the environment needs to +be as realistic as possible. A complete market approximates the real +market where friction, market impact from orders, transaction cost +and asset liquidity exist. However, such highly interactive envi- +ronment is complex to model and computationally demanding to +simulate. Hence, a simplified incomplete market is customarily used +where only transaction cost is considered. Here, we shall adopt the +latter approach and consider transaction costs of 0.1p/therm. In +this environment, agents observe price, technical and fundamental +features at each time step with a look-back window to detect trends. + +Deep Reinforcement Learning for Gas Trading +Conference’17, , +Table 1: Table outlines the technical and fundamental fea- +tures used in the experiment. Technical features include can- +dlestick features and their derived differences, technical in- +dicators and volatility adjusted returns. Fundamental fea- +tures include demand, production, transportation and stor- +age data as well as liquefied natural gas (LNG) terminal data +in the UK. +The agent then outputs an action for this time step, which is evalu- +ated according to a reward function that guides and incentivizes +the learning. +We limit this experiment to a daily trading frequency, and add a +three-day look-back window to aforementioned features. Therefore, +the transition in the observational space moves forward the time- +stamp by a day. We limit the size of look-back window so that +agents consider the most recent information, and the specific choice +of the 3-day look-back window follows from a tradeoff between +model complexity and training efficiency. That is, each day an +agent perceives all the information from the present day and the +three previous days1. As the trading environment is set up for daily +frequency, the agent outputs the action for today based on the +learned policy. For simplicity, the action is taken to be discrete as +buy or sell the maximum amount, or hold. To avoid overfitting to +the data in the replay buffer, a 1% Gaussian noise is added to each +observation. +Learning optimal decisions is guided by the reward function as a +result of the agent’s interaction with the environment through ac- +tions 𝐴𝑡, and the reward𝑟𝑖,𝑡 in form of raw P&L. In addition, for sim- +ulation purposes, we fixed the transaction costs, 𝑡𝑐 = 0.1p/therm +based on the bid-ask spread. We employ reward shaping as follows: +The immediate reward after an action is taken as the P&L, ˆ𝑟𝑖,𝑡 to +guide the agent, and the performance of each episode is assessed at +the end of an episode by the annualized Sharpe Ratio 𝑆𝑅𝑖, where +subscript 𝑡 indicates the time-step in episode 𝑖. +1Note that for the day under consideration the agent can only see the open price of +the day, but not the close, high or low price of the day to prevent leaking future data +that would only be known after trading closes on that day. +ˆ𝑟𝑖,𝑡 = 𝐴𝑡−1𝑟𝑖,𝑡−𝑡𝑐|𝐴𝑡 − 𝐴𝑡−1| +ˆ¯𝑟𝑖 = 1 +𝑛 +𝑛 +∑︁ +𝑡=1 +ˆ𝑟𝑖,𝑡 ; +ˆ𝜎𝑖 = +𝑛 +∑︁ +𝑡=1 +√︂ +1 +𝑛 |ˆ𝑟𝑖,𝑡 − ˆ¯𝑟𝑖 |2 +ˆ𝑆𝑅𝑖 = +√ +252 +ˆ¯𝑟𝑖 +ˆ𝜎𝑖 +. +(1) +3.2 +Implementation +Bons.ai uses DQN Apex by [15] for discrete action spaces and SAC +by [12] for continuous action spaces. Algorithm selection can be +done automatically in the platform based on feature and action +settings. The platform then trains the model, and assessment can +be done separately. Note that the Bons.ai platform can only handle +a single agent. So for any ensemble method requiring interaction +between different realizations, we used a centralized approach for +training before a post-hoc ensembling. +On the other hand, we have built a version of the in-house code +with DQN algorithms, which serves as a verification and comple- +ment to the Bons.ai platform. Its main edge is the fine-grained +control over low-level neural network and simulation architectures, +which benefits the design of our ensemble learning method referred +to as Filtered-Thresholding detailed in Section 3.3. To better match +the high computation speed in Bons.ai, a parallel training scheme +has been designed not only in the local version, but also in a high- +performance-computing version. The empirical experiments show +that an average of 15,000-30,000 episodes are required for successful +training and achieving convergence of a single agent. +3.3 +Ensemble Learning for Virtual Book +During our extensive experiments, the initialization of different +agents has been found to be an influential factor for the learning +trajectory. An exact setup could even result in one agent learn- +ing successfully until converge while another agent being stuck +in a local minimum. Moreover, as the underlying algorithms are +model-free Deep RL, even all “successfully” learned agents will be- +have differently in terms of their underlying trading logic, e.g., one +agent may be relying more on momentum strategy while the other +agent may tend more towards mean-reversion trades. The strategy +preference could be inferred from the difference in holdings across +different agents in the same setup, which has been observed in an +analysis to holding positions in Section 5.1. +To moderate the impact from sub-optimal agents and difference +between underlying strategies, we propose a simple but effective en- +semble learning method, which we refer to as Filtered-Thresholding. +Firstly, 𝑁 number of agents are trained in parallel. Then, based on +the trainings curve, we exclude seemingly sub-optimal agents in +a “Filtering” step. After filtering out bad candidates, we ensemble +the decision-making process between converged agents. Multiple +ensemble methods have been considered, and the thresholding +method was chosen. Each agent at each time-step has three choices +/ positions that it can take, namely Buy, Sell and Hold. If more than +𝑝% of agents agree on an action, then the ensemble agent executes +the decision, otherwise it remains unchanged. In our experiment, +𝑝% = 50% is set for simplicity. This “Thresholding” step avoids large +turnover in transaction costs caused by extensively moving in and +out of positions, it reconciles any dissimilarities and integrates the + +Technical Features +Fundamental Features +Candlestick: +Demand Data +(Open, High, Low and Close price of the day)OHLC +(lndustrial, Gas to Power, Residential) +Volume +Differences: +Production Data +High - Low price of the dayi +(UK production fields) +High - Open price of the day, +Close - previous Close price of the day +Technical Indicators: +LNG Data +MACD +(All 3 UK LNG terminals) +Price RSl, Volume RSI +PCA 1st component +Returns: +Pipeline data +Volatility adjusted 1-month, +(Imports from Norway, NL, BE, IR) +Volatility adjusted 2-month, +Volatility adjusted 3-month, +Storage Data +(All active facilities)Conference’17, , +Y.Wang et al. +advantages of the underlying agents and their respective trading +logic. +4 +RESULTS +4.1 +Results from Bons.ai +In any time-series machine learning problem, the training period is +a crucial parameter that impacts the behaviour and performance of +the algorithm. Besides computational cost, a long training period +recognizes longer-term tendencies, while a short training period +captures more local temporal patterns. This is especially true for +Deep RL. Although prioritised experience replay attempts to empha- +size the most recent pattern, the low signal-to-noise ratio inherent +in financial time series still poses a challenge. Two walk-forward +training schemes are compared in Table 2 and Table 3 for anchored +and sliding-window approaches, respectively. +Table 2: Training and testing performance table for Bons.ai +DQN Apex with anchored window, where rows denote years +and columns denote different versions of the train/test data +split. Each version corresponds to a different split between +training and testing period with the numerical value indicat- +ing the out-of-sample Sharpe Ratio. The yellow and white +slots indicate training year, and the light green slots are the +immediate testing years after training. The light blue slots +provide yet another test data set by testing on future data +that is further out than the subsequent year. +From a 12-year dataset of 2009 to 2020, the anchored window +starts with a minimum 4-year training period, and evaluates per- +formance in the subsequent year. After each evaluation, the year +that has just been used as out of sample test set is added to the +training period for the next walk-forward step. In Table 2, we use +the parallelization in the Bons.ai platform to train eight Bons.ai +brains simultaneously. In the anchored window approach the size of +the training window grows over time, while in the sliding-window +approach the training window length remains fixed, here to four +years of training. +The performance summary of the two walk-forward schemes +with APEX DQN are shown in Table 4 along with the Soft-Actor +Critic(SAC) Deep RL agent. The table compares the performance of +these three agents in terms of cumulative P&L, average Sharpe Ratio +and maximum draw-down. The APEX DQN moving window has an +average Sharpe Ratio of 1.32 and cumulative P&L of 48.32 million +GBP, compared to a slightly lower performance of the anchored +window with a Sharpe Ratio of 1.27 and a cumulative P&L of 44.95 +Table 3: Training and testing performance table for Bons.ai +DQN Apex with sliding window, where rows denote years +and columns denote different versions of the train/test data +split. Each version corresponds to a different split between +training and testing period with the numerical value indicat- +ing the out-of-sample Sharpe Ratio. The yellow and white +slots indicate training year, and the light green slots are the +immediate testing year after training. The light blue slots +provide yet another test data set by testing on future data +that is further out than the subsequent year. +Table 4: Result table summarizing the performance of dif- +ferent Bons.ai agents. The average Sharpe Ratio, maximum +draw-down and cumulative P&L are reported for Bons.ai +APEX DQN with anchored-window and moving-window +training approach as well as for the SAC agent. +million GBP. The above summary statistics suggests a generally +comparable performance of the two walk-forward schemes, and +this is most likely due to the time focus from the aforementioned +prioritised experience replay. In contrast, high volatility results +from the sliding window do imply the benefit of including longer +training periods. Especially for the most recent three years, the +average Sharpe Ratio of the anchored window approach is about +10% better than that of the sliding window approach. This addi- +tional information in training provides our model with more stable +performance and better resilience to loss, observed from the about +7% difference in maximum draw-down. Therefore, both schemes +are valid training approaches, but the sliding window approach has +a shorter training window and accordingly a faster convergence. +To illustrate the superior performance of Deep RL agents, we +present the three classic rule-based trading strategies as bench- +marks and an RL selector with the identical evaluation metrics in +Table 5. The three benchmarks are naive buy and hold of the un- +derlying asset as well as trading based on the 2 separate technical +indicators, MACD and Bollinger Band. The RL selector is a naive +RL agent to predict the most suitable indicator to follow based on +the simulated market environment. + +Year +V1 +V2 +V3 +V4 +V5 +V6 +V7 +V8 +2009 +2.78 +1.54 +2.53 +2.99 +2.02 +1.88 +1.12 +1.88 +2010 +0.43 +0.85 +-0.34 +-0.61 +-0.78 +0.41 +-0.95 +0.41 +2011 +-0.45 +3.07 +0.87 +1.79 +1.02 +1.11 +0.57 +1.11 +2012 +1.20 +1.52 +0.24 +-0.13 +-0.21 +-0.22 +0.41 +-0.22 +2013 +1.95 +1.35 +2.06 +0.71 +2.59 +1.72 +1.75 +1.72 +2014 +0.69 +2.44 +2.15 +2.90 +2.11 +1.53 +2.05 +1.53 +2015 +-0.47 +-0.57 +0.66 +1.19 +0.28 +1.26 +0.18 +1.26 +2016 +-0.13 +-0.84 +1.58 +0.70 +2.15 +2.05 +1.38 +2.05 +2017 +1.78 +0.42 +0.70 +0.67 +1.43 +0.39 +0.53 +0.39 +2018 +0.44 +-0.10 +0.79 +0.68 +-0.63 +1.22 +0.94 +1.22 +2019 +-1.06 +0.11 +0.75 +1.19 +-0.64 +-1.21 +0.27 +-1.21 +2020 +-0.42 +-0.35 +1.09 +-0.26 +1.55 +1.50 +0.27 +1.50Year +V1 +V2 +V3 +V4 +V5 +V6 +V7 +V8 +2009 +1.54 +1.49 +2.45 +2.16 +1.23 +1.36 +-1.56 +1.46 +2010 +0.39 +1.09 +0.64 +0.71 +1.56 +0.63 +-0.19 +1.74 +2011 +0.43 +0.99 +-0.02 +0.58 +-2.04 +-0.97 +-0.41 +0.73 +2012 +1.00 +-0.41 +0.49 +-0.41 +1.35 +-1.28 +0.09 +0.16 +2013 +2.65 +0.69 +2.36 +2.40 +0.65 +1.77 +0.55 +1.30 +2014 +1.00 +2.86 +1.60 +2.62 +2.26 +1.33 +0.31 +-0.52 +2015 +0.33 +0.33 +0.15 +0.80 +1.17 +1.58 +0.11 +-0.28 +2016 +0.56 +-0.40 +2.63 +0.78 +1.73 +1.95 +1.73 +1.69 +2017 +0.43 +1.67 +0.57 +0.35 +1.34 +1.07 +2.83 +0.70 +2018 +0.53 +0.55 +-0.27 +-0.44 +-0.09 +0.08 +-1.32 +0.38 +2019 +-1.55 +-1.70 +0.67 +0.40 +0.15 +1.74 +1.32 +-0.77 +2020 +-0.61 +1.07 +-0.36 +2.76 +-0.29 +0.84 +1.53 +1.40Walk forward +Agent +Avg. Sharpe R. +Max. DD (%) +Cum P&L (ME) +Scheme +anchor-win +1.27 +15.0 +44.95 +Bonsai APEX DQN +1.32 +22.3 +48.32 +Bonsai APEX DQN +moving-win +Bonsai cont. SAC +moving-win +0.97 +28.2 +28.93Deep Reinforcement Learning for Gas Trading +Conference’17, , +Table 5: Result table for rule-based traditional trading meth- +ods served as baseline benchmarks. The average Sharpe Ra- +tio, maximum drawdown and cumulative P&L are reported +for simple Buy&Hold, MACD, BB, and naive RL selector be- +tween MACD and BB. +The negative P&L in all three rule-based strategies is unfortunate, +but unavoidable. These results advocate the ever-increasing compli- +cated trading environment where the once pioneered strategies all +become insufficient. The added RL selector, even though, still poorly +performs, the boosted performance from the model architecture +serves as an indication to consider more sophisticated structures +and signals. Therefore, the three Deep RL agents in Bons.ai with +decent statistics show great potential to be turned into profitable +trading strategies. +4.2 +Results from in-house code +A good complement to the Bons.ai platform is the in-house code +using a DQN agent. The in-house code provides a more granular +level of control. To match the training speed in Bons.ai, we leverage +High Performance Computing (HPC) with identical walk forward +schemes. Additionally, a local version has also been implemented +with an update frequency based on yearly retraining due to limited +local compute resources. Furthermore, for the analysis of techni- +cal and fundamental features, the DQN agent and two traditional +machine learning benchmarks are compared with only technical +features and with technical and fundamental features. All models +use the moving-window training approach, and DQN moves for- +ward every year (identical with Bons.ai version), whereas Linear +Regression and Random Forest models are optimised with four +month move-forward window. The performance of the respective +models is compared in Table 6. +The addition of fundamental features does not seem to improve +strategy performance. For pure technical feature based agents, the +local DQN surpasses the Linear Regression agent, especially in +terms of average Sharpe Ratio (0.96 against 0.53). Yet, Random For- +est seems to be comparable in all three metrics. The two traditional +machine learning agents have a much faster daily retrain update +frequency because of their low computational cost, and the result +suggests Random Forest is a viable alternative to the local DQN +agent. Nevertheless, the demanding HPC DQN with anchored win- +dow dominates the performance table with a 1.14 Sharpe Ratio and +an almost doubled 42.12 million GBP cumulative P&L to the others, +and appears to outperform traditional methods. +To further discuss the role of fundamental features, a box-plot +of Sharpe Ratios based on 15 realizations is presented in Fig. 1, +where one agent is based purely on price features with technical +indicators, and the other agent has in addition also fundamental +features. +Table 6: Result table for in-house DQN with traditional ma- +chine learning baseline models as benchmarks, with tech- +nical features only and with technical plus fundamental +features. All models use moving-window training approach, +and DQN moves forward every year (identical with Bons.ai +version), whereas Linear Regression and Random Forest +models move forward every 4 months. The average Sharpe +Ratio, maximum drawdown and cumulative P&L are re- +ported, with figures after ’±’ sign denoting the respective +standard deviation. Averages and standard deviation are +taken over yearly samples from 2013 to 2020. +Figure 1: Sharpe Ratio box plot for DQN with technical fea- +tures only and with technical plus fundamental features. +Judging from the comparison of average Sharpe Ratios in Table +6 and the median Sharpe Ratio indicated in Fig. 1 by the red line +in the middle of each box, the difference in Sharpe Ratio with and +without fundamental features is subtle. However, including fun- +damental features results in almost double the standard deviation +when compared to the results with technical features only. One pos- +sible explanation could be that more features increase the agents’ +search space, and hence increase the variance of the performance. +5 +MODEL EXPLAINABILITY +To analyse feature importance and their contribution at various +points in time, SHAP plots are used for ad-hoc model explainability +analysis. We analyze the relative feature importance of 2014 and +2020. The top 5 features in 2014 are relative close price between +t and t-1 with no lag (feature 1), spread between close and 63- +day EMA (exponential moving average) with 2-day lag (feature +2), spread between close and 63-day EMA with 1-day lag (feature +3), 3-month volatility-adjusted return with no lag (feature 4) and +12-month volatility-adjusted return with 1-day lag (feature 5). +Feature 1 remains in the dominant position in both years, with +a marginal increased influence in Buy. However, features 2, 3, 4 + +Method +Avg. Sharpe R. +Max. DD (%) +Cum. P&L (ME) +0.55 +79 +8.81 +RL Selector MACD / BB +MACD +0.18 +174 +-1.17 +BB +0.00 +210 +-4.25 +-0.18 +252 +-6.39 +Buy & HoldAgent +Walk forward +Retrain +Avg. Sharpe R. +Max. DD (%) +Cum P&L (ME) +DQN technical +yearly +0.96 ± 0.24 +20 ± 4 +24±4 +moving 1 year +0.94 ± 0.41 +20 ± 6 +DQN tech + fund, +moving 1 year +yearly +24±7 +Linear Reg. technical +moving 4 months +daily +0.53 ± 0.25 +24 ± 4 +22 ± 2 +Lin. Reg. tech + fund. +daily +moving 4 months +0.66 ± 0.36 +21 ± 5 +27 ± 3 +daily +Random Forest tech +moving 4 months +0.91 ± 0.14 +19 ±3 +23±3 +RF tech + fund. +daily +moving 4 months +0.84 ± 0.26 +25±5 +20± 4Technical +2.00 +Technical and Fundamental +175 +150 +125 +1.00 +0.75 +0.50 +0.25 +Technical features only +Tech + fundamental featuresConference’17, , +Y.Wang et al. +become less significant in 2020 compared 2014, while the feature 5 +has dropped out of the top 20 features. This progression suggests +that feature importance is not constant across the time-series, and +regular re-training is necessary for models to reflect most recent +market information. Moreover, it is worth noting that except for +the feature 1 which does not seem to change its mean SHAP value, +the SHAP values of the other features are all smaller in 2020 than +they were in 2014. This distribution may indicate that certain fea- +tures dominate or are highly influential in 2014, while they tend to +contribute equally in 2020. As a result, we would expect a better +performance in feature selection in 2014 than in 2020 since fewer +features are required to approximate the full model. +Furthermore, besides the feature contribution analysis in 2014 +and 2020, a temporal decision plot is also visualized and used for our +analysis. Fig. 2 shows example snapshots for a particular time step +and the top features contributing to decisions at that moment in +2014 and 2020. The full visualization is in the format of a video and +provides details how much a given feature contributes to a potential +Buy, Sell or Hold decision. Not only can it serve as a verification +for the feature selection process, but it can also be used as tool to +explain the model’s rationale to stakeholders and non-technical +parties. +Figure 2: Snapshots of the temporal decision plot in 2014 +(top) and 2020 (bottom). +5.1 +Results from Ensemble Learning +The Filtered-Thresholding ensemble method introduced in Sec- +tion 3.3 eliminates inferior trained agents by filtering based on +training curve. An example of a ’successful’ and a sub-optimal +training curve is presented in Figure 3. In the left sub-plot, the +score improves gradually until saturation, and then remains more +or less constant on average. However, in the right sub-plot, after +an uptrend, the score drops into a local minimum and gets stuck. +These training curves illustrate the criteria for the primitive selec- +tion process in the ensemble learning. An automated systematic +selection criteria can be obtained by combining the rolling average +of the learning curve and monotone convergence theorem given in +[3]. +Each ensemble trains ten instances of the Deep RL agent over +all episodes of the two years of training data preceding the test +data set. Three ensemble learning results in 2018, 2019 and 2020 +are performed. Each ensemble starts with ten realizations before +Figure 3: Examples of a training curve for two instances of +a Deep RL agent trained over all episodes during 2018 and +2019, showing convergence of the trained agent (left) and an- +other agent getting stuck in a local minimum (right). +filtering, and only 1, 0, 2 agents are filtered out due to inferior train- +ing performance in 2018, 2019 and 2020, respectively. Back-testing +performance of 2019 is presented in Figure 4 for exemplification. +Figure 4: Ensemble Learning Results 2019 contains four +blocks, that are the daily close price (top left), cumulative re- +turn (top right), net position for holdings (bottom left), and +realized and unrealized returns (bottom right). +Illustrated in Figure 4, training in 2017 and 2018 produces a back- +testing result with a Sharpe Ratio of 1.71 in 2019. From the top +left subplot, it is obvious that the market in the first half of 2019 +seriously plummeted, while it oscillated in the second half of 2019. +Under this scenario, the ensemble agent first attempted to go long +which resulted in negative returns. Then, it intends to follow the +trend by taking consecutive small holding periods of shorts in the +first half. However, none of these attempts seemed to be effective +enough to bring a positive P&L. In the second half of 2019 before +October, the agents performed multiple good trades by going long, +which has seized most of the opportunities when the oscillations +peaked. Then, both long and short trades have consolidated their +gains at the end of 2019. +This analysis has reviewed certain characteristics of our ensem- +ble trading bot in 2019. Most of its winning trades are based on +mean-reversion behavior in the second half, and it failed trend- +following in the first half. However, it does not seem trivial to infer +a rule-based strategy. Yet, as suggested by Wang [49] and showcased +in Section 5, the market exhibits different behaviours in different +periods, and our agents should learn which features to rely on and +which trading patterns to follow dynamically. + +Close price (line) from :2019 +Sharpe Ratio = 1.71 from 2019 +Cumsum Profit (line) from 2019 +70 +2 +10 +60 +CumsumProfit [MGDP] +4 +2 +30 +06 +2019-03 +2019-12 +2019-03 +2019-04 +2019 +2019-07 +2019- +2019 +Date +Date +Net position (line) from 2019 +Stackplot of Realized value and Unrealized value in :2019 +30000 +12 +Realized value +10 +Unrealizedvalue +20000 +position [kTh] +10000 +Profits [MGBP] +8 +6 +0- +4 +Net +-10000 +2 +20000 +30000 +2019-01 +2019-03 +2019-04 +2019-05 +2019-06 +2019-07 +2019-08 +2019-09 +2019-10 +2019-11 +2019-12 +2019-01 +2019-02 +2019-03 +2019-04 +2019-05 +2019-06 +2019-07 +2019-08 +2019-10 +2019-12 +Date +Datedose +8 +Lo +50 +10D +150 +z +250 +Sell +Do nothing +Bury +0.1 +STO +0.10 +000 +20- +50'0- +OBe +adj_2mr_lag_0 +ens21Spd_lag_1 +T BerJwe [p? +adj_12mr_l8g_1 +o BerJwe [pe +n863Spd_lag_2 +mpd_lag_2 +mBpd_lag_0 +T'BerJwe p? +adj_3mr_log_2 +erne21Spd_lag_1 +o Berwg pe +TBerJwe [pe +adj_12mr_l8g_α +adj_12mr_lag_1 +adj_2mr_lag_0 +shapley +2014 +asop +区 +50 +150 +20D +250 +3ID +Sell +Do nothing +Buy +0.05 +0.10 +0.15 +0.05 +0.10 +50'0- D1'- +50'0- +0.15 +0.15 +DT0 +OBe +zBeJwe pe +'Be]z [pe +merd_lag_2 +adj_ 2mr_lag_2 +z Ber pdsE9Bwa +adj_12mr_l8g_1 +adj_1mr_lag_0 +ern821Spd_lag_0 +enB215pd_lag_2 + Bel JWE [pe +ern8635pdlag_2- +p1mC_leg_0 +rne215pd_lag_0 +Be]t pe +_lag_2 +adj_2mr_lag_2 +shapley +2020 +Cpl +e821125 +120 +100 +100 +75 +80 +50 +core +60 +Score +25 +0 +20 +-25 +0 +-50 +-20 +75 +5000 +10000 +15000 +20000 +25000 +5000 +10000 +15000 +20000 +25000 +Episode# +Episode#Deep Reinforcement Learning for Gas Trading +Conference’17, , +6 +DISCUSSION +The performance of the Deep RL agents, the Linear Regression and +the Random Forest models are compared in Fig. 5. The x-axis repre- +sents the Sharpe Ratio, y-axis represents the maximum drawdown, +and the color represents the cumulative P&L. The best result with a +Sharpe Ratio of 1.32 and a maximum drawdown of 22.3% is achieved +by Bons.ai DQN Apex with moving window for re-training. The +other results obtained generally also demonstrate decent Sharpe +Ratios. We find an average Sharpe Ratio of 1.07 taken across all +versions and algorithms, which outperforms the state-of-the-art +result in Zhang [55] where a Sharpe Ratio of 0.723 for commodity +trading with DQN has been reported. Fig. 5 also displays more con- +ventional ML strategies based on Linear Regression and Random +Forest. However, the Deep RL based strategies appear to be superior +for this gas trading use case. Moreover, by means of ensemble learn- +ing in the form of Filtered-Thresholding we can further improve +the performance. Backtesting yields an average Sharpe Ratio of +1.20 over 2018-2020, a 23% increase from the average Sharpe Ratio +of 0.975 obtained with the in-house DQN method. +Figure 5: The summary plot for all machine learning trad- +ing agents including RL agents and traditional ML agents +assessed over the period from 2013 to 2020. The x-axis rep- +resents the Sharpe Ratio, y-axis represents the maximum +draw-down, and the color represents the cumulative P&L. +Lin.Reg represents linear regression agent, R.F. represents +random forest agent, Tech indicates only technical features +are used, Tech+Fund indicates technical and fundamental +features are used in the in-house agents. All Bons.ai agents +use technical and fundamental features. +Despite the results obtained outperform those reported in state- +of-the-art literature, implementing a Deep RL based trading agent +faces several challenges in practice. First, there is usually still a +drop in performance when going from back-testing to live trading. +Second, extended periods of underwater performance would call +for shutting down an algorithm before it can swing to profitability. +In other words, even if in backtesting it generated a profit over the +entire year of 2019, see Fig. 4, in practice it would not even reach +that point, since it would have been stopped out well before. Third, +the frequency of trades has to fit with the overall strategy of the +trading desk and neither display overly long holding periods nor +too frequent trades / churn. Fourth, model explainability remains a +concern for black-box neural network based models although the +analysis based on SHAP values in Section 5 and feature importance +help to mitigate this point. +7 +CONCLUSIONS +Systematic trading of commodities is a challenging topic in quanti- +tative trading. The low signal-to-noise ratio makes learning models +prone to overfitting. In this paper, we demonstrate our implemen- +tation of a Deep Reinforcement Learning framework for systematic +gas trading with different approaches based on Microsoft Bons.ai +platform as well as in-house code. Our Deep RL agent trained in +Bons.ai has achieved a Sharpe Ratio of 1.32 in back-testing and +thereby outperformed state-of-the-art results from literature. The +proposed ensemble learning scheme for our in-house DQN method +has achieved a Sharpe Ratio of 1.2 with a 23% improvement in +performance over individual DQN agents. +A comparison of models employing only technical features and +those employing both technical and fundamental features suggest +that including fundamental features does not lead to better perfor- +mance here, as it appears that the information gain is offset by the +increased noise in the observation space. +This paper is one of the first applications of model explainability +using Shapley values for Deep Reinforcement Learning applied +to trading and gives insight which feature drives the agent’s buy, +sell or hold decision at a certain point of time. It provides insight +into otherwise black-box neural network based models and thereby +offers a way to analyse the developed rationale of an agent’s trading +strategy. +Despite performance beyond state-of-the-art literature, imple- +menting such Deep RL trading agent in practice faces several chal- +lenges as discussed in detail in Sec. 6 such as a potential drop in +performance between back-testing and live trading, extended peri- +ods of even slight under-performance triggering a shut down of the +algorithm before it can reach profitability, the trading frequency +has to fit with the desk’s overall strategy, and model explainability. +The application of Deep Reinforcement Learning to systematic +gas trading has been the first successful application of Deep RL +in Shell, highlighting that rigorous feature selection, design of the +reward function, model architecture and ensemble learning can +result in improved and robust performance. Ongoing and future +work will consider application of Deep RL to auction-like European +power markets [43] and process optimization [40]. + +30 +Bons.aicont.SAC(moving-window) +28 +45 +26 +R.F.(Tech+Fund) +40 +(%) +Lin.Reg.(Tech) +Cumulative P&L (Mf) +24 +: +Bons.aiAPExDQN(moving-window) +35 +22 +Lin.Reg.(Tech+Fund) +In-house DQN (Tech) +20 +R.F.(Tech) +In-house DQN (Tech+Fund) +30 +: +18 +25 +16 +Bons.aiAPExDQN(anchor-window) +14 +20 +0.4 +0.6 +8:0 +10 +12 +14 +16 +Sharpe RatioConference’17, , +Y.Wang et al. +REFERENCES +[1] Vangelis Bacoyannis, V. Glukhov, Tomoyuki Jin, Jonathan Kochems, and Doo Re +Song. 2018. Idiosyncrasies and challenges of data driven learning in electronic +trading. arXiv: Trading and Market Microstructure (2018). +[2] Dimitris Bertsimas and Andrew W. Lo. 1998. Optimal control of execution costs. +Journal of Financial Markets 1 (1998), 1–50. +[3] John F. Bibby. 1974. Axiomatisations of The Average and a Future Generalization +of Monotonic Sequences. 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Energy Economics 66 (2017), 9–16. + diff --git a/kdE_T4oBgHgl3EQf5hys/content/tmp_files/load_file.txt b/kdE_T4oBgHgl3EQf5hys/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..10b882d292b70f2d0b9bfed9463d3e171616f583 --- /dev/null +++ b/kdE_T4oBgHgl3EQf5hys/content/tmp_files/load_file.txt @@ -0,0 +1,1033 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf,len=1032 +page_content='Deep Reinforcement Learning for Gas Trading Yuanrong Wang University College London London, UK Shell Global Solutions International UK yuanrong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='wang@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='ucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='uk raymond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='wang2@shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='com Yinsen Miao Shell International Exploration and Production USA yinsenm@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='com Alexander CY Wong Shell Global Solutions International The Netherlands alexwong_92@hotmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='com Nikita P Granger Shell Energy North America USA Nikita.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='Granger@shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='com Christian Michler∗ Shell Global Solutions International The Netherlands C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='Michler@shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='com ABSTRACT Deep Reinforcement Learning (Deep RL) has been explored for a number of applications in finance and stock trading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' In this pa- per, we present a practical implementation of Deep RL for trading natural gas futures contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' The Sharpe Ratio obtained exceeds benchmarks given by trend following and mean reversion strategies as well as results reported in literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Moreover, we propose a simple but effective ensemble learning scheme for trading, which significantly improves performance through enhanced model sta- bility and robustness as well as lower turnover and hence lower transaction cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' We discuss the resulting Deep RL strategy in terms of model explainability, trading frequency and risk measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' CCS CONCEPTS Computing methodologies → Reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' KEYWORDS Deep Reinforcement Learning, Gas Trading, Systematic Trading ACM Reference Format: Yuanrong Wang, Yinsen Miao, Alexander CY Wong, Nikita P Granger, and Christian Michler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Deep Reinforcement Learning for Gas Trading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' In Proceedings of (Conference’17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' ACM, New York, NY, USA, 8 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' https: //doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='org/XXXXXXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='XXXXXXX 1 INTRODUCTION The profitability of a trading strategy hinges on a good timing of entering and exiting a position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Researchers, traders and quants have been exploiting fundamental and technical analysis to analyze the market with the aim of predicting future market movements ever 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='$15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='00 https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='org/XXXXXXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='XXXXXXX notoriously low signal-to-noise ratio of financial markets, many expert-designed rule-based strategies fail to cope with changing market conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Recent advancements in Artificial Intelligence and data science start to bring a competitive edge to trading by learning the market dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' With the vast amount of historical data, models’ non-linear learning and inference ability extract pat- terns in time series and exploit volatility and directional signals for position taking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Natural gas is one of the most liquid commodity markets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' En- ergy futures and options contracts are usually traded on deriva- tive marketplaces such as the Chicago Mercantile Exchange (CME) or over-the-counter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Given the high volatility and similar market dynamics between gas and equity markets, the classic technical indicators are considered to still provide analytical signals for the underlying gas price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' However, as commodities are influenced more by supply and demand, the fundamental analysis is very different from stocks that are mostly correlated to the valuation of underly- ing firms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Key contributing factors to the gas price movements are often macro economic, including regional storage and production, global demand, alternative power production as well as weather.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' These differences prevent many stock traders to enter the energy market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Nevertheless, some trading algorithms have shown to be transferable, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' [16, 20, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Broadly speaking, quantitative models in commodity trading focus on the following aspects: Event-driven traders react to sys- tem events and failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Similar to how corporate news influences the price of stocks, a failure in a transmission system or a genera- tor [13, 25] and sometimes in a policy [36] can greatly impact the price of energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Then, technical analysis and fundamental analysis are two main methodologies in the market, where strategies learn from past data for future price forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' An early success story of Moshiri [32] employed a non-linear Artificial Neural Network model to predict crude oil futures which outperformed the tradi- tional econometric-based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Since then, ensemble learning methods by Yu and Jammazi [17, 53] combined Machine Learn- ing (ML) and econometric models by Godarzi and Zhang [9, 54], and also deep learning models by Zhao, Tang, Zhao and Safari [37, 45, 56, 57] have been proposed and designed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' More recently with the advancement in hardware and execution speed, both ag- gressive and passive high frequency trading algorithms have been arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='08359v1 [q-fin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='TR] 19 Jan 2023 Conference’17, , Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' applied in commodity trading, see [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' As the order book structure in natural gas futures on CME and equities on stock exchanges are similar, momentum ignition, order anticipation and arbitrage trading have been explored, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=', Fishe [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Despite the abundance of data and learning ability of ML mod- els, many issues arise from the noisy time-series data, especially in finance, and biased supervised labelling in the trading environ- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' An extremely low signal-to-noise ratio in financial time- series is an inherent impediment despite considerable efforts in time-series filtering by means of, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=', information filtering net- works [26, 27, 46, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Moreover, the general two-step approach in constructing a trading strategy is ill-posed, see [30]: Firstly, a super- vised model forecasts asset price changes with a defined investment horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Then, the forecasts are fed to a strategy module to generate actual trading strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' In the first step, the supervised models are normally labelled by future prices with a defined investment horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' This approach limits generalization of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' In addi- tion, besides the predictions, other features, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=', liquidity, market micro-structure, are usually not included in the second step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' This ideal simplification of market dynamics fails to explicitly address the interaction between a trading strategy and the real market in the form of market impact during execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Reinforcement Learning (RL) agents learn policies (strategies) directly by optimising a numerical reward signal by interacting with a virtual market environment that is usually derived from historical market data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' This has been shown to prevent many of the aforementioned issues in supervised ML by Sutton [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Instead of back-propagating the loss between labelled ground truth and the prediction, a simulation environment is built based on market data for Deep RL agents to explore and exploit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Next, actions are evaluated based on the reward from the interaction with the market environment, and a policy (trading strategy) is learned through it- erative interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' In 2013, Mnih[29] published the seminal work of Deep Q-learning Network (DQN), which marks the transition from Reinforcement Learning to Deep Reinforcement Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Since then, more algorithms have been proposed, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='g, PPO [38], and DDPG [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Early literature has already attempted to apply traditional RL in the financial market, like stock pricing and selec- tion [7, 21, 44], optimal trade execution [2, 19, 34], high frequency trading [39, 48] and portfolio trading [11, 18, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' In the last few years, Spooner [41] has used several Time-Difference methods for market making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' [51] applied DDPG for stock trading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' RoaVicens [35] has embarked on simulating an order book environment in the presence of competitive agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Yang [52] researched ensemble methods between different Deep RL agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Other recent Deep RL research in financial applications has been summarized by Hambly [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Besides the academic community, JP Morgan in 2018 has re- leased a working paper [1] introducing their RL based limit order engine highlighting the potential for applications in trading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' In 2019, Zhang [55] compared DQN, PG, A2C algorithms for equity, FX, fixed income and commodity trading as the current state-of-the-art benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Our work builds on the structure of the DQN applied to commodity trading [55] and combines it with the implementation from Udacity [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' For performance improvement and model explainability, we further incorporate a game-theoretical SHAP (SHapley Additive exPlanations) plot from Lundberg[24] for feature interpretation and selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Furthermore, to address the high compute cost of training Deep RL models discussed in Gudimella [11], we leverage the Microsoft Bons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='ai platform [10] with its containerized simulation environments, high parallelism and automated hyper-parameter tuning which accelerates model training and testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Yang [52] and Carta[5] emphasized that the same model with different initializations can lead to different out- comes due to the long path-dependency especially for problems in trading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' To prevent bias from back-testing and enhance robustness of the model, Borokova [4] present an online ensemble learning method for LSTM in high-frequency equity price prediction, where models are weighted by their recent performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Inspired by their framework, we propose and design an ensemble learning scheme, where agents are trained in parallel before filtering by training per- formance, and then averaged by threshold voting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' The implementa- tion details are presented in Section 3, and results are discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' We will discuss the resulting ensemble learning scheme under several practical aspects in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' 2 PROBLEM DESCRIPTION 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='1 Data We trade the front month NBP UK Natural Gas Futures contract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' The future contracts are for physical delivery through the trans- fer of rights in respect of Natural Gas at the National Balancing Point (NBP) Virtual Trading Point, operated by National Grid, the transmission system operator in the UK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Delivery is made equally each day throughout the delivery period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' The NBP virtual trading point acts as a central exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' The price during a trading day can be characterized by a so-called candle of low, high, open and close price [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Alongside the trading volume of the day, we shall refer to these five features as price features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Other than the price features, technical analysis is employed to construct technical features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' In this experiment, we compute MACD, Price RSI, Volume RSI, PCA 1st component and volatility adjusted returns of 1, 2 and 3 months as additional features [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Furthermore, to investigate the effect of natural gas fundamental features, we also include demand data (industrial, gas to power and residential demand), production data from UK production fields, LNG data in all 3 UK terminals, data of pipeline imports from Norway, Netherlands, Belgium and Ireland as well as storage data in all active facilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' A full list of features is shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' 3 DEEP RL METHODOLOGY 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='1 Set up of virtual market simulation environment In Reinforcement Learning, agents learn policies from interacting with a simulation environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Hence the environment needs to be as realistic as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' A complete market approximates the real market where friction, market impact from orders, transaction cost and asset liquidity exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' However, such highly interactive envi- ronment is complex to model and computationally demanding to simulate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Hence, a simplified incomplete market is customarily used where only transaction cost is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Here, we shall adopt the latter approach and consider transaction costs of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='1p/therm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' In this environment, agents observe price, technical and fundamental features at each time step with a look-back window to detect trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Deep Reinforcement Learning for Gas Trading Conference’17, , Table 1: Table outlines the technical and fundamental fea- tures used in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Technical features include can- dlestick features and their derived differences, technical in- dicators and volatility adjusted returns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Fundamental fea- tures include demand, production, transportation and stor- age data as well as liquefied natural gas (LNG) terminal data in the UK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' The agent then outputs an action for this time step, which is evalu- ated according to a reward function that guides and incentivizes the learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' We limit this experiment to a daily trading frequency, and add a three-day look-back window to aforementioned features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Therefore, the transition in the observational space moves forward the time- stamp by a day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' We limit the size of look-back window so that agents consider the most recent information, and the specific choice of the 3-day look-back window follows from a tradeoff between model complexity and training efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' That is, each day an agent perceives all the information from the present day and the three previous days1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' As the trading environment is set up for daily frequency, the agent outputs the action for today based on the learned policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' For simplicity, the action is taken to be discrete as buy or sell the maximum amount, or hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' To avoid overfitting to the data in the replay buffer, a 1% Gaussian noise is added to each observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Learning optimal decisions is guided by the reward function as a result of the agent’s interaction with the environment through ac- tions 𝐴𝑡, and the reward𝑟𝑖,𝑡 in form of raw P&L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' In addition, for sim- ulation purposes, we fixed the transaction costs, 𝑡𝑐 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='1p/therm based on the bid-ask spread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' We employ reward shaping as follows: The immediate reward after an action is taken as the P&L, ˆ𝑟𝑖,𝑡 to guide the agent, and the performance of each episode is assessed at the end of an episode by the annualized Sharpe Ratio 𝑆𝑅𝑖, where subscript 𝑡 indicates the time-step in episode 𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' 1Note that for the day under consideration the agent can only see the open price of the day, but not the close, high or low price of the day to prevent leaking future data that would only be known after trading closes on that day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' ˆ𝑟𝑖,𝑡 = 𝐴𝑡−1𝑟𝑖,𝑡−𝑡𝑐|𝐴𝑡 − 𝐴𝑡−1| ˆ¯𝑟𝑖 = 1 𝑛 𝑛 ∑︁ 𝑡=1 ˆ𝑟𝑖,𝑡 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' ˆ𝜎𝑖 = 𝑛 ∑︁ 𝑡=1 √︂ 1 𝑛 |ˆ𝑟𝑖,𝑡 − ˆ¯𝑟𝑖 |2 ˆ𝑆𝑅𝑖 = √ 252 ˆ¯𝑟𝑖 ˆ𝜎𝑖 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' (1) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='2 Implementation Bons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='ai uses DQN Apex by [15] for discrete action spaces and SAC by [12] for continuous action spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Algorithm selection can be done automatically in the platform based on feature and action settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' The platform then trains the model, and assessment can be done separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Note that the Bons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='ai platform can only handle a single agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' So for any ensemble method requiring interaction between different realizations, we used a centralized approach for training before a post-hoc ensembling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' On the other hand, we have built a version of the in-house code with DQN algorithms, which serves as a verification and comple- ment to the Bons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='ai platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Its main edge is the fine-grained control over low-level neural network and simulation architectures, which benefits the design of our ensemble learning method referred to as Filtered-Thresholding detailed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' To better match the high computation speed in Bons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='ai, a parallel training scheme has been designed not only in the local version, but also in a high- performance-computing version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' The empirical experiments show that an average of 15,000-30,000 episodes are required for successful training and achieving convergence of a single agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='3 Ensemble Learning for Virtual Book During our extensive experiments, the initialization of different agents has been found to be an influential factor for the learning trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' An exact setup could even result in one agent learn- ing successfully until converge while another agent being stuck in a local minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Moreover, as the underlying algorithms are model-free Deep RL, even all “successfully” learned agents will be- have differently in terms of their underlying trading logic, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=', one agent may be relying more on momentum strategy while the other agent may tend more towards mean-reversion trades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' The strategy preference could be inferred from the difference in holdings across different agents in the same setup, which has been observed in an analysis to holding positions in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' To moderate the impact from sub-optimal agents and difference between underlying strategies, we propose a simple but effective en- semble learning method, which we refer to as Filtered-Thresholding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Firstly, 𝑁 number of agents are trained in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Then, based on the trainings curve, we exclude seemingly sub-optimal agents in a “Filtering” step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' After filtering out bad candidates, we ensemble the decision-making process between converged agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Multiple ensemble methods have been considered, and the thresholding method was chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Each agent at each time-step has three choices / positions that it can take, namely Buy, Sell and Hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' If more than 𝑝% of agents agree on an action, then the ensemble agent executes the decision, otherwise it remains unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' In our experiment, 𝑝% = 50% is set for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' This “Thresholding” step avoids large turnover in transaction costs caused by extensively moving in and out of positions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' it reconciles any dissimilarities and integrates the Technical Features Fundamental Features Candlestick: Demand Data (Open,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' High,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Low and Close price of the day)OHLC (lndustrial,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Gas to Power,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Residential) Volume Differences: Production Data High - Low price of the dayi (UK production fields) High - Open price of the day,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Close - previous Close price of the day Technical Indicators: LNG Data MACD (All 3 UK LNG terminals) Price RSl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Volume RSI PCA 1st component Returns: Pipeline data Volatility adjusted 1-month,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' (Imports from Norway,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' NL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' BE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' IR) Volatility adjusted 2-month,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Volatility adjusted 3-month,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Storage Data (All active facilities)Conference’17,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' advantages of the underlying agents and their respective trading logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' 4 RESULTS 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='1 Results from Bons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='ai In any time-series machine learning problem, the training period is a crucial parameter that impacts the behaviour and performance of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Besides computational cost, a long training period recognizes longer-term tendencies, while a short training period captures more local temporal patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' This is especially true for Deep RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Although prioritised experience replay attempts to empha- size the most recent pattern, the low signal-to-noise ratio inherent in financial time series still poses a challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Two walk-forward training schemes are compared in Table 2 and Table 3 for anchored and sliding-window approaches, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Table 2: Training and testing performance table for Bons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='ai DQN Apex with anchored window, where rows denote years and columns denote different versions of the train/test data split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Each version corresponds to a different split between training and testing period with the numerical value indicat- ing the out-of-sample Sharpe Ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' The yellow and white slots indicate training year, and the light green slots are the immediate testing years after training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' The light blue slots provide yet another test data set by testing on future data that is further out than the subsequent year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' From a 12-year dataset of 2009 to 2020, the anchored window starts with a minimum 4-year training period, and evaluates per- formance in the subsequent year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' After each evaluation, the year that has just been used as out of sample test set is added to the training period for the next walk-forward step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' In Table 2, we use the parallelization in the Bons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='ai platform to train eight Bons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='ai brains simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' In the anchored window approach the size of the training window grows over time, while in the sliding-window approach the training window length remains fixed, here to four years of training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' The performance summary of the two walk-forward schemes with APEX DQN are shown in Table 4 along with the Soft-Actor Critic(SAC) Deep RL agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' The table compares the performance of these three agents in terms of cumulative P&L, average Sharpe Ratio and maximum draw-down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' The APEX DQN moving window has an average Sharpe Ratio of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='32 and cumulative P&L of 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='32 million GBP, compared to a slightly lower performance of the anchored window with a Sharpe Ratio of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='27 and a cumulative P&L of 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='95 Table 3: Training and testing performance table for Bons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='ai DQN Apex with sliding window, where rows denote years and columns denote different versions of the train/test data split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Each version corresponds to a different split between training and testing period with the numerical value indicat- ing the out-of-sample Sharpe Ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' The yellow and white slots indicate training year, and the light green slots are the immediate testing year after training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' The light blue slots provide yet another test data set by testing on future data that is further out than the subsequent year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Table 4: Result table summarizing the performance of dif- ferent Bons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='ai agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' The average Sharpe Ratio, maximum draw-down and cumulative P&L are reported for Bons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='ai APEX DQN with anchored-window and moving-window training approach as well as for the SAC agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' million GBP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' The above summary statistics suggests a generally comparable performance of the two walk-forward schemes, and this is most likely due to the time focus from the aforementioned prioritised experience replay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' In contrast, high volatility results from the sliding window do imply the benefit of including longer training periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Especially for the most recent three years, the average Sharpe Ratio of the anchored window approach is about 10% better than that of the sliding window approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' This addi- tional information in training provides our model with more stable performance and better resilience to loss, observed from the about 7% difference in maximum draw-down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Therefore, both schemes are valid training approaches, but the sliding window approach has a shorter training window and accordingly a faster convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' To illustrate the superior performance of Deep RL agents, we present the three classic rule-based trading strategies as bench- marks and an RL selector with the identical evaluation metrics in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' The three benchmarks are naive buy and hold of the un- derlying asset as well as trading based on the 2 separate technical indicators, MACD and Bollinger Band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' The RL selector is a naive RL agent to predict the most suitable indicator to follow based on the simulated market environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Year V1 V2 V3 V4 V5 V6 V7 V8 2009 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='78 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='54 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='53 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='99 2.' metadata={'source': 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+page_content='3 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='32 Bonsai APEX DQN moving-win Bonsai cont.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' SAC moving-win 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='97 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='2 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='93Deep Reinforcement Learning for Gas Trading Conference’17, , Table 5: Result table for rule-based traditional trading meth- ods served as baseline benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' The average Sharpe Ra- tio, maximum drawdown and cumulative P&L are reported for simple Buy&Hold, MACD, BB, and naive RL selector be- tween MACD and BB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' The negative P&L in all three rule-based strategies is unfortunate, but unavoidable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' These results advocate the ever-increasing compli- cated trading environment where the once pioneered strategies all become insufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' The added RL selector, even though, still poorly performs, the boosted performance from the model architecture serves as an indication to consider more sophisticated structures and signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Therefore, the three Deep RL agents in Bons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='ai with decent statistics show great potential to be turned into profitable trading strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='2 Results from in-house code A good complement to the Bons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='ai platform is the in-house code using a DQN agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' The in-house code provides a more granular level of control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' To match the training speed in Bons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='ai, we leverage High Performance Computing (HPC) with identical walk forward schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Additionally, a local version has also been implemented with an update frequency based on yearly retraining due to limited local compute resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Furthermore, for the analysis of techni- cal and fundamental features, the DQN agent and two traditional machine learning benchmarks are compared with only technical features and with technical and fundamental features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' All models use the moving-window training approach, and DQN moves for- ward every year (identical with Bons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='ai version), whereas Linear Regression and Random Forest models are optimised with four month move-forward window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' The performance of the respective models is compared in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' The addition of fundamental features does not seem to improve strategy performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' For pure technical feature based agents, the local DQN surpasses the Linear Regression agent, especially in terms of average Sharpe Ratio (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='96 against 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='53).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Yet, Random For- est seems to be comparable in all three metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' The two traditional machine learning agents have a much faster daily retrain update frequency because of their low computational cost, and the result suggests Random Forest is a viable alternative to the local DQN agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Nevertheless, the demanding HPC DQN with anchored win- dow dominates the performance table with a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='14 Sharpe Ratio and an almost doubled 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='12 million GBP cumulative P&L to the others, and appears to outperform traditional methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' To further discuss the role of fundamental features, a box-plot of Sharpe Ratios based on 15 realizations is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' 1, where one agent is based purely on price features with technical indicators, and the other agent has in addition also fundamental features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Table 6: Result table for in-house DQN with traditional ma- chine learning baseline models as benchmarks, with tech- nical features only and with technical plus fundamental features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' All models use moving-window training approach, and DQN moves forward every year (identical with Bons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='ai version), whereas Linear Regression and Random Forest models move forward every 4 months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' The average Sharpe Ratio, maximum drawdown and cumulative P&L are re- ported, with figures after ’±’ sign denoting the respective standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Averages and standard deviation are taken over yearly samples from 2013 to 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Figure 1: Sharpe Ratio box plot for DQN with technical fea- tures only and with technical plus fundamental features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Judging from the comparison of average Sharpe Ratios in Table 6 and the median Sharpe Ratio indicated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' 1 by the red line in the middle of each box, the difference in Sharpe Ratio with and without fundamental features is subtle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' However, including fun- damental features results in almost double the standard deviation when compared to the results with technical features only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' One pos- sible explanation could be that more features increase the agents’ search space, and hence increase the variance of the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' 5 MODEL EXPLAINABILITY To analyse feature importance and their contribution at various points in time, SHAP plots are used for ad-hoc model explainability analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' We analyze the relative feature importance of 2014 and 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' The top 5 features in 2014 are relative close price between t and t-1 with no lag (feature 1), spread between close and 63- day EMA (exponential moving average) with 2-day lag (feature 2), spread between close and 63-day EMA with 1-day lag (feature 3), 3-month volatility-adjusted return with no lag (feature 4) and 12-month volatility-adjusted return with 1-day lag (feature 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Feature 1 remains in the dominant position in both years, with a marginal increased influence in Buy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' However, features 2, 3, 4 Method Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Sharpe R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' DD (%) Cum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' P&L (ME) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='55 79 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='81 RL Selector MACD / BB MACD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='18 174 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='17 BB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='00 210 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='18 252 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='39 Buy & HoldAgent Walk forward Retrain Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Sharpe R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' DD (%) Cum P&L (ME) DQN technical yearly 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='96 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='24 20 ± 4 24±4 moving 1 year 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='94 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='41 20 ± 6 DQN tech + fund, moving 1 year yearly 24±7 Linear Reg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' technical moving 4 months daily 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='53 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='25 24 ± 4 22 ± 2 Lin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Reg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' tech + fund.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' daily moving 4 months 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='66 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='36 21 ± 5 27 ± 3 daily Random Forest tech moving 4 months 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='91 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='14 19 ±3 23±3 RF tech + fund.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' daily moving 4 months 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='84 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='26 25±5 20± 4Technical 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='00 Technical and Fundamental 175 150 125 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='25 Technical features only Tech + fundamental featuresConference’17, , Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' become less significant in 2020 compared 2014, while the feature 5 has dropped out of the top 20 features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' This progression suggests that feature importance is not constant across the time-series, and regular re-training is necessary for models to reflect most recent market information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Moreover, it is worth noting that except for the feature 1 which does not seem to change its mean SHAP value, the SHAP values of the other features are all smaller in 2020 than they were in 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' This distribution may indicate that certain fea- tures dominate or are highly influential in 2014, while they tend to contribute equally in 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' As a result, we would expect a better performance in feature selection in 2014 than in 2020 since fewer features are required to approximate the full model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Furthermore, besides the feature contribution analysis in 2014 and 2020, a temporal decision plot is also visualized and used for our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' 2 shows example snapshots for a particular time step and the top features contributing to decisions at that moment in 2014 and 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' The full visualization is in the format of a video and provides details how much a given feature contributes to a potential Buy, Sell or Hold decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Not only can it serve as a verification for the feature selection process, but it can also be used as tool to explain the model’s rationale to stakeholders and non-technical parties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Figure 2: Snapshots of the temporal decision plot in 2014 (top) and 2020 (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='1 Results from Ensemble Learning The Filtered-Thresholding ensemble method introduced in Sec- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='3 eliminates inferior trained agents by filtering based on training curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' An example of a ’successful’ and a sub-optimal training curve is presented in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' In the left sub-plot, the score improves gradually until saturation, and then remains more or less constant on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' However, in the right sub-plot, after an uptrend, the score drops into a local minimum and gets stuck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' These training curves illustrate the criteria for the primitive selec- tion process in the ensemble learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' An automated systematic selection criteria can be obtained by combining the rolling average of the learning curve and monotone convergence theorem given in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Each ensemble trains ten instances of the Deep RL agent over all episodes of the two years of training data preceding the test data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Three ensemble learning results in 2018, 2019 and 2020 are performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Each ensemble starts with ten realizations before Figure 3: Examples of a training curve for two instances of a Deep RL agent trained over all episodes during 2018 and 2019, showing convergence of the trained agent (left) and an- other agent getting stuck in a local minimum (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' filtering, and only 1, 0, 2 agents are filtered out due to inferior train- ing performance in 2018, 2019 and 2020, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Back-testing performance of 2019 is presented in Figure 4 for exemplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Figure 4: Ensemble Learning Results 2019 contains four blocks, that are the daily close price (top left), cumulative re- turn (top right), net position for holdings (bottom left), and realized and unrealized returns (bottom right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Illustrated in Figure 4, training in 2017 and 2018 produces a back- testing result with a Sharpe Ratio of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='71 in 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' From the top left subplot, it is obvious that the market in the first half of 2019 seriously plummeted, while it oscillated in the second half of 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Under this scenario, the ensemble agent first attempted to go long which resulted in negative returns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Then, it intends to follow the trend by taking consecutive small holding periods of shorts in the first half.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' However, none of these attempts seemed to be effective enough to bring a positive P&L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' In the second half of 2019 before October, the agents performed multiple good trades by going long, which has seized most of the opportunities when the oscillations peaked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Then, both long and short trades have consolidated their gains at the end of 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' This analysis has reviewed certain characteristics of our ensem- ble trading bot in 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Most of its winning trades are based on mean-reversion behavior in the second half, and it failed trend- following in the first half.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' However, it does not seem trivial to infer a rule-based strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Yet, as suggested by Wang [49] and showcased in Section 5, the market exhibits different behaviours in different periods, and our agents should learn which features to rely on and which trading patterns to follow dynamically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Close price (line) from :2019 Sharpe Ratio = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='71 from 2019 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='Cumsum Profit (line) from 2019 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='CumsumProfit [MGDP] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='06 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='2019-03 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='2019-12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='2019-03 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='2019-04 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='2019 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='2019-07 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='2019- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='2019 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='Date ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='Date ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='Net position (line) from 2019 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='Stackplot of Realized value and Unrealized value in :2019 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='30000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='Realized value ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='Unrealizedvalue ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='20000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='position [kTh] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='10000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='Profits [MGBP] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='0- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='Net ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='10000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='20000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='30000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='2019-01 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='2019-03 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='2019-04 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='Lo ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='10D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='250 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='Sell ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='Do nothing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='Bury ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='1 STO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content="10 000 20- 50'0- OBe adj_2mr_lag_0 ens21Spd_lag_1 T BerJwe [p?" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=" adj_12mr_l8g_1 o BerJwe [pe n863Spd_lag_2 mpd_lag_2 mBpd_lag_0 T'BerJwe p?" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' adj_3mr_log_2 erne21Spd_lag_1 o Berwg pe TBerJwe [pe adj_12mr_l8g_α adj_12mr_lag_1 adj_2mr_lag_0 shapley 2014 asop 区 50 150 20D 250 3ID Sell Do nothing Buy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content="10 50'0- D1'- 50'0- 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content="15 DT0 OBe zBeJwe pe 'Be]z [pe merd_lag_2 adj_ 2mr_lag_2 z Ber pdsE9Bwa adj_12mr_l8g_1 adj_1mr_lag_0 ern821Spd_lag_0 enB215pd_lag_2 Bel JWE [pe ern8635pdlag_2- p1mC_leg_0 rne215pd_lag_0 Be]t pe _lag_2 adj_2mr_lag_2 shapley 2020 Cpl e821125 120 100 100 75 80 50 core 60 Score 25 0 20 25 0 50 20 75 5000 10000 15000 20000 25000 5000 10000 15000 20000 25000 Episode# Episode#Deep Reinforcement Learning for Gas Trading Conference’17," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' 6 DISCUSSION The performance of the Deep RL agents,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' the Linear Regression and the Random Forest models are compared in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' The x-axis repre- sents the Sharpe Ratio, y-axis represents the maximum drawdown, and the color represents the cumulative P&L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' The best result with a Sharpe Ratio of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='32 and a maximum drawdown of 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='3% is achieved by Bons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='ai DQN Apex with moving window for re-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' The other results obtained generally also demonstrate decent Sharpe Ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' We find an average Sharpe Ratio of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='07 taken across all versions and algorithms, which outperforms the state-of-the-art result in Zhang [55] where a Sharpe Ratio of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='723 for commodity trading with DQN has been reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' 5 also displays more con- ventional ML strategies based on Linear Regression and Random Forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' However, the Deep RL based strategies appear to be superior for this gas trading use case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Moreover, by means of ensemble learn- ing in the form of Filtered-Thresholding we can further improve the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Backtesting yields an average Sharpe Ratio of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='20 over 2018-2020, a 23% increase from the average Sharpe Ratio of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='975 obtained with the in-house DQN method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Figure 5: The summary plot for all machine learning trad- ing agents including RL agents and traditional ML agents assessed over the period from 2013 to 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' The x-axis rep- resents the Sharpe Ratio, y-axis represents the maximum draw-down, and the color represents the cumulative P&L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Lin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='Reg represents linear regression agent, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' represents random forest agent, Tech indicates only technical features are used, Tech+Fund indicates technical and fundamental features are used in the in-house agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' All Bons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='ai agents use technical and fundamental features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Despite the results obtained outperform those reported in state- of-the-art literature, implementing a Deep RL based trading agent faces several challenges in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' First, there is usually still a drop in performance when going from back-testing to live trading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Second, extended periods of underwater performance would call for shutting down an algorithm before it can swing to profitability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' In other words, even if in backtesting it generated a profit over the entire year of 2019, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' 4, in practice it would not even reach that point, since it would have been stopped out well before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Third, the frequency of trades has to fit with the overall strategy of the trading desk and neither display overly long holding periods nor too frequent trades / churn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Fourth, model explainability remains a concern for black-box neural network based models although the analysis based on SHAP values in Section 5 and feature importance help to mitigate this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' 7 CONCLUSIONS Systematic trading of commodities is a challenging topic in quanti- tative trading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' The low signal-to-noise ratio makes learning models prone to overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' In this paper, we demonstrate our implemen- tation of a Deep Reinforcement Learning framework for systematic gas trading with different approaches based on Microsoft Bons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='ai platform as well as in-house code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Our Deep RL agent trained in Bons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='ai has achieved a Sharpe Ratio of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='32 in back-testing and thereby outperformed state-of-the-art results from literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' The proposed ensemble learning scheme for our in-house DQN method has achieved a Sharpe Ratio of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='2 with a 23% improvement in performance over individual DQN agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' A comparison of models employing only technical features and those employing both technical and fundamental features suggest that including fundamental features does not lead to better perfor- mance here, as it appears that the information gain is offset by the increased noise in the observation space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' This paper is one of the first applications of model explainability using Shapley values for Deep Reinforcement Learning applied to trading and gives insight which feature drives the agent’s buy, sell or hold decision at a certain point of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' It provides insight into otherwise black-box neural network based models and thereby offers a way to analyse the developed rationale of an agent’s trading strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Despite performance beyond state-of-the-art literature, imple- menting such Deep RL trading agent in practice faces several chal- lenges as discussed in detail in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' 6 such as a potential drop in performance between back-testing and live trading, extended peri- ods of even slight under-performance triggering a shut down of the algorithm before it can reach profitability, the trading frequency has to fit with the desk’s overall strategy, and model explainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' The application of Deep Reinforcement Learning to systematic gas trading has been the first successful application of Deep RL in Shell, highlighting that rigorous feature selection, design of the reward function, model architecture and ensemble learning can result in improved and robust performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Ongoing and future work will consider application of Deep RL to auction-like European power markets [43] and process optimization [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' 30 Bons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='aicont.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='SAC(moving-window) 28 45 26 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' (Tech+Fund) 40 (%) Lin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='Reg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' (Tech) Cumulative P&L (Mf) 24 : Bons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='aiAPExDQN(moving-window) 35 22 Lin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='Reg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' (Tech+Fund) In-house DQN (Tech) 20 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' (Tech) In-house DQN (Tech+Fund) 30 : 18 25 16 Bons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='aiAPExDQN(anchor-window) 14 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='6 8:0 10 12 14 16 Sharpe RatioConference’17, , Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content='Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' REFERENCES [1] Vangelis Bacoyannis, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' Glukhov, Tomoyuki Jin, Jonathan Kochems, and Doo Re Song.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE_T4oBgHgl3EQf5hys/content/2301.08359v1.pdf'} +page_content=' 2018.' metadata={'source': 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@@ +Improving deep learning precipitation nowcasting +by using prior knowledge +Matej Choma � +Meteopress s.r.o, +Faculty of Information Technology, Czech Technical University in Prague, Czech Republic +Petr Šimánek � � +Faculty of Information Technology, Czech Technical University in Prague, Czech Republic +Jakub Bartel � +Meteopress s.r.o +Abstract +Deep learning methods dominate short-term high-resolution precipitation nowcasting in terms of +prediction error. However, their operational usability is limited by difficulties explaining dynamics +behind the predictions, which are smoothed out and missing the high-frequency features due to +optimizing for mean error loss functions. We experiment with hand-engineering of the advection- +diffusion differential equation into a PhyCell to introduce more accurate physical prior to a PhyDNet +model that disentangles physical and residual dynamics. Results indicate that while PhyCell can +learn the intended dynamics, training of PhyDNet remains driven by loss optimization, resulting in +a model with the same prediction capabilities. +2012 ACM Subject Classification Computing methodologies → Machine learning algorithms +Keywords and phrases Nowcasting, spatio-temporal prediction, ConvLSTM, physics informed NN, +PhyDNet +1 +Introduction +It is normal to adapt day-to-day activities with respect to temperature, wind, and precipitation +outside. Homes are built as a shelter from the weather, and its effects on food production +inspired cultures around the world. Thus, it is beneficial to know in advance what the weather +may be like, and adjust according to it to increase comfort, safety, and profit. However, +weather may sometimes be severe, changing in tens of minutes and destroying anything +standing in its path. The tornado in Moravia, which happened on June 24, 2021, is a tragic +example still in the living memory [12]. In these cases, weather prediction becomes a critical +tool for protection. +Precipitation is not only dictating clothes, transport, or moisture for crops, but in +our latitudes, it accompanies most of the short-term storm-based severe weather as well. +Each time a dark cloud forms on the horizon, a question regarding its future development +and severity arises. +Luckily, precipitation may be monitored in real-time and in high +resolution with weather radars. It may be argued that the observations are sufficient for +taking individual protective measures. Nevertheless, humans have many activities when it is +impossible to monitor their surroundings actively, and a localized short-term prediction may +be game-changing. +We have been exploring the use of deep learning (DL) techniques for short-time high- +resolution rainfall prediction in cooperation with the company Meteopress [5]. Building on the +PhyDNet architecture disentangling physical from unknown dynamics [8], we have achieved +unparalleled quantitative performance of an operational precipitation nowcasting system +[6]. The difficulty of explaining dynamics learned by a DL model lowers the trustworthiness +of the predictions in the eyes of meteorologists. The regression formulation of the learning +arXiv:2301.11707v1 [cs.LG] 27 Jan 2023 + +2 +Improving deep learning precipitation nowcasting by using prior knowledge +problem, guided by mean error loss functions, results in the ignorance of hardly predictable +high-frequency features, which are the most important ones during storm events. Last but +not least, the performance decays quickly with prolonged forecast times. +PhyDNet is a neural network (NN) developed for a general video prediction, where the +underlying dynamics governing the system are unknown. However, with the long history of +weather forecasting [4], this is not the case for precipitation. In this thesis, we aim to progress +in addressing the issues mentioned above by exploiting the prior knowledge of precipitation +physics. This work will explore how the human knowledge of the atmosphere may be used to +enhance the physical part of the prediction in PhyDNet. Subsequently, models incorporating +the proposed changes will be trained on a radar echo dataset and compared to a PhyDNet +baseline. The results will be thoroughly analyzed and discussed. +2 +Related work +Traditional multi-day weather forecasts are computed using numerical weather prediction +(NWP) models, which model physical atmospheric processes on a selected grid-scale as +an initial value problem. Real-time high-resolution radar and satellite observations make +accurate NWP initializations possible. However, the cost of data assimilation and limitations +on the model resolution to maintain computability cause not an optimal use of this data +for short-range 0 − 2 h nowcasting. An accepted approach to this time range is to compute +nowcasts as an extrapolation on a sequence of radar or satellite measurements. [13] +In Lagrangian persistence models, it is assumed that precipitation intensity does not +change. An advection field (optical flow) is estimated from a sequence of past observations, +and the future ones are predicted by advecting the present rainfall. An open-source library +containing these models is rainymotion [3]. There have been advances, building on the +Lagrangian persistence, allowing probabilistic, more accurate nowcasts, such as models from +the library pySTEPS [14]. However, the nowcasting of convective initiation, development, and +decay remains difficult. [13] +“Machine learning provides an opportunity to capture complex non-linear spatio-temporal +patterns and to combine heterogeneous data sources for use in prediction,” [13]. The ConvL- +STM architecture [18] was initially designed for precipitation nowcasting, and improvements +to spatio-temporal predictions were introduced in PredRNN [20]. A Deep Generative Model +may be used to predict high-frequency features in the precipitation [17]. +2.1 +Physics and Deep Learning +Enhancing DL models with a physics prior or a combination of physical modeling and DL +can improve the ability of models to generalize to unseen samples, reduce the size of models +or help training when not enough training data is available. A good overview of the topic +may be found in [19]. The following work, alongside PhyDNet [8], influenced our research. +Physics-informed neural networks [15] are constrained by physical laws, expressed as +general non-linear PDEs. These can learn solutions to supervised training problems +data-efficiently while respecting the given laws. +In [16] the authors present hidden fluid mechanics, a DL framework for inference of +hidden quantities, like fluid pressure and velocity, from spatio-temporal visualizations of +a passive scalar. Passive scalar is transported by the fluid but has no dynamical effect on +the fluid motion. +APHYNITY [21] is a framework for augmenting physical models with DL. The novel +formulation of the learning problem allows the physical model to learn as much of the + +M. Choma and P. Šimánek and J. Bartel +3 +dynamics as possible. +3 +PhyDNet +PhyDNet [8] is a recurrent NN (RNN) designed for a general prediction of future video frames +that learns disentanglement between physical and unknown dynamics governing the system +captured in the video. The approach proposed in [8] builds on the idea of approximation of +partial differential equations (PDEs) with convolutional filters and creates a way to include +the equations in deep learning models. +Given a frame of the video u(t) (for details about dimensions see Appendix C.2), PhyDNet +is trained to predict the following frame u(t+∆), under the assumption that the captured +system can be at least partially described by some physical laws. The design of the architecture +contains two branches. The first branch consists of PhyCell which models some differential +operators and handles physical dynamics in the prediction. +The second one is a deep +ConvLSTM [18] cell handling the residual dynamics. As the differential operators may not +catch all the dynamics at the pixel level of the video, this disentanglement is preceded by an +embedding to a latent space H that is learned end-to-end by deep convolutional encoder E +and decoder D. [8] +PhyCell leverages physical prior to improve generalization and allows the model to +learn some dynamics describable by PDE more effectively with less trainable parameters. +ConvLSTM learns the complex unknown factors necessary for pixel-level prediction. [8] +In the latent space H, the memory of the PhyDNet cell stores learned embedding of a video +up to a time t, in a domain with coordinates x = (x, y), represented as h(t, x) = h(t) ∈ H and +linearly disentangled into physical and residual components as h(t) = h(t) +p + h(t) +r . Dynamics +of the video are then governed by the following PDE: +∂h(t) +∂t += ∂h(t) +p +∂t ++ ∂h(t) +r +∂t +:= Mp(h(t) +p , E(u(t))) + Mr(h(t) +r , E(u(t))), +(1) +where Mp is modeled by PhyCell and Mr by ConvLSTM. Prediction of the next frame, +discretized according to the forward Euler method, is computed as: +�u(t+∆) = D(h(t+∆) +p ++h(t+∆) +r +) = D(h(t) +p +Mp(h(t) +p , E(u(t)))+h(t) +r +Mr(h(t) +r , E(u(t)))), (2) +remembering the newly computed hidden states h(t+∆) +p +and h(t+∆) +r +. [8] +3.1 +Physical Model – PhyCell +PhyCell is a novel ”physically constrained” recurrent cell introduced in [8] that models the +dynamics in two steps: +Mp(hp, E(u)) := Φ(hp) + C(hp, E(u)). +(3) +The first step is prediction in the latent space Φ(hp) (Equation 4) using a linear combination of +spatial derivatives. Then, correction step C(hp, E(u)) (Equation 6) handles the assimilation +of input data into the latent representation similarly as in the Kalman filter [10]. +3.1.1 +Prediction Step +The physical predictor Φ(hp) models a generic class of linear PDEs as +Φ(h(t) +p ) := +� +i,j